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Theorem mulcomprg 6832
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 6727 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 6733 . . . . . . . . 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 6727 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 6733 . . . . . . . . . . . . 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 6635 . . . . . . . . . . . . 13  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( z  .Q  y
)  =  ( y  .Q  z ) )
87eqeq2d 2093 . . . . . . . . . . . 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 2366 . . . . . . . . 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 2366 . . . . 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 2519 . . . . 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 2594 . . 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 6734 . . . . . . . . 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 6734 . . . . . . . . . . . . 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 2366 . . . . . . . . 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 2366 . . . . 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 2519 . . . . 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 2594 . . 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 3586 . 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 6791 . . 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 6791 . 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 2125 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 1285    e. wcel 1434   E.wrex 2350   {crab 2353   <.cop 3409   ` cfv 4932  (class class class)co 5543   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532    .Q cmq 6535   P.cnp 6543    .P. cmp 6546
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-mi 6558  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-mqqs 6602  df-inp 6718  df-imp 6721
This theorem is referenced by:  ltmprr  6894  mulcmpblnrlemg  6979  mulcomsrg  6996  mulasssrg  6997  m1m1sr  7000  recexgt0sr  7012  mulgt0sr  7016  mulextsr1lem  7018  recidpirqlemcalc  7087
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