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Theorem genpcdl 7439
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 7285 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4638 . . . . . 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 7432 . . . . . . . 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 274 . . . . . . 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 3969 . . . . . . . . . . . . 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 408 . . . . . . . . . . 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 362 . . . . . . . . . 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 578 . . . . . . . . 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 2581 . . . . . . 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 963    = wceq 1335    e. wcel 2128   E.wrex 2436   {crab 2439   <.cop 3563   class class class wbr 3965   ` cfv 5170  (class class class)co 5824    e. cmpo 5826   1stc1st 6086   2ndc2nd 6087   Q.cnq 7200    <Q cltq 7205   P.cnp 7211
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-qs 6486  df-ni 7224  df-nqqs 7268  df-ltnqqs 7273  df-inp 7386
This theorem is referenced by:  genprndl  7441
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