MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addpqnq Structured version   Visualization version   GIF version

Theorem addpqnq 10978
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 10954 . . . . 5 +Q = (([Q] ∘ +pQ ) ↾ (Q × Q))
21fveq1i 6907 . . . 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 5722 . . . 4 ((𝐴Q𝐵Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
54fvresd 6926 . . 3 ((𝐴Q𝐵Q) → ((([Q] ∘ +pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ +pQ )‘⟨𝐴, 𝐵⟩))
6 df-plpq 10948 . . . . 5 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
7 opex 5469 . . . . 5 ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩ ∈ V
86, 7fnmpoi 8095 . . . 4 +pQ Fn ((N × N) × (N × N))
9 elpqn 10965 . . . . 5 (𝐴Q𝐴 ∈ (N × N))
10 elpqn 10965 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
11 opelxpi 5722 . . . . 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 2781 . 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 6909 . 2 ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩))
1915, 16, 183eqtr4g 2802 1 ((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cop 4632   × cxp 5683  cres 5687  ccom 5689   Fn wfn 6556  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Ncnpi 10884   +N cpli 10885   ·N cmi 10886   +pQ cplpq 10888  Qcnq 10892  [Q]cerq 10894   +Q cplq 10895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-plpq 10948  df-nq 10952  df-plq 10954
This theorem is referenced by:  addclnq  10985  addcomnq  10991  adderpq  10996  addassnq  10998  distrnq  11001  ltanq  11011  1lt2nq  11013  prlem934  11073
  Copyright terms: Public domain W3C validator