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Theorem genpelxp 6494
Description: Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
Hypothesis
Ref Expression
genpelvl.1  F  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
Assertion
Ref Expression
genpelxp  P.  P.  F  ~P Q.  X.  ~P Q.
Distinct variable groups:   ,,,,,   ,,,,,   , G,,,,
Allowed substitution hints:    F(,,,,)

Proof of Theorem genpelxp
StepHypRef Expression
1 ssrab2 3019 . . . . 5  {  Q.  |  Q. 
Q.  1st `  1st `  G }  C_  Q.
2 nqex 6347 . . . . . 6  Q.  _V
32elpw2 3902 . . . . 5  {  Q.  |  Q.  Q.  1st `  1st `  G }  ~P Q.  {  Q.  |  Q.  Q.  1st `  1st `  G }  C_  Q.
41, 3mpbir 134 . . . 4  {  Q.  |  Q. 
Q.  1st `  1st `  G }  ~P Q.
5 ssrab2 3019 . . . . 5  {  Q.  |  Q. 
Q.  2nd `  2nd `  G }  C_  Q.
62elpw2 3902 . . . . 5  {  Q.  |  Q.  Q.  2nd `  2nd `  G }  ~P Q.  {  Q.  |  Q.  Q.  2nd `  2nd `  G }  C_  Q.
75, 6mpbir 134 . . . 4  {  Q.  |  Q. 
Q.  2nd `  2nd `  G }  ~P Q.
8 opelxpi 4319 . . . 4  {  Q.  |  Q.  Q.  1st `  1st `  G }  ~P Q.  { 
Q.  |  Q. 
Q.  2nd `  2nd `  G }  ~P Q.  <. { 
Q.  |  Q. 
Q.  1st `  1st `  G } ,  {  Q.  |  Q. 
Q.  2nd `  2nd `  G } >.  ~P Q.  X.  ~P Q.
94, 7, 8mp2an 402 . . 3  <. {  Q.  |  Q. 
Q.  1st `  1st `  G } ,  {  Q.  |  Q. 
Q.  2nd `  2nd `  G } >.  ~P Q.  X.  ~P Q.
10 fveq2 5121 . . . . . . . . 9  1st `  1st `
1110eleq2d 2104 . . . . . . . 8  1st `  1st `
12113anbi1d 1210 . . . . . . 7  1st `  1st `  G  1st `  1st `  G
13122rexbidv 2343 . . . . . 6  Q.  Q.  1st `  1st `  G 
Q.  Q.  1st `  1st `  G
1413rabbidv 2543 . . . . 5  {  Q.  |  Q.  Q.  1st `  1st `  G }  {  Q.  |  Q. 
Q.  1st `  1st `  G }
15 fveq2 5121 . . . . . . . . 9  2nd `  2nd `
1615eleq2d 2104 . . . . . . . 8  2nd `  2nd `
17163anbi1d 1210 . . . . . . 7  2nd `  2nd `  G  2nd `  2nd `  G
18172rexbidv 2343 . . . . . 6  Q.  Q.  2nd `  2nd `  G 
Q.  Q.  2nd `  2nd `  G
1918rabbidv 2543 . . . . 5  {  Q.  |  Q.  Q.  2nd `  2nd `  G }  {  Q.  |  Q. 
Q.  2nd `  2nd `  G }
2014, 19opeq12d 3548 . . . 4  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
21 fveq2 5121 . . . . . . . . 9  1st `  1st `
2221eleq2d 2104 . . . . . . . 8  1st `  1st `
23223anbi2d 1211 . . . . . . 7  1st `  1st `  G  1st `  1st `  G
24232rexbidv 2343 . . . . . 6  Q.  Q.  1st `  1st `  G 
Q.  Q.  1st `  1st `  G
2524rabbidv 2543 . . . . 5  {  Q.  |  Q.  Q.  1st `  1st `  G }  {  Q.  |  Q. 
Q.  1st `  1st `  G }
26 fveq2 5121 . . . . . . . . 9  2nd `  2nd `
2726eleq2d 2104 . . . . . . . 8  2nd `  2nd `
28273anbi2d 1211 . . . . . . 7  2nd `  2nd `  G  2nd `  2nd `  G
29282rexbidv 2343 . . . . . 6  Q.  Q.  2nd `  2nd `  G 
Q.  Q.  2nd `  2nd `  G
3029rabbidv 2543 . . . . 5  {  Q.  |  Q.  Q.  2nd `  2nd `  G }  {  Q.  |  Q. 
Q.  2nd `  2nd `  G }
3125, 30opeq12d 3548 . . . 4  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
32 genpelvl.1 . . . 4  F  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
3320, 31, 32ovmpt2g 5577 . . 3  P.  P.  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.  ~P Q.  X.  ~P Q.  F 
<. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
349, 33mp3an3 1220 . 2  P.  P.  F  <. {  Q.  |  Q. 
Q.  1st `  1st `  G } ,  {  Q.  |  Q. 
Q.  2nd `  2nd `  G } >.
3534, 9syl6eqel 2125 1  P.  P.  F  ~P Q.  X.  ~P Q.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390  wrex 2301   {crab 2304    C_ wss 2911   ~Pcpw 3351   <.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-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-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-qs 6048  df-ni 6288  df-nqqs 6332
This theorem is referenced by:  addclpr  6520  mulclpr  6553
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