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Theorem genpdf 6491
Description: Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
Hypothesis
Ref Expression
genpdf.1  F  P. ,  P.  |->  <. { q  Q.  |  r  Q.  s  Q. 
r  1st `  s  1st `  q  r G s } ,  {
q  Q.  |  r  Q.  s  Q. 
r  2nd `  s  2nd `  q  r G s } >.
Assertion
Ref Expression
genpdf  F  P. ,  P.  |->  <. { q  Q.  |  r  1st `  s  1st `  q  r G s } ,  { q  Q.  |  r  2nd `  s  2nd `  q  r G s } >.
Distinct variable group:    r, q, s,,
Allowed substitution hints:    F(,, s, r, q)    G(,, s, r, q)

Proof of Theorem genpdf
StepHypRef Expression
1 genpdf.1 . 2  F  P. ,  P.  |->  <. { q  Q.  |  r  Q.  s  Q. 
r  1st `  s  1st `  q  r G s } ,  {
q  Q.  |  r  Q.  s  Q. 
r  2nd `  s  2nd `  q  r G s } >.
2 prop 6458 . . . . . . 7  P.  <. 1st `  ,  2nd `  >.  P.
3 elprnql 6464 . . . . . . 7 
<. 1st `  ,  2nd `  >.  P.  r  1st `  r  Q.
42, 3sylan 267 . . . . . 6  P.  r  1st `  r  Q.
54adantlr 446 . . . . 5  P.  P.  r  1st `  r  Q.
6 prop 6458 . . . . . . 7  P.  <. 1st `  ,  2nd `  >.  P.
7 elprnql 6464 . . . . . . 7 
<. 1st `  ,  2nd `  >.  P.  s  1st `  s  Q.
86, 7sylan 267 . . . . . 6  P.  s  1st `  s  Q.
98adantll 445 . . . . 5  P.  P.  s  1st `  s  Q.
105, 9genpdflem 6490 . . . 4  P.  P.  { q  Q.  |  r  Q.  s  Q. 
r  1st `  s  1st `  q  r G s }  {
q  Q.  |  r  1st `  s  1st `  q  r G s }
11 elprnqu 6465 . . . . . . 7 
<. 1st `  ,  2nd `  >.  P.  r  2nd `  r  Q.
122, 11sylan 267 . . . . . 6  P.  r  2nd `  r  Q.
1312adantlr 446 . . . . 5  P.  P.  r  2nd `  r  Q.
14 elprnqu 6465 . . . . . . 7 
<. 1st `  ,  2nd `  >.  P.  s  2nd `  s  Q.
156, 14sylan 267 . . . . . 6  P.  s  2nd `  s  Q.
1615adantll 445 . . . . 5  P.  P.  s  2nd `  s  Q.
1713, 16genpdflem 6490 . . . 4  P.  P.  { q  Q.  |  r  Q.  s  Q. 
r  2nd `  s  2nd `  q  r G s }  {
q  Q.  |  r  2nd `  s  2nd `  q  r G s }
1810, 17opeq12d 3548 . . 3  P.  P.  <. { q 
Q.  |  r  Q.  s 
Q.  r  1st `  s  1st `  q 
r G s } ,  {
q  Q.  |  r  Q.  s  Q. 
r  2nd `  s  2nd `  q  r G s } >.  <. { q  Q.  |  r  1st `  s  1st `  q  r G s } ,  {
q  Q.  |  r  2nd `  s  2nd `  q  r G s } >.
1918mpt2eq3ia 5512 . 2  P. ,  P.  |->  <. { q  Q.  |  r  Q.  s  Q.  r  1st `  s  1st `  q 
r G s } ,  {
q  Q.  |  r  Q.  s  Q. 
r  2nd `  s  2nd `  q  r G s } >.  P. ,  P.  |->  <. { q  Q.  |  r  1st `  s  1st `  q  r G s } ,  { q  Q.  |  r  2nd `  s  2nd `  q  r G s } >.
201, 19eqtri 2057 1  F  P. ,  P.  |->  <. { q  Q.  |  r  1st `  s  1st `  q  r G s } ,  { q  Q.  |  r  2nd `  s  2nd `  q  r G s } >.
Colors of variables: wff set class
Syntax hints:   wa 97   w3a 884   wceq 1242   wcel 1390  wrex 2301   {crab 2304   <.cop 3370   ` cfv 4845  (class class class)co 5455    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   P.cnp 6275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-qs 6048  df-ni 6288  df-nqqs 6332  df-inp 6449
This theorem is referenced by:  genipv  6492
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