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Theorem genpcdl 7291
Description: Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-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. )
genpcdl.2  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B
) ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g G h )  ->  x  e.  ( 1st `  ( A F B ) ) ) )
Assertion
Ref Expression
genpcdl  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( 1st `  ( A F B ) )  ->  ( x  <Q  f  ->  x  e.  ( 1st `  ( 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 genpcdl
StepHypRef Expression
1 ltrelnq 7137 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4559 . . . . . 6  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
32simpld 111 . . . . 5  |-  ( x 
<Q  f  ->  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, 5genpelvl 7284 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( 1st `  ( A F B ) )  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B ) f  =  ( g G h ) ) )
76adantr 272 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( f  e.  ( 1st `  ( A F B ) )  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B ) f  =  ( g G h ) ) )
8 breq2 3901 . . . . . . . . . . . . 13  |-  ( f  =  ( g G h )  ->  (
x  <Q  f  <->  x  <Q  ( g G h ) ) )
98biimpd 143 . . . . . . . . . . . 12  |-  ( f  =  ( g G h )  ->  (
x  <Q  f  ->  x  <Q  ( g G h ) ) )
10 genpcdl.2 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B
) ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g G h )  ->  x  e.  ( 1st `  ( A F B ) ) ) )
119, 10sylan9r 405 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  P.  /\  g  e.  ( 1st `  A
) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B
) ) )  /\  x  e.  Q. )  /\  f  =  (
g G h ) )  ->  ( x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) )
1211exp31 359 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B ) ) )  ->  ( x  e. 
Q.  ->  ( f  =  ( g G h )  ->  ( x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) ) ) )
1312an4s 560 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( g  e.  ( 1st `  A )  /\  h  e.  ( 1st `  B ) ) )  ->  (
x  e.  Q.  ->  ( f  =  ( g G h )  -> 
( x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) ) ) )
1413impancom 258 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( ( g  e.  ( 1st `  A
)  /\  h  e.  ( 1st `  B ) )  ->  ( f  =  ( g G h )  ->  (
x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) ) ) )
1514rexlimdvv 2531 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( E. g  e.  ( 1st `  A
) E. h  e.  ( 1st `  B
) f  =  ( g G h )  ->  ( x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) ) )
167, 15sylbid 149 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  x  e.  Q. )  ->  ( f  e.  ( 1st `  ( A F B ) )  ->  ( x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) ) )
1716ex 114 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  Q.  ->  ( f  e.  ( 1st `  ( A F B ) )  ->  ( x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) ) ) )
183, 17syl5 32 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  <Q  f  ->  ( f  e.  ( 1st `  ( A F B ) )  ->  ( x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) ) ) )
1918com34 83 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  <Q  f  ->  ( x  <Q  f  ->  ( f  e.  ( 1st `  ( A F B ) )  ->  x  e.  ( 1st `  ( A F B ) ) ) ) ) )
2019pm2.43d 50 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  <Q  f  ->  ( f  e.  ( 1st `  ( A F B ) )  ->  x  e.  ( 1st `  ( A F B ) ) ) ) )
2120com23 78 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  ( 1st `  ( A F B ) )  ->  ( x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2392   {crab 2395   <.cop 3498   class class class wbr 3897   ` cfv 5091  (class class class)co 5740    e. cmpo 5742   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    <Q cltq 7057   P.cnp 7063
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 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 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
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 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-qs 6401  df-ni 7076  df-nqqs 7120  df-ltnqqs 7125  df-inp 7238
This theorem is referenced by:  genprndl  7293
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