MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addpipq2 Structured version   Visualization version   GIF version

Theorem addpipq2 10211
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 6545 . . . . 5 (𝑥 = 𝐴 → (1st𝑥) = (1st𝐴))
21oveq1d 7038 . . . 4 (𝑥 = 𝐴 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝑦)))
3 fveq2 6545 . . . . 5 (𝑥 = 𝐴 → (2nd𝑥) = (2nd𝐴))
43oveq2d 7039 . . . 4 (𝑥 = 𝐴 → ((1st𝑦) ·N (2nd𝑥)) = ((1st𝑦) ·N (2nd𝐴)))
52, 4oveq12d 7041 . . 3 (𝑥 = 𝐴 → (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))) = (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))))
63oveq1d 7038 . . 3 (𝑥 = 𝐴 → ((2nd𝑥) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝑦)))
75, 6opeq12d 4724 . 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 6545 . . . . 5 (𝑦 = 𝐵 → (2nd𝑦) = (2nd𝐵))
98oveq2d 7039 . . . 4 (𝑦 = 𝐵 → ((1st𝐴) ·N (2nd𝑦)) = ((1st𝐴) ·N (2nd𝐵)))
10 fveq2 6545 . . . . 5 (𝑦 = 𝐵 → (1st𝑦) = (1st𝐵))
1110oveq1d 7038 . . . 4 (𝑦 = 𝐵 → ((1st𝑦) ·N (2nd𝐴)) = ((1st𝐵) ·N (2nd𝐴)))
129, 11oveq12d 7041 . . 3 (𝑦 = 𝐵 → (((1st𝐴) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝐴))) = (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))))
138oveq2d 7039 . . 3 (𝑦 = 𝐵 → ((2nd𝐴) ·N (2nd𝑦)) = ((2nd𝐴) ·N (2nd𝐵)))
1412, 13opeq12d 4724 . 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 10183 . 2 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
16 opex 5255 . 2 ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ ∈ V
177, 14, 15, 16ovmpo 7173 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 1525  wcel 2083  cop 4484   × cxp 5448  cfv 6232  (class class class)co 7023  1st c1st 7550  2nd c2nd 7551  Ncnpi 10119   +N cpli 10120   ·N cmi 10121   +pQ cplpq 10123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-plpq 10183
This theorem is referenced by:  addpipq  10212  addcompq  10225  adderpqlem  10229  addassnq  10233  distrnq  10236  ltanq  10246
  Copyright terms: Public domain W3C validator