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Theorem mulpipq2 10964
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulpipq2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)

Proof of Theorem mulpipq2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6896 . . . 4 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21oveq1d 7434 . . 3 (𝑥 = 𝐴 → ((1st𝑥) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝑦)))
3 fveq2 6896 . . . 4 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
43oveq1d 7434 . . 3 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
52, 4opeq12d 4883 . 2 (𝑥 = 𝐴 → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩)
6 fveq2 6896 . . . 4 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
76oveq2d 7435 . . 3 (𝑦 = 𝐵 → ((1st𝐴) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝐵)))
8 fveq2 6896 . . . 4 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
98oveq2d 7435 . . 3 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
107, 9opeq12d 4883 . 2 (𝑦 = 𝐵 → ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
11 df-mpq 10934 . 2 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
12 opex 5466 . 2 ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V
135, 10, 11, 12ovmpo 7581 1 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4636   × cxp 5676  cfv 6549  (class class class)co 7419  1st c1st 7992  2nd c2nd 7993  Ncnpi 10869   ·N cmi 10871   ·pQ cmpq 10874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-mpq 10934
This theorem is referenced by:  mulpipq  10965  mulcompq  10977  mulerpqlem  10980  mulassnq  10984  distrnq  10986  ltmnq  10997
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