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Theorem genpelxp 7343
Description: Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
Hypothesis
Ref Expression
genpelvl.1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
Assertion
Ref Expression
genpelxp  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  ( ~P Q.  X.  ~P Q. ) )
Distinct variable groups:    x, y, z, w, v, A    x, B, y, z, w, v   
x, G, y, z, w, v
Allowed substitution hints:    F( x, y, z, w, v)

Proof of Theorem genpelxp
StepHypRef Expression
1 ssrab2 3187 . . . . 5  |-  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  B )  /\  x  =  ( y G z ) ) }  C_  Q.
2 nqex 7195 . . . . . 6  |-  Q.  e.  _V
32elpw2 4090 . . . . 5  |-  ( { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  B
)  /\  x  =  ( y G z ) ) }  e.  ~P Q.  <->  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  B
)  /\  x  =  ( y G z ) ) }  C_  Q. )
41, 3mpbir 145 . . . 4  |-  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  B )  /\  x  =  ( y G z ) ) }  e.  ~P Q.
5 ssrab2 3187 . . . . 5  |-  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  B )  /\  x  =  ( y G z ) ) }  C_  Q.
62elpw2 4090 . . . . 5  |-  ( { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  B
)  /\  x  =  ( y G z ) ) }  e.  ~P Q.  <->  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  B
)  /\  x  =  ( y G z ) ) }  C_  Q. )
75, 6mpbir 145 . . . 4  |-  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  B )  /\  x  =  ( y G z ) ) }  e.  ~P Q.
8 opelxpi 4579 . . . 4  |-  ( ( { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  B
)  /\  x  =  ( y G z ) ) }  e.  ~P Q.  /\  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  B )  /\  x  =  ( y G z ) ) }  e.  ~P Q. )  ->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  B )  /\  x  =  ( y G z ) ) } ,  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  B
)  /\  x  =  ( y G z ) ) } >.  e.  ( ~P Q.  X.  ~P Q. ) )
94, 7, 8mp2an 423 . . 3  |-  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  B )  /\  x  =  ( y G z ) ) } ,  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  B
)  /\  x  =  ( y G z ) ) } >.  e.  ( ~P Q.  X.  ~P Q. )
10 fveq2 5429 . . . . . . . . 9  |-  ( w  =  A  ->  ( 1st `  w )  =  ( 1st `  A
) )
1110eleq2d 2210 . . . . . . . 8  |-  ( w  =  A  ->  (
y  e.  ( 1st `  w )  <->  y  e.  ( 1st `  A ) ) )
12113anbi1d 1295 . . . . . . 7  |-  ( w  =  A  ->  (
( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) )  <->  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) ) )
13122rexbidv 2463 . . . . . 6  |-  ( w  =  A  ->  ( E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) )  <->  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) ) )
1413rabbidv 2678 . . . . 5  |-  ( w  =  A  ->  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  w
)  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) }  =  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } )
15 fveq2 5429 . . . . . . . . 9  |-  ( w  =  A  ->  ( 2nd `  w )  =  ( 2nd `  A
) )
1615eleq2d 2210 . . . . . . . 8  |-  ( w  =  A  ->  (
y  e.  ( 2nd `  w )  <->  y  e.  ( 2nd `  A ) ) )
17163anbi1d 1295 . . . . . . 7  |-  ( w  =  A  ->  (
( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) )  <->  ( y  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) ) )
18172rexbidv 2463 . . . . . 6  |-  ( w  =  A  ->  ( E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) )  <->  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) ) )
1918rabbidv 2678 . . . . 5  |-  ( w  =  A  ->  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 2nd `  w
)  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) }  =  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } )
2014, 19opeq12d 3721 . . . 4  |-  ( w  =  A  ->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  w
)  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >.  = 
<. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
21 fveq2 5429 . . . . . . . . 9  |-  ( v  =  B  ->  ( 1st `  v )  =  ( 1st `  B
) )
2221eleq2d 2210 . . . . . . . 8  |-  ( v  =  B  ->  (
z  e.  ( 1st `  v )  <->  z  e.  ( 1st `  B ) ) )
23223anbi2d 1296 . . . . . . 7  |-  ( v  =  B  ->  (
( y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) )  <->  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  B )  /\  x  =  ( y G z ) ) ) )
24232rexbidv 2463 . . . . . 6  |-  ( v  =  B  ->  ( E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) )  <->  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  B )  /\  x  =  ( y G z ) ) ) )
2524rabbidv 2678 . . . . 5  |-  ( v  =  B  ->  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) }  =  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  B
)  /\  x  =  ( y G z ) ) } )
26 fveq2 5429 . . . . . . . . 9  |-  ( v  =  B  ->  ( 2nd `  v )  =  ( 2nd `  B
) )
2726eleq2d 2210 . . . . . . . 8  |-  ( v  =  B  ->  (
z  e.  ( 2nd `  v )  <->  z  e.  ( 2nd `  B ) ) )
28273anbi2d 1296 . . . . . . 7  |-  ( v  =  B  ->  (
( y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) )  <->  ( y  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  B )  /\  x  =  ( y G z ) ) ) )
29282rexbidv 2463 . . . . . 6  |-  ( v  =  B  ->  ( E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) )  <->  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  B )  /\  x  =  ( y G z ) ) ) )
3029rabbidv 2678 . . . . 5  |-  ( v  =  B  ->  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) }  =  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  B
)  /\  x  =  ( y G z ) ) } )
3125, 30opeq12d 3721 . . . 4  |-  ( v  =  B  ->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >.  = 
<. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  B
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  B
)  /\  x  =  ( y G z ) ) } >. )
32 genpelvl.1 . . . 4  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
3320, 31, 32ovmpog 5913 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  A )  /\  z  e.  ( 1st `  B
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  B
)  /\  x  =  ( y G z ) ) } >.  e.  ( ~P Q.  X.  ~P Q. ) )  -> 
( A F B )  =  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  B )  /\  x  =  ( y G z ) ) } ,  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  B
)  /\  x  =  ( y G z ) ) } >. )
349, 33mp3an3 1305 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  =  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
Q.  ( y  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  B )  /\  x  =  ( y G z ) ) } ,  {
x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  B
)  /\  x  =  ( y G z ) ) } >. )
3534, 9eqeltrdi 2231 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  ( ~P Q.  X.  ~P Q. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1332    e. wcel 1481   E.wrex 2418   {crab 2421    C_ wss 3076   ~Pcpw 3515   <.cop 3535    X. cxp 4545   ` cfv 5131  (class class class)co 5782    e. cmpo 5784   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   P.cnp 7123
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-qs 6443  df-ni 7136  df-nqqs 7180
This theorem is referenced by:  addclpr  7369  mulclpr  7404
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