Theorem addcompq 10637

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 10581	. . . 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 10583	. . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴))
3	1, 2	opeq12i 4806	. . 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 10623	. . 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 10623	. . . 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 458	. . 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 2805	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
8		addpqf 10631	. . . 4 ⊢ +pQ :((N × N) × (N × N))⟶(N × N)
9	8	fdmi 6596	. . 3 ⊢ dom +pQ = ((N × N) × (N × N))
10	9	ndmovcom 7437	. 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
11	7, 10	pm2.61i 182	1 ⊢ (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   × cxp 5578  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  Ncnpi 10531   +_N cpli 10532   ·_N cmi 10533   +_pQ cplpq 10535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-oadd 8271  df-omul 8272  df-ni 10559  df-pli 10560  df-mi 10561  df-plpq 10595

This theorem is referenced by:  addcomnq  10638  adderpq  10643