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 Description: Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
addcompr (𝐴 +P 𝐵) = (𝐵 +P 𝐴)

Proof of Theorem addcompr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plpv 9711 . . 3 ((𝐴P𝐵P) → (𝐴 +P 𝐵) = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 +Q 𝑦)})
2 plpv 9711 . . . . 5 ((𝐵P𝐴P) → (𝐵 +P 𝐴) = {𝑥 ∣ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 +Q 𝑧)})
3 addcomnq 9652 . . . . . . . . 9 (𝑦 +Q 𝑧) = (𝑧 +Q 𝑦)
43eqeq2i 2622 . . . . . . . 8 (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦))
542rexbii 3024 . . . . . . 7 (∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑧 +Q 𝑦))
6 rexcom 3080 . . . . . . 7 (∃𝑦𝐵𝑧𝐴 𝑥 = (𝑧 +Q 𝑦) ↔ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 +Q 𝑦))
75, 6bitri 263 . . . . . 6 (∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 +Q 𝑦))
87abbii 2726 . . . . 5 {𝑥 ∣ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 +Q 𝑧)} = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 +Q 𝑦)}
92, 8syl6eq 2660 . . . 4 ((𝐵P𝐴P) → (𝐵 +P 𝐴) = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 +Q 𝑦)})
109ancoms 468 . . 3 ((𝐴P𝐵P) → (𝐵 +P 𝐴) = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 +Q 𝑦)})
111, 10eqtr4d 2647 . 2 ((𝐴P𝐵P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))
12 dmplp 9713 . . 3 dom +P = (P × P)
1312ndmovcom 6719 . 2 (¬ (𝐴P𝐵P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))
1411, 13pm2.61i 175 1 (𝐴 +P 𝐵) = (𝐵 +P 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  (class class class)co 6549   +Q cplq 9556  Pcnp 9560   +P cpp 9562 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  ax-inf2 8421 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-rmo 2904  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-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-1nq 9617  df-np 9682  df-plp 9684 This theorem is referenced by:  enrer  9765  addcmpblnr  9769  mulcmpblnrlem  9770  ltsrpr  9777  addcomsr  9787  mulcomsr  9789  mulasssr  9790  distrsr  9791  ltsosr  9794  0lt1sr  9795  0idsr  9797  1idsr  9798  ltasr  9800  recexsrlem  9803  mulgt0sr  9805  ltpsrpr  9809  map2psrpr  9810
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