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Theorem addpipq2 10868
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 6839 . . . . 5 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21oveq1d 7368 . . . 4 (𝑥 = 𝐴 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝑦)))
3 fveq2 6839 . . . . 5 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
43oveq2d 7369 . . . 4 (𝑥 = 𝐴 → ((1st𝑦) ·N (2nd𝑥)) = ((1st𝑦) ·N (2nd𝐴)))
52, 4oveq12d 7371 . . 3 (𝑥 = 𝐴 → (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))) = (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))))
63oveq1d 7368 . . 3 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
75, 6opeq12d 4836 . 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 6839 . . . . 5 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
98oveq2d 7369 . . . 4 (𝑦 = 𝐵 → ((1st𝐴) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝐵)))
10 fveq2 6839 . . . . 5 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
1110oveq1d 7368 . . . 4 (𝑦 = 𝐵 → ((1st𝑦) ·N (2nd𝐴)) = ((1st𝐵) ·N (2nd𝐴)))
129, 11oveq12d 7371 . . 3 (𝑦 = 𝐵 → (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))) = (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))))
138oveq2d 7369 . . 3 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
1412, 13opeq12d 4836 . 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 10840 . 2 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
16 opex 5419 . 2 ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V
177, 14, 15, 16ovmpo 7511 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 396   = wceq 1541  wcel 2106  cop 4590   × cxp 5629  cfv 6493  (class class class)co 7353  1st c1st 7915  2nd c2nd 7916  Ncnpi 10776   +N cpli 10777   ·N cmi 10778   +pQ cplpq 10780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-plpq 10840
This theorem is referenced by:  addpipq  10869  addcompq  10882  adderpqlem  10886  addassnq  10890  distrnq  10893  ltanq  10903
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