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Mirrors > Home > MPE Home > Th. List > mulcompi | Structured version Visualization version GIF version |
Description: Multiplication of positive integers is commutative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulcompi | ⊢ (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 10903 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 10903 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nnmcom 8647 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)) | |
4 | 1, 2, 3 | syl2an 594 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)) |
5 | mulpiord 10910 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) | |
6 | mulpiord 10910 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 ·N 𝐴) = (𝐵 ·o 𝐴)) | |
7 | 6 | ancoms 457 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 ·N 𝐴) = (𝐵 ·o 𝐴)) |
8 | 4, 5, 7 | 3eqtr4d 2775 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
9 | dmmulpi 10916 | . . 3 ⊢ dom ·N = (N × N) | |
10 | 9 | ndmovcom 7608 | . 2 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
11 | 8, 10 | pm2.61i 182 | 1 ⊢ (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ωcom 7871 ·o comu 8485 Ncnpi 10869 ·N cmi 10871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-oadd 8491 df-omul 8492 df-ni 10897 df-mi 10899 |
This theorem is referenced by: enqbreq2 10945 enqer 10946 nqereu 10954 addcompq 10975 mulcompq 10977 adderpqlem 10979 mulerpqlem 10980 addassnq 10983 mulcanenq 10985 distrnq 10986 recmulnq 10989 ltsonq 10994 lterpq 10995 ltanq 10996 ltmnq 10997 ltexnq 11000 |
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