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Theorem addpipq2 11005
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 6920 . . . . 5 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21oveq1d 7463 . . . 4 (𝑥 = 𝐴 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝑦)))
3 fveq2 6920 . . . . 5 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
43oveq2d 7464 . . . 4 (𝑥 = 𝐴 → ((1st𝑦) ·N (2nd𝑥)) = ((1st𝑦) ·N (2nd𝐴)))
52, 4oveq12d 7466 . . 3 (𝑥 = 𝐴 → (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))) = (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))))
63oveq1d 7463 . . 3 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
75, 6opeq12d 4905 . 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 6920 . . . . 5 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
98oveq2d 7464 . . . 4 (𝑦 = 𝐵 → ((1st𝐴) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝐵)))
10 fveq2 6920 . . . . 5 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
1110oveq1d 7463 . . . 4 (𝑦 = 𝐵 → ((1st𝑦) ·N (2nd𝐴)) = ((1st𝐵) ·N (2nd𝐴)))
129, 11oveq12d 7466 . . 3 (𝑦 = 𝐵 → (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))) = (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))))
138oveq2d 7464 . . 3 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
1412, 13opeq12d 4905 . 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 10977 . 2 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
16 opex 5484 . 2 ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V
177, 14, 15, 16ovmpo 7610 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 395   = wceq 1537  wcel 2108  cop 4654   × cxp 5698  cfv 6573  (class class class)co 7448  1st c1st 8028  2nd c2nd 8029  Ncnpi 10913   +N cpli 10914   ·N cmi 10915   +pQ cplpq 10917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-plpq 10977
This theorem is referenced by:  addpipq  11006  addcompq  11019  adderpqlem  11023  addassnq  11027  distrnq  11030  ltanq  11040
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