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Mirrors > Home > ILE Home > Th. List > mulcompig | Unicode version |
Description: Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Ref | Expression |
---|---|
mulcompig |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7229 | . . 3 | |
2 | pinn 7229 | . . 3 | |
3 | nnmcom 6436 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 |
5 | mulpiord 7237 | . 2 | |
6 | mulpiord 7237 | . . 3 | |
7 | 6 | ancoms 266 | . 2 |
8 | 4, 5, 7 | 3eqtr4d 2200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 com 4549 (class class class)co 5824 comu 6361 cnpi 7192 cmi 7194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-oadd 6367 df-omul 6368 df-ni 7224 df-mi 7226 |
This theorem is referenced by: dfplpq2 7274 enqbreq2 7277 enqer 7278 addcmpblnq 7287 mulcmpblnq 7288 ordpipqqs 7294 addcomnqg 7301 addassnqg 7302 mulcomnqg 7303 mulcanenq 7305 distrnqg 7307 mulidnq 7309 recexnq 7310 nqtri3or 7316 ltsonq 7318 ltanqg 7320 ltmnqg 7321 ltexnqq 7328 archnqq 7337 prarloclemarch2 7339 ltnnnq 7343 prarloclemlt 7413 |
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