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 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))⟩)

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6650 . . . . 5 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21oveq1d 7155 . . . 4 (𝑥 = 𝐴 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝑦)))
3 fveq2 6650 . . . . 5 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
43oveq2d 7156 . . . 4 (𝑥 = 𝐴 → ((1st𝑦) ·N (2nd𝑥)) = ((1st𝑦) ·N (2nd𝐴)))
52, 4oveq12d 7158 . . 3 (𝑥 = 𝐴 → (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))) = (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))))
63oveq1d 7155 . . 3 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
75, 6opeq12d 4774 . 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 6650 . . . . 5 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
98oveq2d 7156 . . . 4 (𝑦 = 𝐵 → ((1st𝐴) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝐵)))
10 fveq2 6650 . . . . 5 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
1110oveq1d 7155 . . . 4 (𝑦 = 𝐵 → ((1st𝑦) ·N (2nd𝐴)) = ((1st𝐵) ·N (2nd𝐴)))
129, 11oveq12d 7158 . . 3 (𝑦 = 𝐵 → (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))) = (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))))
138oveq2d 7156 . . 3 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
1412, 13opeq12d 4774 . 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 10326 . 2 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
16 opex 5322 . 2 ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V
177, 14, 15, 16ovmpo 7295 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 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   × cxp 5518  ‘cfv 6327  (class class class)co 7140  1st c1st 7676  2nd c2nd 7677  Ncnpi 10262   +N cpli 10263   ·N cmi 10264   +pQ cplpq 10266 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6286  df-fun 6329  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-plpq 10326 This theorem is referenced by:  addpipq  10355  addcompq  10368  adderpqlem  10372  addassnq  10376  distrnq  10379  ltanq  10389
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