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Theorem genpdf 7284
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 7251 . . . . . . 7  |-  ( w  e.  P.  ->  <. ( 1st `  w ) ,  ( 2nd `  w
) >.  e.  P. )
3 elprnql 7257 . . . . . . 7  |-  ( (
<. ( 1st `  w
) ,  ( 2nd `  w ) >.  e.  P.  /\  r  e.  ( 1st `  w ) )  -> 
r  e.  Q. )
42, 3sylan 281 . . . . . 6  |-  ( ( w  e.  P.  /\  r  e.  ( 1st `  w ) )  -> 
r  e.  Q. )
54adantlr 468 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  r  e.  ( 1st `  w ) )  ->  r  e.  Q. )
6 prop 7251 . . . . . . 7  |-  ( v  e.  P.  ->  <. ( 1st `  v ) ,  ( 2nd `  v
) >.  e.  P. )
7 elprnql 7257 . . . . . . 7  |-  ( (
<. ( 1st `  v
) ,  ( 2nd `  v ) >.  e.  P.  /\  s  e.  ( 1st `  v ) )  -> 
s  e.  Q. )
86, 7sylan 281 . . . . . 6  |-  ( ( v  e.  P.  /\  s  e.  ( 1st `  v ) )  -> 
s  e.  Q. )
98adantll 467 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  s  e.  ( 1st `  v ) )  ->  s  e.  Q. )
105, 9genpdflem 7283 . . . 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 7258 . . . . . . 7  |-  ( (
<. ( 1st `  w
) ,  ( 2nd `  w ) >.  e.  P.  /\  r  e.  ( 2nd `  w ) )  -> 
r  e.  Q. )
122, 11sylan 281 . . . . . 6  |-  ( ( w  e.  P.  /\  r  e.  ( 2nd `  w ) )  -> 
r  e.  Q. )
1312adantlr 468 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  r  e.  ( 2nd `  w ) )  ->  r  e.  Q. )
14 elprnqu 7258 . . . . . . 7  |-  ( (
<. ( 1st `  v
) ,  ( 2nd `  v ) >.  e.  P.  /\  s  e.  ( 2nd `  v ) )  -> 
s  e.  Q. )
156, 14sylan 281 . . . . . 6  |-  ( ( v  e.  P.  /\  s  e.  ( 2nd `  v ) )  -> 
s  e.  Q. )
1615adantll 467 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  s  e.  ( 2nd `  v ) )  ->  s  e.  Q. )
1713, 16genpdflem 7283 . . . 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 3683 . . 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 5804 . 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 2138 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 103    /\ w3a 947    = wceq 1316    e. wcel 1465   E.wrex 2394   {crab 2397   <.cop 3500   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056   P.cnp 7067
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-qs 6403  df-ni 7080  df-nqqs 7124  df-inp 7242
This theorem is referenced by:  genipv  7285
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