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Theorem genpdf 7504
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 7471 . . . . . . 7  |-  ( w  e.  P.  ->  <. ( 1st `  w ) ,  ( 2nd `  w
) >.  e.  P. )
3 elprnql 7477 . . . . . . 7  |-  ( (
<. ( 1st `  w
) ,  ( 2nd `  w ) >.  e.  P.  /\  r  e.  ( 1st `  w ) )  -> 
r  e.  Q. )
42, 3sylan 283 . . . . . 6  |-  ( ( w  e.  P.  /\  r  e.  ( 1st `  w ) )  -> 
r  e.  Q. )
54adantlr 477 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  r  e.  ( 1st `  w ) )  ->  r  e.  Q. )
6 prop 7471 . . . . . . 7  |-  ( v  e.  P.  ->  <. ( 1st `  v ) ,  ( 2nd `  v
) >.  e.  P. )
7 elprnql 7477 . . . . . . 7  |-  ( (
<. ( 1st `  v
) ,  ( 2nd `  v ) >.  e.  P.  /\  s  e.  ( 1st `  v ) )  -> 
s  e.  Q. )
86, 7sylan 283 . . . . . 6  |-  ( ( v  e.  P.  /\  s  e.  ( 1st `  v ) )  -> 
s  e.  Q. )
98adantll 476 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  s  e.  ( 1st `  v ) )  ->  s  e.  Q. )
105, 9genpdflem 7503 . . . 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 7478 . . . . . . 7  |-  ( (
<. ( 1st `  w
) ,  ( 2nd `  w ) >.  e.  P.  /\  r  e.  ( 2nd `  w ) )  -> 
r  e.  Q. )
122, 11sylan 283 . . . . . 6  |-  ( ( w  e.  P.  /\  r  e.  ( 2nd `  w ) )  -> 
r  e.  Q. )
1312adantlr 477 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  r  e.  ( 2nd `  w ) )  ->  r  e.  Q. )
14 elprnqu 7478 . . . . . . 7  |-  ( (
<. ( 1st `  v
) ,  ( 2nd `  v ) >.  e.  P.  /\  s  e.  ( 2nd `  v ) )  -> 
s  e.  Q. )
156, 14sylan 283 . . . . . 6  |-  ( ( v  e.  P.  /\  s  e.  ( 2nd `  v ) )  -> 
s  e.  Q. )
1615adantll 476 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  s  e.  ( 2nd `  v ) )  ->  s  e.  Q. )
1713, 16genpdflem 7503 . . . 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 3786 . . 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 ) } >. )
1918mpoeq3ia 5937 . 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 2198 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3595   ` cfv 5215  (class class class)co 5872    e. cmpo 5874   1stc1st 6136   2ndc2nd 6137   Q.cnq 7276   P.cnp 7287
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-iinf 4586
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-qs 6538  df-ni 7300  df-nqqs 7344  df-inp 7462
This theorem is referenced by:  genipv  7505
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