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Mirrors > Home > MPE Home > Th. List > mulcompq | Structured version Visualization version GIF version |
Description: Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulcompq | ⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcompi 10318 | . . . 4 ⊢ ((1st ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (1st ‘𝐴)) | |
2 | mulcompi 10318 | . . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴)) | |
3 | 1, 2 | opeq12i 4808 | . . 3 ⊢ 〈((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉 = 〈((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))〉 |
4 | mulpipq2 10361 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = 〈((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉) | |
5 | mulpipq2 10361 | . . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = 〈((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))〉) | |
6 | 5 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = 〈((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))〉) |
7 | 3, 4, 6 | 3eqtr4a 2882 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)) |
8 | mulpqf 10368 | . . . 4 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | |
9 | 8 | fdmi 6524 | . . 3 ⊢ dom ·pQ = ((N × N) × (N × N)) |
10 | 9 | ndmovcom 7335 | . 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)) |
11 | 7, 10 | pm2.61i 184 | 1 ⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 × cxp 5553 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 Ncnpi 10266 ·N cmi 10268 ·pQ cmpq 10271 |
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 df-mpq 10331 |
This theorem is referenced by: mulcomnq 10375 mulerpq 10379 |
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