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Theorem addpipq2 9796
Description: Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
addpipq2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)

Proof of Theorem addpipq2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . . 5 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21oveq1d 6705 . . . 4 (𝑥 = 𝐴 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝑦)))
3 fveq2 6229 . . . . 5 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
43oveq2d 6706 . . . 4 (𝑥 = 𝐴 → ((1st𝑦) ·N (2nd𝑥)) = ((1st𝑦) ·N (2nd𝐴)))
52, 4oveq12d 6708 . . 3 (𝑥 = 𝐴 → (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))) = (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))))
63oveq1d 6705 . . 3 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
75, 6opeq12d 4441 . 2 (𝑥 = 𝐴 → ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩ = ⟨(((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝑦))⟩)
8 fveq2 6229 . . . . 5 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
98oveq2d 6706 . . . 4 (𝑦 = 𝐵 → ((1st𝐴) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝐵)))
10 fveq2 6229 . . . . 5 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
1110oveq1d 6705 . . . 4 (𝑦 = 𝐵 → ((1st𝑦) ·N (2nd𝐴)) = ((1st𝐵) ·N (2nd𝐴)))
129, 11oveq12d 6708 . . 3 (𝑦 = 𝐵 → (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))) = (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))))
138oveq2d 6706 . . 3 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
1412, 13opeq12d 4441 . 2 (𝑦 = 𝐵 → ⟨(((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝑦))⟩ = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
15 df-plpq 9768 . 2 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
16 opex 4962 . 2 ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V
177, 14, 15, 16ovmpt2 6838 1 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cop 4216   × cxp 5141  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Ncnpi 9704   +N cpli 9705   ·N cmi 9706   +pQ cplpq 9708
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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
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-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  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-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-plpq 9768
This theorem is referenced by:  addpipq  9797  addcompq  9810  adderpqlem  9814  addassnq  9818  distrnq  9821  ltanq  9831
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