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Theorem genipv 6491
Description: Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
Hypotheses
Ref Expression
genp.1  F  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
genp.2  Q.  Q.  G  Q.
Assertion
Ref Expression
genipv  P.  P.  F  <. { q  Q.  |  r  1st `  s  1st `  q 
r G s } ,  { q  Q.  |  r  2nd `  s  2nd `  q 
r G s } >.
Distinct variable groups:   ,,, q, r, s,   ,,,, q, r, s   ,,, G,,, q, r, s
Allowed substitution hints:   (,)   (,)    F(,,,,, s, r, q)

Proof of Theorem genipv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5462 . . . 4  F  F
2 fveq2 5121 . . . . . . 7  1st `  1st `
32rexeqdv 2506 . . . . . 6  1st `  1st `  G  1st `  1st `  G
43rabbidv 2543 . . . . 5  {  Q.  |  1st `  1st `  G }  {  Q.  |  1st `  1st `  G }
5 fveq2 5121 . . . . . . 7  2nd `  2nd `
65rexeqdv 2506 . . . . . 6  2nd `  2nd `  G  2nd `  2nd `  G
76rabbidv 2543 . . . . 5  {  Q.  |  2nd `  2nd `  G }  {  Q.  |  2nd `  2nd `  G }
84, 7opeq12d 3548 . . . 4  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.  <. { 
Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
91, 8eqeq12d 2051 . . 3  F  <. { 
Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.  F  <. { 
Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
10 oveq2 5463 . . . 4  F  F
11 fveq2 5121 . . . . . . . 8  1st `  1st `
1211rexeqdv 2506 . . . . . . 7  1st `  G  1st `  G
1312rexbidv 2321 . . . . . 6  1st `  1st `  G  1st `  1st `  G
1413rabbidv 2543 . . . . 5  {  Q.  |  1st `  1st `  G }  {  Q.  |  1st `  1st `  G }
15 fveq2 5121 . . . . . . . 8  2nd `  2nd `
1615rexeqdv 2506 . . . . . . 7  2nd `  G  2nd `  G
1716rexbidv 2321 . . . . . 6  2nd `  2nd `  G  2nd `  2nd `  G
1817rabbidv 2543 . . . . 5  {  Q.  |  2nd `  2nd `  G }  {  Q.  |  2nd `  2nd `  G }
1914, 18opeq12d 3548 . . . 4  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.  <. { 
Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
2010, 19eqeq12d 2051 . . 3  F  <. { 
Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.  F  <. { 
Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
21 nqex 6347 . . . . . . 7  Q.  _V
2221a1i 9 . . . . . 6  P.  P.  Q.  _V
23 rabssab 3021 . . . . . . 7  {  Q.  |  1st `  1st `  G }  C_ 
{  |  1st `  1st `  G }
24 prop 6457 . . . . . . . . . . . 12  P.  <. 1st `  ,  2nd `  >.  P.
25 elprnql 6463 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
2624, 25sylan 267 . . . . . . . . . . 11  P.  1st `  Q.
27 prop 6457 . . . . . . . . . . . 12  P.  <. 1st `  ,  2nd `  >.  P.
28 elprnql 6463 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
2927, 28sylan 267 . . . . . . . . . . 11  P.  1st `  Q.
30 genp.2 . . . . . . . . . . . 12  Q.  Q.  G  Q.
31 eleq1 2097 . . . . . . . . . . . 12  G  Q.  G 
Q.
3230, 31syl5ibrcom 146 . . . . . . . . . . 11  Q.  Q.  G  Q.
3326, 29, 32syl2an 273 . . . . . . . . . 10  P.  1st `  P.  1st `  G  Q.
3433an4s 522 . . . . . . . . 9  P.  P.  1st `  1st `  G  Q.
3534rexlimdvva 2434 . . . . . . . 8  P.  P.  1st `  1st `  G  Q.
3635abssdv 3008 . . . . . . 7  P.  P.  {  |  1st `  1st `  G }  C_ 
Q.
3723, 36syl5ss 2950 . . . . . 6  P.  P.  {  Q.  |  1st `  1st `  G }  C_  Q.
3822, 37ssexd 3888 . . . . 5  P.  P.  {  Q.  |  1st `  1st `  G }  _V
39 rabssab 3021 . . . . . . 7  {  Q.  |  2nd `  2nd `  G }  C_ 
{  |  2nd `  2nd `  G }
40 elprnqu 6464 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  2nd `  Q.
4124, 40sylan 267 . . . . . . . . . . 11  P.  2nd `  Q.
42 elprnqu 6464 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  2nd `  Q.
4327, 42sylan 267 . . . . . . . . . . 11  P.  2nd `  Q.
4441, 43, 32syl2an 273 . . . . . . . . . 10  P.  2nd `  P.  2nd `  G  Q.
4544an4s 522 . . . . . . . . 9  P.  P.  2nd `  2nd `  G  Q.
4645rexlimdvva 2434 . . . . . . . 8  P.  P.  2nd `  2nd `  G  Q.
