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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 7438 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·o 𝐵) = (𝐴 ·o 𝐵)) | |
2 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐴)) | |
3 | 1, 2 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))) |
4 | 3 | imbi2d 340 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))) |
5 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ·o 𝐵) = (∅ ·o 𝐵)) | |
6 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅)) | |
7 | 5, 6 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = ∅ → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (∅ ·o 𝐵) = (𝐵 ·o ∅))) |
8 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ·o 𝐵) = (𝑦 ·o 𝐵)) | |
9 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦)) | |
10 | 8, 9 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝑦 ·o 𝐵) = (𝐵 ·o 𝑦))) |
11 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ·o 𝐵) = (suc 𝑦 ·o 𝐵)) | |
12 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦)) | |
13 | 11, 12 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦))) |
14 | nnm0r 8647 | . . . . 5 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅) | |
15 | nnm0 8642 | . . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 ·o ∅) = ∅) | |
16 | 14, 15 | eqtr4d 2778 | . . . 4 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = (𝐵 ·o ∅)) |
17 | oveq1 7438 | . . . . . 6 ⊢ ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → ((𝑦 ·o 𝐵) +o 𝐵) = ((𝐵 ·o 𝑦) +o 𝐵)) | |
18 | nnmsucr 8662 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·o 𝐵) = ((𝑦 ·o 𝐵) +o 𝐵)) | |
19 | nnmsuc 8644 | . . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵)) | |
20 | 19 | ancoms 458 | . . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵)) |
21 | 18, 20 | eqeq12d 2751 | . . . . . 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 246 | . . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦))) |
23 | 22 | ex 412 | . . . 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 7921 | . . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥))) |
25 | 4, 24 | vtoclga 3577 | . 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))) |
26 | 25 | imp 406 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∅c0 4339 suc csuc 6388 (class class class)co 7431 ωcom 7887 +o coa 8502 ·o comu 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-oadd 8509 df-omul 8510 |
This theorem is referenced by: nnmwordri 8673 nn2m 8691 omopthlem1 8696 mulcompi 10934 |
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