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Theorem mulcompr 9724
 Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulcompr (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴)

Proof of Theorem mulcompr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mpv 9712 . . 3 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦)})
2 mpv 9712 . . . . 5 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 ·Q 𝑧)})
3 mulcomnq 9654 . . . . . . . . 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 dmmp 9714 . . 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 cmq 9557  Pcnp 9560   ·P cmp 9563 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-mi 9575  df-lti 9576  df-mpq 9610  df-enq 9612  df-nq 9613  df-erq 9614  df-mq 9616  df-1nq 9617  df-np 9682  df-mp 9685 This theorem is referenced by:  mulcmpblnrlem  9770  mulcomsr  9789  mulasssr  9790  m1m1sr  9793  recexsrlem  9803  mulgt0sr  9805
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