Theorem addcompq 10564

Description: Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
addcompq	⊢ (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)

Proof of Theorem addcompq

Step	Hyp	Ref	Expression
1		addcompi 10508	. . . 4 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((1st ‘𝐵) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐵)))
2		mulcompi 10510	. . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴))
3	1, 2	opeq12i 4789	. . 3 ⊢ ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ = ⟨(((1st ‘𝐵) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩
4		addpipq2 10550	. . 3 ⊢ ((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))⟩)
5		addpipq2 10550	. . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 +pQ 𝐴) = ⟨(((1st ‘𝐵) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩)
6	5	ancoms 462	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 +pQ 𝐴) = ⟨(((1st ‘𝐵) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐴))⟩)
7	3, 4, 6	3eqtr4a 2804	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
8		addpqf 10558	. . . 4 ⊢ +pQ :((N × N) × (N × N))⟶(N × N)
9	8	fdmi 6557	. . 3 ⊢ dom +pQ = ((N × N) × (N × N))
10	9	ndmovcom 7395	. 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
11	7, 10	pm2.61i 185	1 ⊢ (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)

Syntax hints:   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ⟨cop 4547   × cxp 5549  ‘cfv 6380  (class class class)co 7213  1^st c1st 7759  2^nd c2nd 7760  Ncnpi 10458   +_N cpli 10459   ·_N cmi 10460   +_pQ cplpq 10462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-oadd 8206  df-omul 8207  df-ni 10486  df-pli 10487  df-mi 10488  df-plpq 10522

This theorem is referenced by:  addcomnq  10565  adderpq  10570