Theorem addcompq 10873

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 10817	. . . 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 10819	. . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴))
3	1, 2	opeq12i 4822	. . 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 10859	. . 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 10859	. . . 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 2798	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
8		addpqf 10867	. . . 4 ⊢ +pQ :((N × N) × (N × N))⟶(N × N)
9	8	fdmi 6680	. . 3 ⊢ dom +pQ = ((N × N) × (N × N))
10	9	ndmovcom 7554	. 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 1542   ∈ wcel 2114  ⟨cop 4574   × cxp 5629  ‘cfv 6499  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  Ncnpi 10767   +_N cpli 10768   ·_N cmi 10769   +_pQ cplpq 10771

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 5232  ax-nul 5242  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-reu 3344  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-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  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-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-oadd 8409  df-omul 8410  df-ni 10795  df-pli 10796  df-mi 10797  df-plpq 10831

This theorem is referenced by:  addcomnq  10874  adderpq  10879