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 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)

StepHypRef Expression
1 addcompi 9595 . . . 4 (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵)))
2 mulcompi 9597 . . . 4 ((2nd𝐴) ·N (2nd𝐵)) = ((2nd𝐵) ·N (2nd𝐴))
31, 2opeq12i 4345 . . 3 ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ = ⟨(((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐴))⟩
4 addpipq2 9637 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
5 addpipq2 9637 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 +pQ 𝐴) = ⟨(((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐴))⟩)
65ancoms 468 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 +pQ 𝐴) = ⟨(((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐴))⟩)
73, 4, 63eqtr4a 2670 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
8 addpqf 9645 . . . 4 +pQ :((N × N) × (N × N))⟶(N × N)
98fdmi 5965 . . 3 dom +pQ = ((N × N) × (N × N))
109ndmovcom 6719 . 2 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
117, 10pm2.61i 175 1 (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Ncnpi 9545   +N cpli 9546   ·N cmi 9547   +pQ cplpq 9549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452  df-ni 9573  df-pli 9574  df-mi 9575  df-plpq 9609 This theorem is referenced by:  addcomnq  9652  adderpq  9657
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