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Theorem mulcomprg 7502
Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
Assertion
Ref Expression
mulcomprg  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )

Proof of Theorem mulcomprg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7397 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 7403 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  z  e.  ( 1st `  B ) )  -> 
z  e.  Q. )
31, 2sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  z  e.  ( 1st `  B ) )  -> 
z  e.  Q. )
4 prop 7397 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7403 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
64, 5sylan 281 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
7 mulcomnqg 7305 . . . . . . . . . . . . 13  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( z  .Q  y
)  =  ( y  .Q  z ) )
87eqeq2d 2169 . . . . . . . . . . . 12  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( x  =  ( z  .Q  y )  <-> 
x  =  ( y  .Q  z ) ) )
96, 8sylan2 284 . . . . . . . . . . 11  |-  ( ( z  e.  Q.  /\  ( A  e.  P.  /\  y  e.  ( 1st `  A ) ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
109anassrs 398 . . . . . . . . . 10  |-  ( ( ( z  e.  Q.  /\  A  e.  P. )  /\  y  e.  ( 1st `  A ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
1110rexbidva 2454 . . . . . . . . 9  |-  ( ( z  e.  Q.  /\  A  e.  P. )  ->  ( E. y  e.  ( 1st `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 1st `  A ) x  =  ( y  .Q  z ) ) )
1211ancoms 266 . . . . . . . 8  |-  ( ( A  e.  P.  /\  z  e.  Q. )  ->  ( E. y  e.  ( 1st `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 1st `  A ) x  =  ( y  .Q  z ) ) )
133, 12sylan2 284 . . . . . . 7  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  z  e.  ( 1st `  B ) ) )  ->  ( E. y  e.  ( 1st `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 1st `  A ) x  =  ( y  .Q  z ) ) )
1413anassrs 398 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  z  e.  ( 1st `  B ) )  ->  ( E. y  e.  ( 1st `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 1st `  A ) x  =  ( y  .Q  z ) ) )
1514rexbidva 2454 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. z  e.  ( 1st `  B
) E. y  e.  ( 1st `  A
) x  =  ( z  .Q  y )  <->  E. z  e.  ( 1st `  B ) E. y  e.  ( 1st `  A ) x  =  ( y  .Q  z
) ) )
16 rexcom 2621 . . . . 5  |-  ( E. z  e.  ( 1st `  B ) E. y  e.  ( 1st `  A
) x  =  ( y  .Q  z )  <->  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y  .Q  z
) )
1715, 16bitrdi 195 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. z  e.  ( 1st `  B
) E. y  e.  ( 1st `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y  .Q  z
) ) )
1817rabbidv 2701 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  { x  e.  Q.  |  E. z  e.  ( 1st `  B ) E. y  e.  ( 1st `  A ) x  =  ( z  .Q  y ) }  =  { x  e. 
Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  B
) x  =  ( y  .Q  z ) } )
19 elprnqu 7404 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  z  e.  ( 2nd `  B ) )  -> 
z  e.  Q. )
201, 19sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  z  e.  ( 2nd `  B ) )  -> 
z  e.  Q. )
21 elprnqu 7404 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
224, 21sylan 281 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2322, 8sylan2 284 . . . . . . . . . . 11  |-  ( ( z  e.  Q.  /\  ( A  e.  P.  /\  y  e.  ( 2nd `  A ) ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
2423anassrs 398 . . . . . . . . . 10  |-  ( ( ( z  e.  Q.  /\  A  e.  P. )  /\  y  e.  ( 2nd `  A ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
2524rexbidva 2454 . . . . . . . . 9  |-  ( ( z  e.  Q.  /\  A  e.  P. )  ->  ( E. y  e.  ( 2nd `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 2nd `  A ) x  =  ( y  .Q  z ) ) )
2625ancoms 266 . . . . . . . 8  |-  ( ( A  e.  P.  /\  z  e.  Q. )  ->  ( E. y  e.  ( 2nd `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 2nd `  A ) x  =  ( y  .Q  z ) ) )
2720, 26sylan2 284 . . . . . . 7  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  z  e.  ( 2nd `  B ) ) )  ->  ( E. y  e.  ( 2nd `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 2nd `  A ) x  =  ( y  .Q  z ) ) )
2827anassrs 398 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  z  e.  ( 2nd `  B ) )  ->  ( E. y  e.  ( 2nd `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 2nd `  A ) x  =  ( y  .Q  z ) ) )
2928rexbidva 2454 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. z  e.  ( 2nd `  B
) E. y  e.  ( 2nd `  A
) x  =  ( z  .Q  y )  <->  E. z  e.  ( 2nd `  B ) E. y  e.  ( 2nd `  A ) x  =  ( y  .Q  z
) ) )
30 rexcom 2621 . . . . 5  |-  ( E. z  e.  ( 2nd `  B ) E. y  e.  ( 2nd `  A
) x  =  ( y  .Q  z )  <->  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y  .Q  z
) )
3129, 30bitrdi 195 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. z  e.  ( 2nd `  B
) E. y  e.  ( 2nd `  A
) x  =  ( z  .Q  y )  <->  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y  .Q  z
) ) )
3231rabbidv 2701 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  { x  e.  Q.  |  E. z  e.  ( 2nd `  B ) E. y  e.  ( 2nd `  A ) x  =  ( z  .Q  y ) }  =  { x  e. 
Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  B
) x  =  ( y  .Q  z ) } )
3318, 32opeq12d 3751 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
<. { x  e.  Q.  |  E. z  e.  ( 1st `  B ) E. y  e.  ( 1st `  A ) x  =  ( z  .Q  y ) } ,  { x  e. 
Q.  |  E. z  e.  ( 2nd `  B
) E. y  e.  ( 2nd `  A
) x  =  ( z  .Q  y ) } >.  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y  .Q  z
) } ,  {
x  e.  Q.  |  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y  .Q  z
) } >. )
34 mpvlu 7461 . . 3  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  .P.  A
)  =  <. { x  e.  Q.  |  E. z  e.  ( 1st `  B
) E. y  e.  ( 1st `  A
) x  =  ( z  .Q  y ) } ,  { x  e.  Q.  |  E. z  e.  ( 2nd `  B
) E. y  e.  ( 2nd `  A
) x  =  ( z  .Q  y ) } >. )
3534ancoms 266 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( B  .P.  A
)  =  <. { x  e.  Q.  |  E. z  e.  ( 1st `  B
) E. y  e.  ( 1st `  A
) x  =  ( z  .Q  y ) } ,  { x  e.  Q.  |  E. z  e.  ( 2nd `  B
) E. y  e.  ( 2nd `  A
) x  =  ( z  .Q  y ) } >. )
36 mpvlu 7461 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A
) E. z  e.  ( 1st `  B
) x  =  ( y  .Q  z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  B
) x  =  ( y  .Q  z ) } >. )
3733, 35, 363eqtr4rd 2201 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   E.wrex 2436   {crab 2439   <.cop 3564   ` cfv 5172  (class class class)co 5826   1stc1st 6088   2ndc2nd 6089   Q.cnq 7202    .Q cmq 7205   P.cnp 7213    .P. cmp 7216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-mi 7228  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-mqqs 7272  df-inp 7388  df-imp 7391
This theorem is referenced by:  ltmprr  7564  mulcmpblnrlemg  7662  mulcomsrg  7679  mulasssrg  7680  m1m1sr  7683  recexgt0sr  7695  mulgt0sr  7700  mulextsr1lem  7702  recidpirqlemcalc  7779
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