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Theorem genpdf 7046
Description: Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
Hypothesis
Ref Expression
genpdf.1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 1st `  w )  /\  s  e.  ( 1st `  v
)  /\  q  =  ( r G s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 2nd `  w )  /\  s  e.  ( 2nd `  v
)  /\  q  =  ( r G s ) ) } >. )
Assertion
Ref Expression
genpdf  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { q  e.  Q.  |  E. r  e.  ( 1st `  w ) E. s  e.  ( 1st `  v ) q  =  ( r G s ) } ,  { q  e. 
Q.  |  E. r  e.  ( 2nd `  w
) E. s  e.  ( 2nd `  v
) q  =  ( r G s ) } >. )
Distinct variable group:    r, q, s, v, w
Allowed substitution hints:    F( w, v, s, r, q)    G( w, v, s, r, q)

Proof of Theorem genpdf
StepHypRef Expression
1 genpdf.1 . 2  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 1st `  w )  /\  s  e.  ( 1st `  v
)  /\  q  =  ( r G s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 2nd `  w )  /\  s  e.  ( 2nd `  v
)  /\  q  =  ( r G s ) ) } >. )
2 prop 7013 . . . . . . 7  |-  ( w  e.  P.  ->  <. ( 1st `  w ) ,  ( 2nd `  w
) >.  e.  P. )
3 elprnql 7019 . . . . . . 7  |-  ( (
<. ( 1st `  w
) ,  ( 2nd `  w ) >.  e.  P.  /\  r  e.  ( 1st `  w ) )  -> 
r  e.  Q. )
42, 3sylan 277 . . . . . 6  |-  ( ( w  e.  P.  /\  r  e.  ( 1st `  w ) )  -> 
r  e.  Q. )
54adantlr 461 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  r  e.  ( 1st `  w ) )  ->  r  e.  Q. )
6 prop 7013 . . . . . . 7  |-  ( v  e.  P.  ->  <. ( 1st `  v ) ,  ( 2nd `  v
) >.  e.  P. )
7 elprnql 7019 . . . . . . 7  |-  ( (
<. ( 1st `  v
) ,  ( 2nd `  v ) >.  e.  P.  /\  s  e.  ( 1st `  v ) )  -> 
s  e.  Q. )
86, 7sylan 277 . . . . . 6  |-  ( ( v  e.  P.  /\  s  e.  ( 1st `  v ) )  -> 
s  e.  Q. )
98adantll 460 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  s  e.  ( 1st `  v ) )  ->  s  e.  Q. )
105, 9genpdflem 7045 . . . 4  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 1st `  w )  /\  s  e.  ( 1st `  v
)  /\  q  =  ( r G s ) ) }  =  { q  e.  Q.  |  E. r  e.  ( 1st `  w ) E. s  e.  ( 1st `  v ) q  =  ( r G s ) } )
11 elprnqu 7020 . . . . . . 7  |-  ( (
<. ( 1st `  w
) ,  ( 2nd `  w ) >.  e.  P.  /\  r  e.  ( 2nd `  w ) )  -> 
r  e.  Q. )
122, 11sylan 277 . . . . . 6  |-  ( ( w  e.  P.  /\  r  e.  ( 2nd `  w ) )  -> 
r  e.  Q. )
1312adantlr 461 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  r  e.  ( 2nd `  w ) )  ->  r  e.  Q. )
14 elprnqu 7020 . . . . . . 7  |-  ( (
<. ( 1st `  v
) ,  ( 2nd `  v ) >.  e.  P.  /\  s  e.  ( 2nd `  v ) )  -> 
s  e.  Q. )
156, 14sylan 277 . . . . . 6  |-  ( ( v  e.  P.  /\  s  e.  ( 2nd `  v ) )  -> 
s  e.  Q. )
1615adantll 460 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  s  e.  ( 2nd `  v ) )  ->  s  e.  Q. )
1713, 16genpdflem 7045 . . . 4  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 2nd `  w )  /\  s  e.  ( 2nd `  v
)  /\  q  =  ( r G s ) ) }  =  { q  e.  Q.  |  E. r  e.  ( 2nd `  w ) E. s  e.  ( 2nd `  v ) q  =  ( r G s ) } )
1810, 17opeq12d 3625 . . 3  |-  ( ( w  e.  P.  /\  v  e.  P. )  -> 
<. { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 1st `  w )  /\  s  e.  ( 1st `  v
)  /\  q  =  ( r G s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 2nd `  w )  /\  s  e.  ( 2nd `  v
)  /\  q  =  ( r G s ) ) } >.  = 
<. { q  e.  Q.  |  E. r  e.  ( 1st `  w ) E. s  e.  ( 1st `  v ) q  =  ( r G s ) } ,  { q  e. 
Q.  |  E. r  e.  ( 2nd `  w
) E. s  e.  ( 2nd `  v
) q  =  ( r G s ) } >. )
1918mpt2eq3ia 5696 . 2  |-  ( w  e.  P. ,  v  e.  P.  |->  <. { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  ( r  e.  ( 1st `  w
)  /\  s  e.  ( 1st `  v )  /\  q  =  ( r G s ) ) } ,  {
q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 2nd `  w )  /\  s  e.  ( 2nd `  v
)  /\  q  =  ( r G s ) ) } >. )  =  ( w  e. 
P. ,  v  e. 
P.  |->  <. { q  e. 
Q.  |  E. r  e.  ( 1st `  w
) E. s  e.  ( 1st `  v
) q  =  ( r G s ) } ,  { q  e.  Q.  |  E. r  e.  ( 2nd `  w ) E. s  e.  ( 2nd `  v
) q  =  ( r G s ) } >. )
201, 19eqtri 2108 1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { q  e.  Q.  |  E. r  e.  ( 1st `  w ) E. s  e.  ( 1st `  v ) q  =  ( r G s ) } ,  { q  e. 
Q.  |  E. r  e.  ( 2nd `  w
) E. s  e.  ( 2nd `  v
) q  =  ( r G s ) } >. )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360   {crab 2363   <.cop 3444   ` cfv 5002  (class class class)co 5634    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892   Q.cnq 6818   P.cnp 6829
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 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  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-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-qs 6278  df-ni 6842  df-nqqs 6886  df-inp 7004
This theorem is referenced by:  genipv  7047
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