MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulpqnq Structured version   Visualization version   GIF version

Theorem mulpqnq 9801
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 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))

Proof of Theorem mulpqnq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mq 9775 . . . . 5 ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
21fveq1i 6230 . . . 4 ( ·Q ‘⟨𝐴, 𝐵⟩) = ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩)
32a1i 11 . . 3 ((𝐴Q𝐵Q) → ( ·Q ‘⟨𝐴, 𝐵⟩) = ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩))
4 opelxpi 5182 . . . 4 ((𝐴Q𝐵Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
5 fvres 6245 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (Q × Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩))
64, 5syl 17 . . 3 ((𝐴Q𝐵Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩))
7 df-mpq 9769 . . . . 5 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
8 opex 4962 . . . . 5 ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ V
97, 8fnmpt2i 7284 . . . 4 ·pQ Fn ((N × N) × (N × N))
10 elpqn 9785 . . . . 5 (𝐴Q𝐴 ∈ (N × N))
11 elpqn 9785 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
12 opelxpi 5182 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
1310, 11, 12syl2an 493 . . . 4 ((𝐴Q𝐵Q) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
14 fvco2 6312 . . . 4 (( ·pQ Fn ((N × N) × (N × N)) ∧ ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N))) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
159, 13, 14sylancr 696 . . 3 ((𝐴Q𝐵Q) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
163, 6, 153eqtrd 2689 . 2 ((𝐴Q𝐵Q) → ( ·Q ‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
17 df-ov 6693 . 2 (𝐴 ·Q 𝐵) = ( ·Q ‘⟨𝐴, 𝐵⟩)
18 df-ov 6693 . . 3 (𝐴 ·pQ 𝐵) = ( ·pQ ‘⟨𝐴, 𝐵⟩)
1918fveq2i 6232 . 2 ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩))
2016, 17, 193eqtr4g 2710 1 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cop 4216   × cxp 5141  cres 5145  ccom 5147   Fn wfn 5921  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Ncnpi 9704   ·N cmi 9706   ·pQ cmpq 9709  Qcnq 9712  [Q]cerq 9714   ·Q cmq 9716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-mpq 9769  df-nq 9772  df-mq 9775
This theorem is referenced by:  mulclnq  9807  mulcomnq  9813  mulerpq  9817  mulassnq  9819  distrnq  9821  mulidnq  9823  ltmnq  9832
  Copyright terms: Public domain W3C validator