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Theorem genpelxp 7535
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 3255 . . . . 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 7387 . . . . . 6  |-  Q.  e.  _V
32elpw2 4172 . . . . 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 146 . . . 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 3255 . . . . 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 4172 . . . . 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 146 . . . 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 4673 . . . 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 426 . . 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 5531 . . . . . . . . 9  |-  ( w  =  A  ->  ( 1st `  w )  =  ( 1st `  A
) )
1110eleq2d 2259 . . . . . . . 8  |-  ( w  =  A  ->  (
y  e.  ( 1st `  w )  <->  y  e.  ( 1st `  A ) ) )
12113anbi1d 1327 . . . . . . 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 2515 . . . . . 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 2741 . . . . 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 5531 . . . . . . . . 9  |-  ( w  =  A  ->  ( 2nd `  w )  =  ( 2nd `  A
) )
1615eleq2d 2259 . . . . . . . 8  |-  ( w  =  A  ->  (
y  e.  ( 2nd `  w )  <->  y  e.  ( 2nd `  A ) ) )
17163anbi1d 1327 . . . . . . 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 2515 . . . . . 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 2741 . . . . 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 3801 . . . 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 5531 . . . . . . . . 9  |-  ( v  =  B  ->  ( 1st `  v )  =  ( 1st `  B
) )
2221eleq2d 2259 . . . . . . . 8  |-  ( v  =  B  ->  (
z  e.  ( 1st `  v )  <->  z  e.  ( 1st `  B ) ) )
23223anbi2d 1328 . . . . . . 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 2515 . . . . . 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 2741 . . . . 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 5531 . . . . . . . . 9  |-  ( v  =  B  ->  ( 2nd `  v )  =  ( 2nd `  B
) )
2726eleq2d 2259 . . . . . . . 8  |-  ( v  =  B  ->  (
z  e.  ( 2nd `  v )  <->  z  e.  ( 2nd `  B ) ) )
28273anbi2d 1328 . . . . . . 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 2515 . . . . . 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 2741 . . . . 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 3801 . . . 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 6027 . . 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 1337 . 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 2280 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 104    /\ w3a 980    = wceq 1364    e. wcel 2160   E.wrex 2469   {crab 2472    C_ wss 3144   ~Pcpw 3590   <.cop 3610    X. cxp 4639   ` cfv 5232  (class class class)co 5892    e. cmpo 5894   1stc1st 6158   2ndc2nd 6159   Q.cnq 7304   P.cnp 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-qs 6560  df-ni 7328  df-nqqs 7372
This theorem is referenced by:  addclpr  7561  mulclpr  7596
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