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Theorem genpdf 6664
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 6631 . . . . . . 7  |-  ( w  e.  P.  ->  <. ( 1st `  w ) ,  ( 2nd `  w
) >.  e.  P. )
3 elprnql 6637 . . . . . . 7  |-  ( (
<. ( 1st `  w
) ,  ( 2nd `  w ) >.  e.  P.  /\  r  e.  ( 1st `  w ) )  -> 
r  e.  Q. )
42, 3sylan 271 . . . . . 6  |-  ( ( w  e.  P.  /\  r  e.  ( 1st `  w ) )  -> 
r  e.  Q. )
54adantlr 454 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  r  e.  ( 1st `  w ) )  ->  r  e.  Q. )
6 prop 6631 . . . . . . 7  |-  ( v  e.  P.  ->  <. ( 1st `  v ) ,  ( 2nd `  v
) >.  e.  P. )
7 elprnql 6637 . . . . . . 7  |-  ( (
<. ( 1st `  v
) ,  ( 2nd `  v ) >.  e.  P.  /\  s  e.  ( 1st `  v ) )  -> 
s  e.  Q. )
86, 7sylan 271 . . . . . 6  |-  ( ( v  e.  P.  /\  s  e.  ( 1st `  v ) )  -> 
s  e.  Q. )
98adantll 453 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  s  e.  ( 1st `  v ) )  ->  s  e.  Q. )
105, 9genpdflem 6663 . . . 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 6638 . . . . . . 7  |-  ( (
<. ( 1st `  w
) ,  ( 2nd `  w ) >.  e.  P.  /\  r  e.  ( 2nd `  w ) )  -> 
r  e.  Q. )
122, 11sylan 271 . . . . . 6  |-  ( ( w  e.  P.  /\  r  e.  ( 2nd `  w ) )  -> 
r  e.  Q. )
1312adantlr 454 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  r  e.  ( 2nd `  w ) )  ->  r  e.  Q. )
14 elprnqu 6638 . . . . . . 7  |-  ( (
<. ( 1st `  v
) ,  ( 2nd `  v ) >.  e.  P.  /\  s  e.  ( 2nd `  v ) )  -> 
s  e.  Q. )
156, 14sylan 271 . . . . . 6  |-  ( ( v  e.  P.  /\  s  e.  ( 2nd `  v ) )  -> 
s  e.  Q. )
1615adantll 453 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  s  e.  ( 2nd `  v ) )  ->  s  e.  Q. )
1713, 16genpdflem 6663 . . . 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 3585 . . 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 5598 . 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 2076 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 101    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   {crab 2327   <.cop 3406   ` cfv 4930  (class class class)co 5540    |-> cmpt2 5542   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436   P.cnp 6447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-qs 6143  df-ni 6460  df-nqqs 6504  df-inp 6622
This theorem is referenced by:  genipv  6665
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