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Theorem addcompq 10988
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
StepHypRef Expression
1 addcompi 10932 . . . 4 (((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵)))
2 mulcompi 10934 . . . 4 ((2nd𝐴) ·N (2nd𝐵)) = ((2nd𝐵) ·N (2nd𝐴))
31, 2opeq12i 4883 . . 3 ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩ = ⟨(((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐴))⟩
4 addpipq2 10974 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
5 addpipq2 10974 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 +pQ 𝐴) = ⟨(((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐴))⟩)
65ancoms 458 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 +pQ 𝐴) = ⟨(((1st𝐵) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐵))), ((2nd𝐵) ·N (2nd𝐴))⟩)
73, 4, 63eqtr4a 2801 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
8 addpqf 10982 . . . 4 +pQ :((N × N) × (N × N))⟶(N × N)
98fdmi 6748 . . 3 dom +pQ = ((N × N) × (N × N))
109ndmovcom 7620 . 2 (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴))
117, 10pm2.61i 182 1 (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  cop 4637   × cxp 5687  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Ncnpi 10882   +N cpli 10883   ·N cmi 10884   +pQ cplpq 10886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509  df-omul 8510  df-ni 10910  df-pli 10911  df-mi 10912  df-plpq 10946
This theorem is referenced by:  addcomnq  10989  adderpq  10994
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