Theorem mulpqnq 10972

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 10946	. . . . 5 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
2	1	fveq1i 6903	. . . 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 5719	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
5	4	fvresd 6922	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩))
6		df-mpq 10940	. . . . 5 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
7		opex 5470	. . . . 5 ⊢ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ V
8	6, 7	fnmpoi 8080	. . . 4 ⊢ ·pQ Fn ((N × N) × (N × N))
9		elpqn 10956	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
10		elpqn 10956	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
11		opelxpi 5719	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
12	9, 10, 11	syl2an 594	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
13		fvco2 7000	. . . 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 585	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
15	3, 5, 14	3eqtrd 2772	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
16		df-ov 7429	. 2 ⊢ (𝐴 ·Q 𝐵) = ( ·Q ‘⟨𝐴, 𝐵⟩)
17		df-ov 7429	. . 3 ⊢ (𝐴 ·pQ 𝐵) = ( ·pQ ‘⟨𝐴, 𝐵⟩)
18	17	fveq2i 6905	. 2 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩))
19	15, 16, 18	3eqtr4g 2793	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ⟨cop 4638   × cxp 5680   ↾ cres 5684   ∘ ccom 5686   Fn wfn 6548  ‘cfv 6553  (class class class)co 7426  1^st c1st 7997  2^nd c2nd 7998  Ncnpi 10875   ·_N cmi 10877   ·_pQ cmpq 10880  Qcnq 10883  [Q]cerq 10885   ·_Q cmq 10887

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-mpq 10940  df-nq 10943  df-mq 10946

This theorem is referenced by:  mulclnq  10978  mulcomnq  10984  mulerpq  10988  mulassnq  10990  distrnq  10992  mulidnq  10994  ltmnq  11003