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Theorem mulpqnq 10979
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 10953 . . . . 5 ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
21fveq1i 6908 . . . 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 5726 . . . 4 ((𝐴Q𝐵Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
54fvresd 6927 . . 3 ((𝐴Q𝐵Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩))
6 df-mpq 10947 . . . . 5 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
7 opex 5475 . . . . 5 ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ V
86, 7fnmpoi 8094 . . . 4 ·pQ Fn ((N × N) × (N × N))
9 elpqn 10963 . . . . 5 (𝐴Q𝐴 ∈ (N × N))
10 elpqn 10963 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
11 opelxpi 5726 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
129, 10, 11syl2an 596 . . . 4 ((𝐴Q𝐵Q) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
13 fvco2 7006 . . . 4 (( ·pQ Fn ((N × N) × (N × N)) ∧ ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N))) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
148, 12, 13sylancr 587 . . 3 ((𝐴Q𝐵Q) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
153, 5, 143eqtrd 2779 . 2 ((𝐴Q𝐵Q) → ( ·Q ‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
16 df-ov 7434 . 2 (𝐴 ·Q 𝐵) = ( ·Q ‘⟨𝐴, 𝐵⟩)
17 df-ov 7434 . . 3 (𝐴 ·pQ 𝐵) = ( ·pQ ‘⟨𝐴, 𝐵⟩)
1817fveq2i 6910 . 2 ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩))
1915, 16, 183eqtr4g 2800 1 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cop 4637   × cxp 5687  cres 5691  ccom 5693   Fn wfn 6558  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Ncnpi 10882   ·N cmi 10884   ·pQ cmpq 10887  Qcnq 10890  [Q]cerq 10892   ·Q cmq 10894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-mpq 10947  df-nq 10950  df-mq 10953
This theorem is referenced by:  mulclnq  10985  mulcomnq  10991  mulerpq  10995  mulassnq  10997  distrnq  10999  mulidnq  11001  ltmnq  11010
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