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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 10300 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 10300 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nnmcom 8252 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)) | |
4 | 1, 2, 3 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)) |
5 | mulpiord 10307 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) | |
6 | mulpiord 10307 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 ·N 𝐴) = (𝐵 ·o 𝐴)) | |
7 | 6 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 ·N 𝐴) = (𝐵 ·o 𝐴)) |
8 | 4, 5, 7 | 3eqtr4d 2866 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
9 | dmmulpi 10313 | . . 3 ⊢ dom ·N = (N × N) | |
10 | 9 | ndmovcom 7335 | . 2 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
11 | 8, 10 | pm2.61i 184 | 1 ⊢ (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ωcom 7580 ·o comu 8100 Ncnpi 10266 ·N cmi 10268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-oadd 8106 df-omul 8107 df-ni 10294 df-mi 10296 |
This theorem is referenced by: enqbreq2 10342 enqer 10343 nqereu 10351 addcompq 10372 mulcompq 10374 adderpqlem 10376 mulerpqlem 10377 addassnq 10380 mulcanenq 10382 distrnq 10383 recmulnq 10386 ltsonq 10391 lterpq 10392 ltanq 10393 ltmnq 10394 ltexnq 10397 |
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