ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mulcomprg Unicode version

Theorem mulcomprg 7542
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 7437 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 elprnql 7443 . . . . . . . . 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 7437 . . . . . . . . . . . . 13  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7443 . . . . . . . . . . . . 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 7345 . . . . . . . . . . . . 13  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( z  .Q  y
)  =  ( y  .Q  z ) )
87eqeq2d 2182 . . . . . . . . . . . 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 2467 . . . . . . . . 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 2467 . . . . 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 2634 . . . . 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 2719 . . 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 7444 . . . . . . . . 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 7444 . . . . . . . . . . . . 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 2467 . . . . . . . . 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 2467 . . . . 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 2634 . . . . 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 2719 . . 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 3773 . 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 7501 . . 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 7501 . 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 2214 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 1348    e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3586   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242    .Q cmq 7245   P.cnp 7253    .P. cmp 7256
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-mi 7268  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-mqqs 7312  df-inp 7428  df-imp 7431
This theorem is referenced by:  ltmprr  7604  mulcmpblnrlemg  7702  mulcomsrg  7719  mulasssrg  7720  m1m1sr  7723  recexgt0sr  7735  mulgt0sr  7740  mulextsr1lem  7742  recidpirqlemcalc  7819
  Copyright terms: Public domain W3C validator