Theorem addpipq2 10939

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) = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)

Proof of Theorem addpipq2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6888	. . . . 5 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
2	1	oveq1d 7438	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝑦)))
3		fveq2 6888	. . . . 5 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
4	3	oveq2d 7439	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑦) ·N (2nd ‘𝑥)) = ((1st ‘𝑦) ·N (2nd ‘𝐴)))
5	2, 4	oveq12d 7441	. . 3 ⊢ (𝑥 = 𝐴 → (((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))) = (((1st ‘𝐴) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝐴))))
6	3	oveq1d 7438	. . 3 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝑦)))
7	5, 6	opeq12d 4851	. 2 ⊢ (𝑥 = 𝐴 → ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ = ⟨(((1st ‘𝐴) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝑦))⟩)
8		fveq2 6888	. . . . 5 ⊢ (𝑦 = 𝐵 → (2nd ‘𝑦) = (2nd ‘𝐵))
9	8	oveq2d 7439	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝐴) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝐵)))
10		fveq2 6888	. . . . 5 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵))
11	10	oveq1d 7438	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝑦) ·N (2nd ‘𝐴)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))
12	9, 11	oveq12d 7441	. . 3 ⊢ (𝑦 = 𝐵 → (((1st ‘𝐴) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝐴))) = (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
13	8	oveq2d 7439	. . 3 ⊢ (𝑦 = 𝐵 → ((2nd ‘𝐴) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
14	12, 13	opeq12d 4851	. 2 ⊢ (𝑦 = 𝐵 → ⟨(((1st ‘𝐴) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝑦))⟩ = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
15		df-plpq 10911	. 2 ⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
16		opex 5450	. 2 ⊢ ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V
17	7, 14, 15, 16	ovmpo 7583	1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ⟨cop 4600   × cxp 5664  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994  Ncnpi 10847   +_N cpli 10848   ·_N cmi 10849   +_pQ cplpq 10851

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-plpq 10911

This theorem is used by:  addpipq  10940  addcompq  10953  adderpqlem  10957  addassnq  10961  distrnq  10964  ltanq  10974