Theorem mulpipq2 10887

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 6856	. . . 4 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
2	1	oveq1d 7400	. . 3 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ·N (1st ‘𝑦)) = ((1st ‘𝐴) ·N (1st ‘𝑦)))
3		fveq2 6856	. . . 4 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
4	3	oveq1d 7400	. . 3 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝑦)))
5	2, 4	opeq12d 4833	. 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 6856	. . . 4 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵))
7	6	oveq2d 7401	. . 3 ⊢ (𝑦 = 𝐵 → ((1st ‘𝐴) ·N (1st ‘𝑦)) = ((1st ‘𝐴) ·N (1st ‘𝐵)))
8		fveq2 6856	. . . 4 ⊢ (𝑦 = 𝐵 → (2nd ‘𝑦) = (2nd ‘𝐵))
9	8	oveq2d 7401	. . 3 ⊢ (𝑦 = 𝐵 → ((2nd ‘𝐴) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
10	7, 9	opeq12d 4833	. 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 10857	. 2 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
12		opex 5425	. 2 ⊢ ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V
13	5, 10, 11, 12	ovmpo 7545	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 398   = wceq 1554   ∈ wcel 2136  ⟨cop 4582   × cxp 5638  ‘cfv 6510  (class class class)co 7385  1^st c1st 7957  2^nd c2nd 7958  Ncnpi 10792   ·_N cmi 10794   ·_pQ cmpq 10797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6512  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-mpq 10857

This theorem is referenced by:  mulpipq  10888  mulcompq  10900  mulerpqlem  10903  mulassnq  10907  distrnq  10909  ltmnq  10920