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Theorem mulcompr 9883
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 9871 . . 3 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦)})
2 mpv 9871 . . . . 5 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 ·Q 𝑧)})
3 mulcomnq 9813 . . . . . . . . 9 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
43eqeq2i 2663 . . . . . . . 8 (𝑥 = (𝑦 ·Q 𝑧) ↔ 𝑥 = (𝑧 ·Q 𝑦))
542rexbii 3071 . . . . . . 7 (∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑧 ·Q 𝑦))
6 rexcom 3128 . . . . . . 7 (∃𝑦𝐵𝑧𝐴 𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦))
75, 6bitri 264 . . . . . 6 (∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦))
87abbii 2768 . . . . 5 {𝑥 ∣ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 ·Q 𝑧)} = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦)}
92, 8syl6eq 2701 . . . 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 2688 . 2 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
12 dmmp 9873 . . 3 dom ·P = (P × P)
1312ndmovcom 6863 . 2 (¬ (𝐴P𝐵P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
1411, 13pm2.61i 176 1 (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  (class class class)co 6690   ·Q cmq 9716  Pcnp 9719   ·P cmp 9722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-mi 9734  df-lti 9735  df-mpq 9769  df-enq 9771  df-nq 9772  df-erq 9773  df-mq 9775  df-1nq 9776  df-np 9841  df-mp 9844
This theorem is referenced by:  mulcmpblnrlem  9929  mulcomsr  9948  mulasssr  9949  m1m1sr  9952  recexsrlem  9962  mulgt0sr  9964
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