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Theorem addpqnq 9798
Description: Addition 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
addpqnq ((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))

Proof of Theorem addpqnq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-plq 9774 . . . . 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-plpq 9768 . . . . 5 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
8 opex 4962 . . . . 5 ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((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 cpli 9705   ·N cmi 9706   +pQ cplpq 9708  Qcnq 9712  [Q]cerq 9714   +Q cplq 9715
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-plpq 9768  df-nq 9772  df-plq 9774
This theorem is referenced by:  addclnq  9805  addcomnq  9811  adderpq  9816  addassnq  9818  distrnq  9821  ltanq  9831  1lt2nq  9833  prlem934  9893
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