Theorem mulpqnq 10901

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 10875	. . . . 5 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
2	1	fveq1i 6862	. . . 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 5678	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
5	4	fvresd 6881	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩))
6		df-mpq 10869	. . . . 5 ⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
7		opex 5427	. . . . 5 ⊢ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ V
8	6, 7	fnmpoi 8052	. . . 4 ⊢ ·pQ Fn ((N × N) × (N × N))
9		elpqn 10885	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
10		elpqn 10885	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
11		opelxpi 5678	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
12	9, 10, 11	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
13		fvco2 6961	. . . 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 587	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
15	3, 5, 14	3eqtrd 2769	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( ·Q ‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
16		df-ov 7393	. 2 ⊢ (𝐴 ·Q 𝐵) = ( ·Q ‘⟨𝐴, 𝐵⟩)
17		df-ov 7393	. . 3 ⊢ (𝐴 ·pQ 𝐵) = ( ·pQ ‘⟨𝐴, 𝐵⟩)
18	17	fveq2i 6864	. 2 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩))
19	15, 16, 18	3eqtr4g 2790	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4598   × cxp 5639   ↾ cres 5643   ∘ ccom 5645   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  Ncnpi 10804   ·_N cmi 10806   ·_pQ cmpq 10809  Qcnq 10812  [Q]cerq 10814   ·_Q cmq 10816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-mpq 10869  df-nq 10872  df-mq 10875

This theorem is referenced by:  mulclnq  10907  mulcomnq  10913  mulerpq  10917  mulassnq  10919  distrnq  10921  mulidnq  10923  ltmnq  10932