Theorem addpqnq 10850

Description: Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
addpqnq	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))

Proof of Theorem addpqnq

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

Step	Hyp	Ref	Expression
1		df-plq 10826	. . . . 5 ⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q))
2	1	fveq1i 6833	. . . 4 ⊢ ( +Q ‘⟨𝐴, 𝐵⟩) = ((([Q] ∘ +pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩)
3	2	a1i 11	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( +Q ‘⟨𝐴, 𝐵⟩) = ((([Q] ∘ +pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩))
4		opelxpi 5659	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
5	4	fvresd 6852	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ +pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ +pQ )‘⟨𝐴, 𝐵⟩))
6		df-plpq 10820	. . . . 5 ⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
7		opex 5409	. . . . 5 ⊢ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ V
8	6, 7	fnmpoi 8014	. . . 4 ⊢ +pQ Fn ((N × N) × (N × N))
9		elpqn 10837	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
10		elpqn 10837	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
11		opelxpi 5659	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
12	9, 10, 11	syl2an 597	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
13		fvco2 6929	. . . 4 ⊢ (( +pQ Fn ((N × N) × (N × N)) ∧ ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N))) → (([Q] ∘ +pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩)))
14	8, 12, 13	sylancr 588	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ +pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩)))
15	3, 5, 14	3eqtrd 2776	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( +Q ‘⟨𝐴, 𝐵⟩) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩)))
16		df-ov 7361	. 2 ⊢ (𝐴 +Q 𝐵) = ( +Q ‘⟨𝐴, 𝐵⟩)
17		df-ov 7361	. . 3 ⊢ (𝐴 +pQ 𝐵) = ( +pQ ‘⟨𝐴, 𝐵⟩)
18	17	fveq2i 6835	. 2 ⊢ ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩))
19	15, 16, 18	3eqtr4g 2797	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4574   × cxp 5620   ↾ cres 5624   ∘ ccom 5626   Fn wfn 6485  ‘cfv 6490  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Ncnpi 10756   +_N cpli 10757   ·_N cmi 10758   +_pQ cplpq 10760  Qcnq 10764  [Q]cerq 10766   +_Q cplq 10767

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-plpq 10820  df-nq 10824  df-plq 10826

This theorem is referenced by:  addclnq  10857  addcomnq  10863  adderpq  10868  addassnq  10870  distrnq  10873  ltanq  10883  1lt2nq  10885  prlem934  10945