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Theorem genpelxp 7264
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 3148 . . . . 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 7116 . . . . . 6  |-  Q.  e.  _V
32elpw2 4042 . . . . 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 3148 . . . . 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 4042 . . . . 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 4531 . . . 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 420 . . 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 5375 . . . . . . . . 9  |-  ( w  =  A  ->  ( 1st `  w )  =  ( 1st `  A
) )
1110eleq2d 2184 . . . . . . . 8  |-  ( w  =  A  ->  (
y  e.  ( 1st `  w )  <->  y  e.  ( 1st `  A ) ) )
12113anbi1d 1277 . . . . . . 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 2434 . . . . . 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 2646 . . . . 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 5375 . . . . . . . . 9  |-  ( w  =  A  ->  ( 2nd `  w )  =  ( 2nd `  A
) )
1615eleq2d 2184 . . . . . . . 8  |-  ( w  =  A  ->  (
y  e.  ( 2nd `  w )  <->  y  e.  ( 2nd `  A ) ) )
17163anbi1d 1277 . . . . . . 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 2434 . . . . . 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 2646 . . . . 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 3679 . . . 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 5375 . . . . . . . . 9  |-  ( v  =  B  ->  ( 1st `  v )  =  ( 1st `  B
) )
2221eleq2d 2184 . . . . . . . 8  |-  ( v  =  B  ->  (
z  e.  ( 1st `  v )  <->  z  e.  ( 1st `  B ) ) )
23223anbi2d 1278 . . . . . . 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 2434 . . . . . 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 2646 . . . . 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 5375 . . . . . . . . 9  |-  ( v  =  B  ->  ( 2nd `  v )  =  ( 2nd `  B
) )
2726eleq2d 2184 . . . . . . . 8  |-  ( v  =  B  ->  (
z  e.  ( 2nd `  v )  <->  z  e.  ( 2nd `  B ) ) )
28273anbi2d 1278 . . . . . . 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 2434 . . . . . 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 2646 . . . . 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 3679 . . . 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 5859 . . 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 1287 . 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, 9syl6eqel 2205 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 945    = wceq 1314    e. wcel 1463   E.wrex 2391   {crab 2394    C_ wss 3037   ~Pcpw 3476   <.cop 3496    X. cxp 4497   ` cfv 5081  (class class class)co 5728    e. cmpo 5730   1stc1st 5990   2ndc2nd 5991   Q.cnq 7033   P.cnp 7044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-qs 6389  df-ni 7057  df-nqqs 7101
This theorem is referenced by:  addclpr  7290  mulclpr  7325
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