Theorem mulpqnq 10862

Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
mulpqnq	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))

Proof of Theorem mulpqnq

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

Step	Hyp	Ref	Expression
1		df-mq 10836	. . . . 5 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
2	1	fveq1i 6835	. . . 4 ⊢ ( ·Q ‘⟨𝐴, 𝐵⟩) = ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩)
3	2	a1i 11	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘⟨𝐴, 𝐵⟩) = ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩))
4		opelxpi 5662	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
5	4	fvresd 6854	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩))
6		df-mpq 10830	. . . . 5 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
7		opex 5410	. . . . 5 ⊢ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ V
8	6, 7	fnmpoi 8019	. . . 4 ⊢ ·pQ Fn ((N × N) × (N × N))
9		elpqn 10846	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
10		elpqn 10846	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
11		opelxpi 5662	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
12	9, 10, 11	syl2an 602	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
13		fvco2 6931	. . . 4 ⊢ (( ·pQ Fn ((N × N) × (N × N)) ∧ ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N))) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
14	8, 12, 13	sylancr 593	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
15	3, 5, 14	3eqtrd 2779	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
16		df-ov 7366	. 2 ⊢ (𝐴 ·Q 𝐵) = ( ·Q ‘⟨𝐴, 𝐵⟩)
17		df-ov 7366	. . 3 ⊢ (𝐴 ·pQ 𝐵) = ( ·pQ ‘⟨𝐴, 𝐵⟩)
18	17	fveq2i 6837	. 2 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩))
19	15, 16, 18	3eqtr4g 2800	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ⟨cop 4568   × cxp 5623   ↾ cres 5627   ∘ ccom 5629   Fn wfn 6487  ‘cfv 6492  (class class class)co 7363  1^st c1st 7936  2^nd c2nd 7937  Ncnpi 10765   ·_N cmi 10767   ·_pQ cmpq 10770  Qcnq 10773  [Q]cerq 10775   ·_Q cmq 10777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-mpq 10830  df-nq 10833  df-mq 10836

This theorem is referenced by:  mulclnq  10868  mulcomnq  10874  mulerpq  10878  mulassnq  10880  distrnq  10882  mulidnq  10884  ltmnq  10893