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Theorem mulpipq2 10364
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 6673 . . . 4 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21oveq1d 7174 . . 3 (𝑥 = 𝐴 → ((1st𝑥) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝑦)))
3 fveq2 6673 . . . 4 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
43oveq1d 7174 . . 3 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
52, 4opeq12d 4814 . 2 (𝑥 = 𝐴 → ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩)
6 fveq2 6673 . . . 4 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
76oveq2d 7175 . . 3 (𝑦 = 𝐵 → ((1st𝐴) ·N (1st𝑦)) = ((1st𝐴) ·N (1st𝐵)))
8 fveq2 6673 . . . 4 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
98oveq2d 7175 . . 3 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
107, 9opeq12d 4814 . 2 (𝑦 = 𝐵 → ⟨((1st𝐴) ·N (1st𝑦)), ((2nd𝐴) ·N (2nd𝑦))⟩ = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
11 df-mpq 10334 . 2 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
12 opex 5359 . 2 ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V
135, 10, 11, 12ovmpo 7313 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 398   = wceq 1536  wcel 2113  cop 4576   × cxp 5556  cfv 6358  (class class class)co 7159  1st c1st 7690  2nd c2nd 7691  Ncnpi 10269   ·N cmi 10271   ·pQ cmpq 10274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-mpq 10334
This theorem is referenced by:  mulpipq  10365  mulcompq  10377  mulerpqlem  10380  mulassnq  10384  distrnq  10386  ltmnq  10397
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