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Theorem mulpqnq 10161
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 10135 . . . . 5 ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
21fveq1i 6500 . . . 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 5444 . . . 4 ((𝐴Q𝐵Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
54fvresd 6519 . . 3 ((𝐴Q𝐵Q) → ((([Q] ∘ ·pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩))
6 df-mpq 10129 . . . . 5 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
7 opex 5213 . . . . 5 ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ V
86, 7fnmpoi 7576 . . . 4 ·pQ Fn ((N × N) × (N × N))
9 elpqn 10145 . . . . 5 (𝐴Q𝐴 ∈ (N × N))
10 elpqn 10145 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
11 opelxpi 5444 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
129, 10, 11syl2an 586 . . . 4 ((𝐴Q𝐵Q) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
13 fvco2 6586 . . . 4 (( ·pQ Fn ((N × N) × (N × N)) ∧ ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N))) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
148, 12, 13sylancr 578 . . 3 ((𝐴Q𝐵Q) → (([Q] ∘ ·pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
153, 5, 143eqtrd 2818 . 2 ((𝐴Q𝐵Q) → ( ·Q ‘⟨𝐴, 𝐵⟩) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩)))
16 df-ov 6979 . 2 (𝐴 ·Q 𝐵) = ( ·Q ‘⟨𝐴, 𝐵⟩)
17 df-ov 6979 . . 3 (𝐴 ·pQ 𝐵) = ( ·pQ ‘⟨𝐴, 𝐵⟩)
1817fveq2i 6502 . 2 ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘( ·pQ ‘⟨𝐴, 𝐵⟩))
1915, 16, 183eqtr4g 2839 1 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  cop 4447   × cxp 5405  cres 5409  ccom 5411   Fn wfn 6183  cfv 6188  (class class class)co 6976  1st c1st 7499  2nd c2nd 7500  Ncnpi 10064   ·N cmi 10066   ·pQ cmpq 10069  Qcnq 10072  [Q]cerq 10074   ·Q cmq 10076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-mpq 10129  df-nq 10132  df-mq 10135
This theorem is referenced by:  mulclnq  10167  mulcomnq  10173  mulerpq  10177  mulassnq  10179  distrnq  10181  mulidnq  10183  ltmnq  10192
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