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 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)))

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-plq 10330 . . . . 5 +Q = (([Q] ∘ +pQ ) ↾ (Q × Q))
21fveq1i 6665 . . . 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 5586 . . . 4 ((𝐴Q𝐵Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
54fvresd 6684 . . 3 ((𝐴Q𝐵Q) → ((([Q] ∘ +pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ +pQ )‘⟨𝐴, 𝐵⟩))
6 df-plpq 10324 . . . . 5 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
7 opex 5348 . . . . 5 ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ V
86, 7fnmpoi 7762 . . . 4 +pQ Fn ((N × N) × (N × N))
9 elpqn 10341 . . . . 5 (𝐴Q𝐴 ∈ (N × N))
10 elpqn 10341 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
11 opelxpi 5586 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
129, 10, 11syl2an 597 . . . 4 ((𝐴Q𝐵Q) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
13 fvco2 6752 . . . 4 (( +pQ Fn ((N × N) × (N × N)) ∧ ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N))) → (([Q] ∘ +pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩)))
148, 12, 13sylancr 589 . . 3 ((𝐴Q𝐵Q) → (([Q] ∘ +pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩)))
153, 5, 143eqtrd 2860 . 2 ((𝐴Q𝐵Q) → ( +Q ‘⟨𝐴, 𝐵⟩) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩)))
16 df-ov 7153 . 2 (𝐴 +Q 𝐵) = ( +Q ‘⟨𝐴, 𝐵⟩)
17 df-ov 7153 . . 3 (𝐴 +pQ 𝐵) = ( +pQ ‘⟨𝐴, 𝐵⟩)
1817fveq2i 6667 . 2 ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩))
1915, 16, 183eqtr4g 2881 1 ((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ⟨cop 4566   × cxp 5547   ↾ cres 5551   ∘ ccom 5553   Fn wfn 6344  ‘cfv 6349  (class class class)co 7150  1st c1st 7681  2nd c2nd 7682  Ncnpi 10260   +N cpli 10261   ·N cmi 10262   +pQ cplpq 10264  Qcnq 10268  [Q]cerq 10270   +Q cplq 10271 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-plpq 10324  df-nq 10328  df-plq 10330 This theorem is referenced by:  addclnq  10361  addcomnq  10367  adderpq  10372  addassnq  10374  distrnq  10377  ltanq  10387  1lt2nq  10389  prlem934  10449
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