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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 10475 | . . . 4 ⊢ ((1st ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (1st ‘𝐴)) | |
2 | mulcompi 10475 | . . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴)) | |
3 | 1, 2 | opeq12i 4775 | . . 3 ⊢ 〈((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉 = 〈((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))〉 |
4 | mulpipq2 10518 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = 〈((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉) | |
5 | mulpipq2 10518 | . . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = 〈((1st ‘𝐵) ·N (1st ‘𝐴)), ((2nd ‘𝐵) ·N (2nd ‘𝐴))〉) | |
6 | 5 | ancoms 462 | . . 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 2797 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)) |
8 | mulpqf 10525 | . . . 4 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | |
9 | 8 | fdmi 6535 | . . 3 ⊢ dom ·pQ = ((N × N) × (N × N)) |
10 | 9 | ndmovcom 7373 | . 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)) |
11 | 7, 10 | pm2.61i 185 | 1 ⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2112 〈cop 4533 × cxp 5534 ‘cfv 6358 (class class class)co 7191 1st c1st 7737 2nd c2nd 7738 Ncnpi 10423 ·N cmi 10425 ·pQ cmpq 10428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-oadd 8184 df-omul 8185 df-ni 10451 df-mi 10453 df-mpq 10488 |
This theorem is referenced by: mulcomnq 10532 mulerpq 10536 |
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