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Theorem mulcompq 9718
Description: Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulcompq (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)

Proof of Theorem mulcompq
StepHypRef Expression
1 mulcompi 9662 . . . 4 ((1st𝐴) ·N (1st𝐵)) = ((1st𝐵) ·N (1st𝐴))
2 mulcompi 9662 . . . 4 ((2nd𝐴) ·N (2nd𝐵)) = ((2nd𝐵) ·N (2nd𝐴))
31, 2opeq12i 4375 . . 3 ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ = ⟨((1st𝐵) ·N (1st𝐴)), ((2nd𝐵) ·N (2nd𝐴))⟩
4 mulpipq2 9705 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
5 mulpipq2 9705 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = ⟨((1st𝐵) ·N (1st𝐴)), ((2nd𝐵) ·N (2nd𝐴))⟩)
65ancoms 469 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 ·pQ 𝐴) = ⟨((1st𝐵) ·N (1st𝐴)), ((2nd𝐵) ·N (2nd𝐴))⟩)
73, 4, 63eqtr4a 2681 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴))
8 mulpqf 9712 . . . 4 ·pQ :((N × N) × (N × N))⟶(N × N)
98fdmi 6009 . . 3 dom ·pQ = ((N × N) × (N × N))
109ndmovcom 6774 . 2 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴))
117, 10pm2.61i 176 1 (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  cop 4154   × cxp 5072  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  Ncnpi 9610   ·N cmi 9612   ·pQ cmpq 9615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-omul 7510  df-ni 9638  df-mi 9640  df-mpq 9675
This theorem is referenced by:  mulcomnq  9719  mulerpq  9723
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