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Theorem mulcomprg 7593
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 7488 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 7494 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  z  e.  ( 1st `  B ) )  -> 
z  e.  Q. )
31, 2sylan 283 . . . . . . . 8  |-  ( ( B  e.  P.  /\  z  e.  ( 1st `  B ) )  -> 
z  e.  Q. )
4 prop 7488 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7494 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
64, 5sylan 283 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
7 mulcomnqg 7396 . . . . . . . . . . . . 13  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( z  .Q  y
)  =  ( y  .Q  z ) )
87eqeq2d 2199 . . . . . . . . . . . 12  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( x  =  ( z  .Q  y )  <-> 
x  =  ( y  .Q  z ) ) )
96, 8sylan2 286 . . . . . . . . . . 11  |-  ( ( z  e.  Q.  /\  ( A  e.  P.  /\  y  e.  ( 1st `  A ) ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
109anassrs 400 . . . . . . . . . 10  |-  ( ( ( z  e.  Q.  /\  A  e.  P. )  /\  y  e.  ( 1st `  A ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
1110rexbidva 2484 . . . . . . . . 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 268 . . . . . . . 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 286 . . . . . . 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 400 . . . . . 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 2484 . . . . 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 2651 . . . . 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 196 . . . 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 2738 . . 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 7495 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  z  e.  ( 2nd `  B ) )  -> 
z  e.  Q. )
201, 19sylan 283 . . . . . . . 8  |-  ( ( B  e.  P.  /\  z  e.  ( 2nd `  B ) )  -> 
z  e.  Q. )
21 elprnqu 7495 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
224, 21sylan 283 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
2322, 8sylan2 286 . . . . . . . . . . 11  |-  ( ( z  e.  Q.  /\  ( A  e.  P.  /\  y  e.  ( 2nd `  A ) ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
2423anassrs 400 . . . . . . . . . 10  |-  ( ( ( z  e.  Q.  /\  A  e.  P. )  /\  y  e.  ( 2nd `  A ) )  ->  ( x  =  ( z  .Q  y
)  <->  x  =  (
y  .Q  z ) ) )
2524rexbidva 2484 . . . . . . . . 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 268 . . . . . . . 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 286 . . . . . . 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 400 . . . . . 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 2484 . . . . 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 2651 . . . . 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 196 . . . 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 2738 . . 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 3798 . 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 7552 . . 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 268 . 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 7552 . 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 2231 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  =  ( B  .P.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363    e. wcel 2158   E.wrex 2466   {crab 2469   <.cop 3607   ` cfv 5228  (class class class)co 5888   1stc1st 6153   2ndc2nd 6154   Q.cnq 7293    .Q cmq 7296   P.cnp 7304    .P. cmp 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-mi 7319  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-mqqs 7363  df-inp 7479  df-imp 7482
This theorem is referenced by:  ltmprr  7655  mulcmpblnrlemg  7753  mulcomsrg  7770  mulasssrg  7771  m1m1sr  7774  recexgt0sr  7786  mulgt0sr  7791  mulextsr1lem  7793  recidpirqlemcalc  7870
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