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Theorem genpelxp 7473
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 3232 . . . . 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 7325 . . . . . 6  |-  Q.  e.  _V
32elpw2 4143 . . . . 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 3232 . . . . 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 4143 . . . . 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 4643 . . . 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 424 . . 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 5496 . . . . . . . . 9  |-  ( w  =  A  ->  ( 1st `  w )  =  ( 1st `  A
) )
1110eleq2d 2240 . . . . . . . 8  |-  ( w  =  A  ->  (
y  e.  ( 1st `  w )  <->  y  e.  ( 1st `  A ) ) )
12113anbi1d 1311 . . . . . . 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 2495 . . . . . 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 2719 . . . . 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 5496 . . . . . . . . 9  |-  ( w  =  A  ->  ( 2nd `  w )  =  ( 2nd `  A
) )
1615eleq2d 2240 . . . . . . . 8  |-  ( w  =  A  ->  (
y  e.  ( 2nd `  w )  <->  y  e.  ( 2nd `  A ) ) )
17163anbi1d 1311 . . . . . . 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 2495 . . . . . 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 2719 . . . . 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 3773 . . . 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 5496 . . . . . . . . 9  |-  ( v  =  B  ->  ( 1st `  v )  =  ( 1st `  B
) )
2221eleq2d 2240 . . . . . . . 8  |-  ( v  =  B  ->  (
z  e.  ( 1st `  v )  <->  z  e.  ( 1st `  B ) ) )
23223anbi2d 1312 . . . . . . 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 2495 . . . . . 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 2719 . . . . 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 5496 . . . . . . . . 9  |-  ( v  =  B  ->  ( 2nd `  v )  =  ( 2nd `  B
) )
2726eleq2d 2240 . . . . . . . 8  |-  ( v  =  B  ->  (
z  e.  ( 2nd `  v )  <->  z  e.  ( 2nd `  B ) ) )
28273anbi2d 1312 . . . . . . 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 2495 . . . . . 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 2719 . . . . 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 3773 . . . 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 5987 . . 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 1321 . 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 2261 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 973    = wceq 1348    e. wcel 2141   E.wrex 2449   {crab 2452    C_ wss 3121   ~Pcpw 3566   <.cop 3586    X. cxp 4609   ` cfv 5198  (class class class)co 5853    e. cmpo 5855   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   P.cnp 7253
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-qs 6519  df-ni 7266  df-nqqs 7310
This theorem is referenced by:  addclpr  7499  mulclpr  7534
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