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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 10031 . . . 4 (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵)))
2 mulcompi 10033 . . . 4 ((2nd𝐴) ·N (2nd𝐵)) = ((2nd𝐵) ·N (2nd𝐴))
31, 2opeq12i 4628 . . 3 ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ = ⟨(((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐴))⟩
4 addpipq2 10073 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
5 addpipq2 10073 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 +pQ 𝐴) = ⟨(((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐴))⟩)
65ancoms 452 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 +pQ 𝐴) = ⟨(((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐴))⟩)
73, 4, 63eqtr4a 2887 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
8 addpqf 10081 . . . 4 +pQ :((N × N) × (N × N))⟶(N × N)
98fdmi 6288 . . 3 dom +pQ = ((N × N) × (N × N))
109ndmovcom 7081 . 2 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
117, 10pm2.61i 177 1 (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ⟨cop 4403   × cxp 5340  ‘cfv 6123  (class class class)co 6905  1st c1st 7426  2nd c2nd 7427  Ncnpi 9981   +N cpli 9982   ·N cmi 9983   +pQ cplpq 9985 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-oadd 7830  df-omul 7831  df-ni 10009  df-pli 10010  df-mi 10011  df-plpq 10045 This theorem is referenced by:  addcomnq  10088  adderpq  10093
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