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Mirrors > Home > MPE Home > Th. List > nnmcom | Structured version Visualization version 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 6697 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐵)) | |
2 | oveq2 6698 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐴)) | |
3 | 1, 2 | eqeq12d 2666 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))) |
4 | 3 | imbi2d 329 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)))) |
5 | oveq1 6697 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ·𝑜 𝐵) = (∅ ·𝑜 𝐵)) | |
6 | oveq2 6698 | . . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅)) | |
7 | 5, 6 | eqeq12d 2666 | . . . 4 ⊢ (𝑥 = ∅ → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (∅ ·𝑜 𝐵) = (𝐵 ·𝑜 ∅))) |
8 | oveq1 6697 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ·𝑜 𝐵) = (𝑦 ·𝑜 𝐵)) | |
9 | oveq2 6698 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦)) | |
10 | 8, 9 | eqeq12d 2666 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦))) |
11 | oveq1 6697 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ·𝑜 𝐵) = (suc 𝑦 ·𝑜 𝐵)) | |
12 | oveq2 6698 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦)) | |
13 | 11, 12 | eqeq12d 2666 | . . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦))) |
14 | nnm0r 7735 | . . . . 5 ⊢ (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅) | |
15 | nnm0 7730 | . . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 ·𝑜 ∅) = ∅) | |
16 | 14, 15 | eqtr4d 2688 | . . . 4 ⊢ (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = (𝐵 ·𝑜 ∅)) |
17 | oveq1 6697 | . . . . . 6 ⊢ ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → ((𝑦 ·𝑜 𝐵) +𝑜 𝐵) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) | |
18 | nnmsucr 7750 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·𝑜 𝐵) = ((𝑦 ·𝑜 𝐵) +𝑜 𝐵)) | |
19 | nnmsuc 7732 | . . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) | |
20 | 19 | ancoms 468 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) |
21 | 18, 20 | eqeq12d 2666 | . . . . . 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 236 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦))) |
23 | 22 | ex 449 | . . . 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 7136 | . . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥))) |
25 | 4, 24 | vtoclga 3303 | . 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))) |
26 | 25 | imp 444 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∅c0 3948 suc csuc 5763 (class class class)co 6690 ωcom 7107 +𝑜 coa 7602 ·𝑜 comu 7603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-oadd 7609 df-omul 7610 |
This theorem is referenced by: nnmwordri 7761 nn2m 7775 omopthlem1 7780 mulcompi 9756 |
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