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Theorem addpqnq 10976
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 10952 . . . . 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-plpq 10946 . . . . 5 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
7 opex 5475 . . . . 5 ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((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 cpli 10883   ·N cmi 10884   +pQ cplpq 10886  Qcnq 10890  [Q]cerq 10892   +Q cplq 10893
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-plpq 10946  df-nq 10950  df-plq 10952
This theorem is referenced by:  addclnq  10983  addcomnq  10989  adderpq  10994  addassnq  10996  distrnq  10999  ltanq  11009  1lt2nq  11011  prlem934  11071
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