HomeRelated theorems
Unicode version
| Home |
Metamath Proof Explorer |
< Previous
Next >
Related theorems Unicode version |
| Description: Closure law for addition. |
| Ref | Expression |
|---|---|
| axi.1 |
    |
| axi.2 |
 |
| Ref | Expression |
|---|---|
| addcl |
 
 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | axi.1 |
. 2
| |
| 2 | axi.2 |
. 2
| |
| 3 | axaddcl 5243 |
. 2
  | |
| 4 | 1, 2, 3 | mp2an 695 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wcel 955 (class class class)co 3948
cc 5204
caddc 5209 |
| This theorem is referenced by: 0cn 5300 negsub 5353 negdi 5420 divdir 5710 binom2 6575 sqeqor 6578 nnesq 6592 nn0opthlem1 6594 nn0opth2 6597 crulem 6666 crmul 6671 crrecz 6672 cjadd 6723 cjmul 6724 addcj 6733 abstri 6829 bcpasc2 6905 binomlem6 7009 binom 7010 fnsmnt 7161 efaddlem27 7306 ef4p 7340 sinadd 7393 cosadd 7394 ipid 8297 ip1ilem 8416 pythi 8441 normlem2 8898 normlem3 8899 normlem7 8903 normlem9 8905 bcseq 8907 norm-ii 8925 normpyth 8930 normpar 8942 polid2 8945 projlem3 9104 lnopunilem1 9850 lnophmlem2 9857 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 959 ax-gen 960 ax-8 961 ax-9 962 ax-10 963 ax-11 964 ax-12 965 ax-13 966 ax-14 967 ax-17 968 ax-4 970 ax-5o 972 ax-6o 975 ax-9o 1119 ax-10o 1136 ax-16 1206 ax-11o 1213 ax-ext 1452 ax-rep 2683 ax-sep 2693 ax-nul 2700 ax-pow 2732 ax-pr 2769 ax-un 2857 ax-inf2 4597 |
| This theorem depends on definitions: df-bi 147 df-or 224 df-an 225 df-3or 774 df-3an 775 df-ex 978 df-sb 1168 df-eu 1375 df-mo 1376 df-clab 1457 df-cleq 1462 df-clel 1465 df-ne 1579 df-ral 1641 df-rex 1642 df-reu 1643 df-rab 1644 df-v 1803 df-sbc 1932 df-csb 1992 df-dif 2039 df-un 2040 df-in 2041 df-ss 2043 df-pss 2045 df-nul 2271 df-if 2352 df-pw 2392 df-sn 2402 df-pr 2403 df-tp 2405 df-op 2406 df-uni 2494 df-int 2524 df-iun 2558 df-br 2610 df-opab 2657 df-tr 2671 df-eprel 2821 df-id 2824 df-po 2831 df-so 2841 df-fr 2907 df-we 2924 df-ord 2941 df-on 2942 df-lim 2943 df-suc 2944 df-om 3122 df-xp 3174 df-rel 3175 df-cnv 3176 df-co 3177 df-dm 3178 df-rn 3179 df-res 3180 df-ima 3181 df-fun 3182 df-fn 3183 df-f 3184 df-fv 3188 df-rdg 3917 df-opr 3950 df-oprab 3951 df-1st 4063 df-2nd 4064 df-1o 4117 df-oadd 4119 df-omul 4120 df-er 4245 df-ec 4247 df-qs 4250 df-ni 4972 df-pli 4973 df-mi 4974 df-lti 4975 df-plpq 5007 df-mpq 5008 df-enq 5009 df-nq 5010 df-plq 5011 df-mq 5012 df-rq 5013 df-ltq 5014 df-1q 5015 df-np 5058 df-plp 5060 df-ltp 5062 df-plpr 5136 df-enr 5138 df-nr 5139 df-plr 5140 df-c 5212 df-plus 5217 |