Theorem mulpipq2 10933

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) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)

Proof of Theorem mulpipq2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6884	. . . 4 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
2	1	oveq1d 7419	. . 3 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ·N (1st ‘𝑦)) = ((1st ‘𝐴) ·N (1st ‘𝑦)))
3		fveq2 6884	. . . 4 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
4	3	oveq1d 7419	. . 3 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝑦)))
5	2, 4	opeq12d 4876	. 2 ⊢ (𝑥 = 𝐴 → ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ = ⟨((1st ‘𝐴) ·N (1st ‘𝑦)), ((2nd ‘𝐴) ·N (2nd ‘𝑦))⟩)
6		fveq2 6884	. . . 4 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵))
7	6	oveq2d 7420	. . 3 ⊢ (𝑦 = 𝐵 → ((1st ‘𝐴) ·N (1st ‘𝑦)) = ((1st ‘𝐴) ·N (1st ‘𝐵)))
8		fveq2 6884	. . . 4 ⊢ (𝑦 = 𝐵 → (2nd ‘𝑦) = (2nd ‘𝐵))
9	8	oveq2d 7420	. . 3 ⊢ (𝑦 = 𝐵 → ((2nd ‘𝐴) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
10	7, 9	opeq12d 4876	. 2 ⊢ (𝑦 = 𝐵 → ⟨((1st ‘𝐴) ·N (1st ‘𝑦)), ((2nd ‘𝐴) ·N (2nd ‘𝑦))⟩ = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
11		df-mpq 10903	. 2 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
12		opex 5457	. 2 ⊢ ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V
13	5, 10, 11, 12	ovmpo 7563	1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ⟨cop 4629   × cxp 5667  ‘cfv 6536  (class class class)co 7404  1^st c1st 7969  2^nd c2nd 7970  Ncnpi 10838   ·_N cmi 10840   ·_pQ cmpq 10843

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-mpq 10903

This theorem is referenced by:  mulpipq  10934  mulcompq  10946  mulerpqlem  10949  mulassnq  10953  distrnq  10955  ltmnq  10966