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Theorem genpcdl 7481
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 7327 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4663 . . . . . 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 7474 . . . . . . . 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 3993 . . . . . . . . . . . . 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 583 . . . . . . . . 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 2594 . . . . . . 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 973    = wceq 1348    e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853    e. cmpo 5855   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242    <Q cltq 7247   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-1st 6119  df-2nd 6120  df-qs 6519  df-ni 7266  df-nqqs 7310  df-ltnqqs 7315  df-inp 7428
This theorem is referenced by:  genprndl  7483
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