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Theorem mulcompr 10167
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 10155 . . 3 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦)})
2 mpv 10155 . . . . 5 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 ·Q 𝑧)})
3 mulcomnq 10097 . . . . . . . . 9 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
43eqeq2i 2837 . . . . . . . 8 (𝑥 = (𝑦 ·Q 𝑧) ↔ 𝑥 = (𝑧 ·Q 𝑦))
542rexbii 3252 . . . . . . 7 (∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑧 ·Q 𝑦))
6 rexcom 3309 . . . . . . 7 (∃𝑦𝐵𝑧𝐴 𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦))
75, 6bitri 267 . . . . . 6 (∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦))
87abbii 2944 . . . . 5 {𝑥 ∣ ∃𝑦𝐵𝑧𝐴 𝑥 = (𝑦 ·Q 𝑧)} = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦)}
92, 8syl6eq 2877 . . . 4 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦)})
109ancoms 452 . . 3 ((𝐴P𝐵P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑧𝐴𝑦𝐵 𝑥 = (𝑧 ·Q 𝑦)})
111, 10eqtr4d 2864 . 2 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
12 dmmp 10157 . . 3 dom ·P = (P × P)
1312ndmovcom 7086 . 2 (¬ (𝐴P𝐵P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
1411, 13pm2.61i 177 1 (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1656  wcel 2164  {cab 2811  wrex 3118  (class class class)co 6910   ·Q cmq 10000  Pcnp 10003   ·P cmp 10006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  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-rmo 3125  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 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-omul 7836  df-er 8014  df-ni 10016  df-mi 10018  df-lti 10019  df-mpq 10053  df-enq 10055  df-nq 10056  df-erq 10057  df-mq 10059  df-1nq 10060  df-np 10125  df-mp 10128
This theorem is referenced by:  mulcmpblnrlem  10214  mulcomsr  10233  mulasssr  10234  m1m1sr  10237  recexsrlem  10247  mulgt0sr  10249
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