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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 5884 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·o 𝐵) = (𝐴 ·o 𝐵)) | |
2 | oveq2 5885 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐴)) | |
3 | 1, 2 | eqeq12d 2192 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))) |
4 | 3 | imbi2d 230 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))) |
5 | oveq1 5884 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ·o 𝐵) = (∅ ·o 𝐵)) | |
6 | oveq2 5885 | . . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅)) | |
7 | 5, 6 | eqeq12d 2192 | . . . 4 ⊢ (𝑥 = ∅ → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (∅ ·o 𝐵) = (𝐵 ·o ∅))) |
8 | oveq1 5884 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ·o 𝐵) = (𝑦 ·o 𝐵)) | |
9 | oveq2 5885 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦)) | |
10 | 8, 9 | eqeq12d 2192 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝑦 ·o 𝐵) = (𝐵 ·o 𝑦))) |
11 | oveq1 5884 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ·o 𝐵) = (suc 𝑦 ·o 𝐵)) | |
12 | oveq2 5885 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦)) | |
13 | 11, 12 | eqeq12d 2192 | . . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦))) |
14 | nnm0r 6482 | . . . . 5 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅) | |
15 | nnm0 6478 | . . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 ·o ∅) = ∅) | |
16 | 14, 15 | eqtr4d 2213 | . . . 4 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = (𝐵 ·o ∅)) |
17 | oveq1 5884 | . . . . . 6 ⊢ ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → ((𝑦 ·o 𝐵) +o 𝐵) = ((𝐵 ·o 𝑦) +o 𝐵)) | |
18 | nnmsucr 6491 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·o 𝐵) = ((𝑦 ·o 𝐵) +o 𝐵)) | |
19 | nnmsuc 6480 | . . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵)) | |
20 | 19 | ancoms 268 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵)) |
21 | 18, 20 | eqeq12d 2192 | . . . . . 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 | imbitrrid 156 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦))) |
23 | 22 | ex 115 | . . . 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 4602 | . . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥))) |
25 | 4, 24 | vtoclga 2805 | . 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))) |
26 | 25 | imp 124 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∅c0 3424 suc csuc 4367 ωcom 4591 (class class class)co 5877 +o coa 6416 ·o comu 6417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-oadd 6423 df-omul 6424 |
This theorem is referenced by: nndir 6493 nn2m 6530 mulcompig 7332 enq0sym 7433 enq0ref 7434 enq0tr 7435 addcmpblnq0 7444 mulcmpblnq0 7445 mulcanenq0ec 7446 nnanq0 7459 distrnq0 7460 mulcomnq0 7461 addassnq0 7463 nq02m 7466 |
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