Theorem addcompq 10706

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 10650	. . . 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 10652	. . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴))
3	1, 2	opeq12i 4809	. . 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 10692	. . 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 10692	. . . 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 459	. . 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 10700	. . . 4 ⊢ +pQ :((N × N) × (N × N))⟶(N × N)
9	8	fdmi 6612	. . 3 ⊢ dom +pQ = ((N × N) × (N × N))
10	9	ndmovcom 7459	. 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 396   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   × cxp 5587  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  Ncnpi 10600   +_N cpli 10601   ·_N cmi 10602   +_pQ cplpq 10604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-oadd 8301  df-omul 8302  df-ni 10628  df-pli 10629  df-mi 10630  df-plpq 10664

This theorem is referenced by:  addcomnq  10707  adderpq  10712