Theorem mulpipq2 7294

Description: Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)

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		xp1st 6116	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
2		xp1st 6116	. . . 4 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
3		mulclpi 7251	. . . 4 ⊢ (((1st ‘𝐴) ∈ N ∧ (1st ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
4	1, 2, 3	syl2an 287	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N)
5		xp2nd 6117	. . . 4 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
6		xp2nd 6117	. . . 4 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
7		mulclpi 7251	. . . 4 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
8	5, 6, 7	syl2an 287	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
9		opexg 4191	. . 3 ⊢ ((((1st ‘𝐴) ·N (1st ‘𝐵)) ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N) → ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V)
10	4, 8, 9	syl2anc 409	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V)
11		fveq2 5471	. . . . 5 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
12	11	oveq1d 5842	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ·N (1st ‘𝑦)) = ((1st ‘𝐴) ·N (1st ‘𝑦)))
13		fveq2 5471	. . . . 5 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
14	13	oveq1d 5842	. . . 4 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝑦)))
15	12, 14	opeq12d 3751	. . 3 ⊢ (𝑥 = 𝐴 → ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ = ⟨((1st ‘𝐴) ·N (1st ‘𝑦)), ((2nd ‘𝐴) ·N (2nd ‘𝑦))⟩)
16		fveq2 5471	. . . . 5 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵))
17	16	oveq2d 5843	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝐴) ·N (1st ‘𝑦)) = ((1st ‘𝐴) ·N (1st ‘𝐵)))
18		fveq2 5471	. . . . 5 ⊢ (𝑦 = 𝐵 → (2nd ‘𝑦) = (2nd ‘𝐵))
19	18	oveq2d 5843	. . . 4 ⊢ (𝑦 = 𝐵 → ((2nd ‘𝐴) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
20	17, 19	opeq12d 3751	. . 3 ⊢ (𝑦 = 𝐵 → ⟨((1st ‘𝐴) ·N (1st ‘𝑦)), ((2nd ‘𝐴) ·N (2nd ‘𝑦))⟩ = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
21		df-mpq 7268	. . 3 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
22	15, 20, 21	ovmpog 5958	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V) → (𝐴 ·pQ 𝐵) = ⟨((1st ‘𝐴) ·N (1st ‘𝐵)), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
23	10, 22	mpd3an3 1320	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 103   = wceq 1335   ∈ wcel 2128  Vcvv 2712  ⟨cop 3564   × cxp 4587  ‘cfv 5173  (class class class)co 5827  1^st c1st 6089  2^nd c2nd 6090  Ncnpi 7195   ·_N cmi 7197   ·_pQ cmpq 7200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-iord 4329  df-on 4331  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-irdg 6320  df-oadd 6370  df-omul 6371  df-ni 7227  df-mi 7229  df-mpq 7268

This theorem is referenced by:  mulpipq  7295