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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 9989 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 9989 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nnmcom 7947 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)) | |
4 | 1, 2, 3 | syl2an 590 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)) |
5 | mulpiord 9996 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵)) | |
6 | mulpiord 9996 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 ·N 𝐴) = (𝐵 ·𝑜 𝐴)) | |
7 | 6 | ancoms 451 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 ·N 𝐴) = (𝐵 ·𝑜 𝐴)) |
8 | 4, 5, 7 | 3eqtr4d 2844 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
9 | dmmulpi 10002 | . . 3 ⊢ dom ·N = (N × N) | |
10 | 9 | ndmovcom 7056 | . 2 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
11 | 8, 10 | pm2.61i 177 | 1 ⊢ (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 = wceq 1653 ∈ wcel 2157 (class class class)co 6879 ωcom 7300 ·𝑜 comu 7798 Ncnpi 9955 ·N cmi 9957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-oadd 7804 df-omul 7805 df-ni 9983 df-mi 9985 |
This theorem is referenced by: enqbreq2 10031 enqer 10032 nqereu 10040 addcompq 10061 mulcompq 10063 adderpqlem 10065 mulerpqlem 10066 addassnq 10069 mulcanenq 10071 distrnq 10072 recmulnq 10075 ltsonq 10080 lterpq 10081 ltanq 10082 ltmnq 10083 ltexnq 10086 |
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