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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 5519 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐵)) | |
2 | oveq2 5520 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐴)) | |
3 | 1, 2 | eqeq12d 2054 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))) |
4 | 3 | imbi2d 219 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)))) |
5 | oveq1 5519 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ·𝑜 𝐵) = (∅ ·𝑜 𝐵)) | |
6 | oveq2 5520 | . . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅)) | |
7 | 5, 6 | eqeq12d 2054 | . . . 4 ⊢ (𝑥 = ∅ → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (∅ ·𝑜 𝐵) = (𝐵 ·𝑜 ∅))) |
8 | oveq1 5519 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ·𝑜 𝐵) = (𝑦 ·𝑜 𝐵)) | |
9 | oveq2 5520 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦)) | |
10 | 8, 9 | eqeq12d 2054 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦))) |
11 | oveq1 5519 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ·𝑜 𝐵) = (suc 𝑦 ·𝑜 𝐵)) | |
12 | oveq2 5520 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦)) | |
13 | 11, 12 | eqeq12d 2054 | . . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦))) |
14 | nnm0r 6058 | . . . . 5 ⊢ (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅) | |
15 | nnm0 6054 | . . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 ·𝑜 ∅) = ∅) | |
16 | 14, 15 | eqtr4d 2075 | . . . 4 ⊢ (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = (𝐵 ·𝑜 ∅)) |
17 | oveq1 5519 | . . . . . 6 ⊢ ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → ((𝑦 ·𝑜 𝐵) +𝑜 𝐵) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) | |
18 | nnmsucr 6067 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·𝑜 𝐵) = ((𝑦 ·𝑜 𝐵) +𝑜 𝐵)) | |
19 | nnmsuc 6056 | . . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) | |
20 | 19 | ancoms 255 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) |
21 | 18, 20 | eqeq12d 2054 | . . . . . 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 145 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦))) |
23 | 22 | ex 108 | . . . 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 4324 | . . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥))) |
25 | 4, 24 | vtoclga 2619 | . 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))) |
26 | 25 | imp 115 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∅c0 3224 suc csuc 4102 ωcom 4313 (class class class)co 5512 +𝑜 coa 5998 ·𝑜 comu 5999 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 |
This theorem is referenced by: nndir 6069 nn2m 6099 mulcompig 6429 enq0sym 6530 enq0ref 6531 enq0tr 6532 addcmpblnq0 6541 mulcmpblnq0 6542 mulcanenq0ec 6543 nnanq0 6556 distrnq0 6557 mulcomnq0 6558 addassnq0 6560 nq02m 6563 |
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