4746abssdv 3008 . . . . . . 7  P.  P.  {  |  2nd `  2nd `  G }  C_ 
Q.
4839, 47syl5ss 2950 . . . . . 6  P.  P.  {  Q.  |  2nd `  2nd `  G }  C_  Q.
4922, 48ssexd 3888 . . . . 5  P.  P.  {  Q.  |  2nd `  2nd `  G }  _V
50 opelxp 4317 . . . . 5  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.  _V  X.  _V  {  Q.  |  1st `  1st `  G }  _V  {  Q.  |  2nd `  2nd `  G }  _V
5138, 49, 50sylanbrc 394 . . . 4  P.  P.  <. { 
Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.  _V  X.  _V
52 fveq2 5121 . . . . . . . 8  1st `  1st `
5352rexeqdv 2506 . . . . . . 7  1st `  1st `  G  1st `  1st `  G
5453rabbidv 2543 . . . . . 6  {  Q.  |  1st `  1st `  G }  {  Q.  |  1st `  1st `  G }
55 fveq2 5121 . . . . . . . 8  2nd `  2nd `
5655rexeqdv 2506 . . . . . . 7  2nd `  2nd `  G  2nd `  2nd `  G
5756rabbidv 2543 . . . . . 6  {  Q.  |  2nd `  2nd `  G }  {  Q.  |  2nd `  2nd `  G }
5854, 57opeq12d 3548 . . . . 5  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.  <. { 
Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
59 fveq2 5121 . . . . . . . . 9  1st `  1st `
6059rexeqdv 2506 . . . . . . . 8  1st `  G  1st `  G
6160rexbidv 2321 . . . . . . 7  1st `  1st `  G  1st `  1st `  G
6261rabbidv 2543 . . . . . 6  {  Q.  |  1st `  1st `  G }  {  Q.  |  1st `  1st `  G }
63 fveq2 5121 . . . . . . . . 9  2nd `  2nd `
6463rexeqdv 2506 . . . . . . . 8  2nd `  G  2nd `  G
6564rexbidv 2321 . . . . . . 7  2nd `  2nd `  G  2nd `  2nd `  G
6665rabbidv 2543 . . . . . 6  {  Q.  |  2nd `  2nd `  G }  {  Q.  |  2nd `  2nd `  G }
6762, 66opeq12d 3548 . . . . 5  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.  <. { 
Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
68 genp.1 . . . . . 6  F  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
6968genpdf 6490 . . . . 5  F  P. ,  P.  |->  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
7058, 67, 69ovmpt2g 5577 . . . 4  P.  P.  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.  _V  X.  _V  F 
<. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
7151, 70mpd3an3 1232 . . 3  P.  P.  F  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
729, 20, 71vtocl2ga 2615 . 2  P.  P.  F  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >.
73 eqeq1 2043 . . . . . 6  q  G  q  G
74732rexbidv 2343 . . . . 5  q  1st `  1st `  G  1st `  1st `  q  G
75 oveq1 5462 . . . . . . 7  r  G  r G
7675eqeq2d 2048 . . . . . 6  r 
q  G  q  r G
77 oveq2 5463 . . . . . . 7  s 
r G  r G s
7877eqeq2d 2048 . . . . . 6  s 
q  r G  q  r G s
7976, 78cbvrex2v 2536 . . . . 5  1st `  1st `  q  G  r  1st `  s  1st `  q  r G s
8074, 79syl6bb 185 . . . 4  q  1st `  1st `  G  r  1st `  s  1st `  q  r G s
8180cbvrabv 2550 . . 3  {  Q.  |  1st `  1st `  G }  { q  Q.  |  r  1st `  s  1st `  q  r G s }
82732rexbidv 2343 . . . . 5  q  2nd `  2nd `  G  2nd `  2nd `  q  G
8376, 78cbvrex2v 2536 . . . . 5  2nd `  2nd `  q  G  r  2nd `  s  2nd `  q  r G s
8482, 83syl6bb 185 . . . 4  q  2nd `  2nd `  G  r  2nd `  s  2nd `  q  r G s
8584cbvrabv 2550 . . 3  {  Q.  |  2nd `  2nd `  G }  { q  Q.  |  r  2nd `  s  2nd `  q  r G s }
8681, 85opeq12i 3545 . 2  <. {  Q.  |  1st `  1st `  G } ,  {  Q.  |  2nd `  2nd `  G } >. 
<. { q  Q.  |  r  1st `  s  1st `  q  r G s } ,  { q  Q.  |  r  2nd `  s  2nd `  q  r G s } >.
8772, 86syl6eq 2085 1  P.  P.  F  <. { 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:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390   {cab 2023  wrex 2301   {crab 2304   _Vcvv 2551   <.cop 3370    X. cxp 4286   ` 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-bnd 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-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  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-ne 2203  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-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-qs 6048  df-ni 6288  df-nqqs 6332  df-inp 6448
This theorem is referenced by:  genpelvl  6494  genpelvu  6495  plpvlu  6520  mpvlu  6521
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