Theorem addpqnq 10349

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 10325	. . . . 5 ⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q))
2	1	fveq1i 6646	. . . 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 5556	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ (Q × Q))
5	4	fvresd 6665	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((([Q] ∘ +pQ ) ↾ (Q × Q))‘⟨𝐴, 𝐵⟩) = (([Q] ∘ +pQ )‘⟨𝐴, 𝐵⟩))
6		df-plpq 10319	. . . . 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 5321	. . . . 5 ⊢ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ ∈ V
8	6, 7	fnmpoi 7750	. . . 4 ⊢ +pQ Fn ((N × N) × (N × N))
9		elpqn 10336	. . . . 5 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
10		elpqn 10336	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
11		opelxpi 5556	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
12	9, 10, 11	syl2an 598	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨𝐴, 𝐵⟩ ∈ ((N × N) × (N × N)))
13		fvco2 6735	. . . 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 590	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (([Q] ∘ +pQ )‘⟨𝐴, 𝐵⟩) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩)))
15	3, 5, 14	3eqtrd 2837	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ( +Q ‘⟨𝐴, 𝐵⟩) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩)))
16		df-ov 7138	. 2 ⊢ (𝐴 +Q 𝐵) = ( +Q ‘⟨𝐴, 𝐵⟩)
17		df-ov 7138	. . 3 ⊢ (𝐴 +pQ 𝐵) = ( +pQ ‘⟨𝐴, 𝐵⟩)
18	17	fveq2i 6648	. 2 ⊢ ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘( +pQ ‘⟨𝐴, 𝐵⟩))
19	15, 16, 18	3eqtr4g 2858	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   × cxp 5517   ↾ cres 5521   ∘ ccom 5523   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135  1^st c1st 7669  2^nd c2nd 7670  Ncnpi 10255   +_N cpli 10256   ·_N cmi 10257   +_pQ cplpq 10259  Qcnq 10263  [Q]cerq 10265   +_Q cplq 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 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  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 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-plpq 10319  df-nq 10323  df-plq 10325

This theorem is referenced by:  addclnq  10356  addcomnq  10362  adderpq  10367  addassnq  10369  distrnq  10372  ltanq  10382  1lt2nq  10384  prlem934  10444