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Theorem mulcomprg 7137
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 7032 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 7038 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  z  e.  ( 1st `  B ) )  -> 
z  e.  Q. )
31, 2sylan 277 . . . . . . . 8  |-  ( ( B  e.  P.  /\  z  e.  ( 1st `  B ) )  -> 
z  e.  Q. )
4 prop 7032 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7038 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
64, 5sylan 277 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
7 mulcomnqg 6940 . . . . . . . . . . . . 13  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( z  .Q  y
)  =  ( y  .Q  z ) )
87eqeq2d 2099 . . . . . . . . . . . 12  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( x  =  ( z  .Q  y )  <-> 
x  =  ( y  .Q  z ) ) )
96, 8sylan2 280 . . . . . . . . . . 11  |-  ( ( z  e.  Q.  /\  ( A  e.  P.  /\  y  e.  ( 1st `  A ) ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
109anassrs 392 . . . . . . . . . 10  |-  ( ( ( z  e.  Q.  /\  A  e.  P. )  /\  y  e.  ( 1st `  A ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
1110rexbidva 2377 . . . . . . . . 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 264 . . . . . . . 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 280 . . . . . . 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 392 . . . . . 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 2377 . . . . 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 2531 . . . . 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, 16syl6bb 194 . . . 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 2608 . . 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 7039 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  z  e.  ( 2nd `  B ) )  -> 
z  e.  Q. )
201, 19sylan 277 . . . . . . . 8  |-  ( ( B  e.  P.  /\  z  e.  ( 2nd `  B ) )  -> 
z  e.  Q. )
21 elprnqu 7039 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
224, 21sylan 277 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2322, 8sylan2 280 . . . . . . . . . . 11  |-  ( ( z  e.  Q.  /\  ( A  e.  P.  /\  y  e.  ( 2nd `  A ) ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
2423anassrs 392 . . . . . . . . . 10  |-  ( ( ( z  e.  Q.  /\  A  e.  P. )  /\  y  e.  ( 2nd `  A ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
2524rexbidva 2377 . . . . . . . . 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 264 . . . . . . . 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 280 . . . . . . 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 392 . . . . . 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 2377 . . . . 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 2531 . . . . 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, 30syl6bb 194 . . . 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 2608 . . 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 3630 . 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 7096 . . 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 264 . 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 7096 . 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 2131 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E.wrex 2360   {crab 2363   <.cop 3449   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   Q.cnq 6837    .Q cmq 6840   P.cnp 6848    .P. cmp 6851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-mi 6863  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-mqqs 6907  df-inp 7023  df-imp 7026
This theorem is referenced by:  ltmprr  7199  mulcmpblnrlemg  7284  mulcomsrg  7301  mulasssrg  7302  m1m1sr  7305  recexgt0sr  7317  mulgt0sr  7321  mulextsr1lem  7323  recidpirqlemcalc  7392
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