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Mirrors > Home > ILE Home > Th. List > mulcompig | GIF version |
Description: Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Ref | Expression |
---|---|
mulcompig | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 6561 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 6561 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nnmcom 6133 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)) | |
4 | 1, 2, 3 | syl2an 283 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)) |
5 | mulpiord 6569 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵)) | |
6 | mulpiord 6569 | . . 3 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 ·N 𝐴) = (𝐵 ·𝑜 𝐴)) | |
7 | 6 | ancoms 264 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 ·N 𝐴) = (𝐵 ·𝑜 𝐴)) |
8 | 4, 5, 7 | 3eqtr4d 2124 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ωcom 4339 (class class class)co 5543 ·𝑜 comu 6063 Ncnpi 6524 ·N cmi 6526 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-id 4056 df-iord 4129 df-on 4131 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-irdg 6019 df-oadd 6069 df-omul 6070 df-ni 6556 df-mi 6558 |
This theorem is referenced by: dfplpq2 6606 enqbreq2 6609 enqer 6610 addcmpblnq 6619 mulcmpblnq 6620 ordpipqqs 6626 addcomnqg 6633 addassnqg 6634 mulcomnqg 6635 mulcanenq 6637 distrnqg 6639 mulidnq 6641 recexnq 6642 nqtri3or 6648 ltsonq 6650 ltanqg 6652 ltmnqg 6653 ltexnqq 6660 archnqq 6669 prarloclemarch2 6671 ltnnnq 6675 prarloclemlt 6745 |
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