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Mirrors > Home > ILE Home > Th. List > nnmcom | GIF version |
Description: Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nnmcom | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5539 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐵)) | |
2 | oveq2 5540 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐴)) | |
3 | 1, 2 | eqeq12d 2095 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))) |
4 | 3 | imbi2d 228 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)))) |
5 | oveq1 5539 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ·𝑜 𝐵) = (∅ ·𝑜 𝐵)) | |
6 | oveq2 5540 | . . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅)) | |
7 | 5, 6 | eqeq12d 2095 | . . . 4 ⊢ (𝑥 = ∅ → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (∅ ·𝑜 𝐵) = (𝐵 ·𝑜 ∅))) |
8 | oveq1 5539 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ·𝑜 𝐵) = (𝑦 ·𝑜 𝐵)) | |
9 | oveq2 5540 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦)) | |
10 | 8, 9 | eqeq12d 2095 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦))) |
11 | oveq1 5539 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ·𝑜 𝐵) = (suc 𝑦 ·𝑜 𝐵)) | |
12 | oveq2 5540 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦)) | |
13 | 11, 12 | eqeq12d 2095 | . . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦))) |
14 | nnm0r 6081 | . . . . 5 ⊢ (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅) | |
15 | nnm0 6077 | . . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 ·𝑜 ∅) = ∅) | |
16 | 14, 15 | eqtr4d 2116 | . . . 4 ⊢ (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = (𝐵 ·𝑜 ∅)) |
17 | oveq1 5539 | . . . . . 6 ⊢ ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → ((𝑦 ·𝑜 𝐵) +𝑜 𝐵) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) | |
18 | nnmsucr 6090 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·𝑜 𝐵) = ((𝑦 ·𝑜 𝐵) +𝑜 𝐵)) | |
19 | nnmsuc 6079 | . . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) | |
20 | 19 | ancoms 264 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) |
21 | 18, 20 | eqeq12d 2095 | . . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦) ↔ ((𝑦 ·𝑜 𝐵) +𝑜 𝐵) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))) |
22 | 17, 21 | syl5ibr 154 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦))) |
23 | 22 | ex 113 | . . . 4 ⊢ (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦)))) |
24 | 7, 10, 13, 16, 23 | finds2 4342 | . . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥))) |
25 | 4, 24 | vtoclga 2664 | . 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))) |
26 | 25 | imp 122 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∅c0 3251 suc csuc 4120 ωcom 4331 (class class class)co 5532 +𝑜 coa 6021 ·𝑜 comu 6022 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 |
This theorem is referenced by: nndir 6092 nn2m 6122 mulcompig 6521 enq0sym 6622 enq0ref 6623 enq0tr 6624 addcmpblnq0 6633 mulcmpblnq0 6634 mulcanenq0ec 6635 nnanq0 6648 distrnq0 6649 mulcomnq0 6650 addassnq0 6652 nq02m 6655 |
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