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Theorem genpelxp 7462
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 7314 . . . . . 6  |-  Q.  e.  _V
32elpw2 4141 . . . . 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 4141 . . . . 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 4641 . . . 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 5494 . . . . . . . . 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 5494 . . . . . . . . 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 3771 . . . 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 5494 . . . . . . . . 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 5494 . . . . . . . . 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 3771 . . . 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 5985 . . 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 3564   <.cop 3584    X. cxp 4607   ` cfv 5196  (class class class)co 5851    e. cmpo 5853   1stc1st 6115   2ndc2nd 6116   Q.cnq 7231   P.cnp 7242
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 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-qs 6516  df-ni 7255  df-nqqs 7299
This theorem is referenced by:  addclpr  7488  mulclpr  7523
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