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Theorem genpcuu 7296
Description: Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)
Hypotheses
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 ) ) } >. )
genpelvl.2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
genpcuu.2  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B
) ) )  /\  x  e.  Q. )  ->  ( ( g G h )  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) )
Assertion
Ref Expression
genpcuu  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( 2nd `  ( A F B ) )  ->  ( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
Distinct variable groups:    x, y, z, f, g, h, w, v, A    x, B, y, z, f, g, h, w, v    x, G, y, z, f, g, h, w, v    f, F, g, h
Allowed substitution hints:    F( x, y, z, w, v)

Proof of Theorem genpcuu
StepHypRef Expression
1 ltrelnq 7141 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4561 . . . . . 6  |-  ( f 
<Q  x  ->  ( f  e.  Q.  /\  x  e.  Q. ) )
32simprd 113 . . . . 5  |-  ( f 
<Q  x  ->  x  e. 
Q. )
4 genpelvl.1 . . . . . . . . 9  |-  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 ) ) } >. )
5 genpelvl.2 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
64, 5genpelvu 7289 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( 2nd `  ( A F B ) )  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) f  =  ( g G h ) ) )
76adantr 274 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( f  e.  ( 2nd `  ( A F B ) )  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) f  =  ( g G h ) ) )
8 breq1 3902 . . . . . . . . . . . . 13  |-  ( f  =  ( g G h )  ->  (
f  <Q  x  <->  ( g G h )  <Q  x ) )
98biimpd 143 . . . . . . . . . . . 12  |-  ( f  =  ( g G h )  ->  (
f  <Q  x  ->  (
g G h ) 
<Q  x ) )
10 genpcuu.2 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B
) ) )  /\  x  e.  Q. )  ->  ( ( g G h )  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) )
119, 10sylan9r 407 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  P.  /\  g  e.  ( 2nd `  A
) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B
) ) )  /\  x  e.  Q. )  /\  f  =  (
g G h ) )  ->  ( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) )
1211exp31 361 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  g  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B ) ) )  ->  ( x  e. 
Q.  ->  ( f  =  ( g G h )  ->  ( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) ) )
1312an4s 562 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 2nd `  A )  /\  h  e.  ( 2nd `  B ) ) )  ->  (
x  e.  Q.  ->  ( f  =  ( g G h )  -> 
( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) ) )
1413impancom 258 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( ( g  e.  ( 2nd `  A
)  /\  h  e.  ( 2nd `  B ) )  ->  ( f  =  ( g G h )  ->  (
f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) ) )
1514rexlimdvv 2533 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( E. g  e.  ( 2nd `  A
) E. h  e.  ( 2nd `  B
) f  =  ( g G h )  ->  ( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
167, 15sylbid 149 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( f  e.  ( 2nd `  ( A F B ) )  ->  ( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
1716ex 114 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  Q.  ->  ( f  e.  ( 2nd `  ( A F B ) )  ->  ( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) ) )
183, 17syl5 32 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  <Q  x  ->  ( f  e.  ( 2nd `  ( A F B ) )  ->  ( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) ) )
1918com34 83 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  <Q  x  ->  ( f  <Q  x  ->  ( f  e.  ( 2nd `  ( A F B ) )  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) ) )
2019pm2.43d 50 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  <Q  x  ->  ( f  e.  ( 2nd `  ( A F B ) )  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
2120com23 78 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( 2nd `  ( A F B ) )  ->  ( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    <Q cltq 7061   P.cnp 7067
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-qs 6403  df-ni 7080  df-nqqs 7124  df-ltnqqs 7129  df-inp 7242
This theorem is referenced by:  genprndu  7298
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