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Theorem cdlemk19u1 33988
Description: cdlemk19 33888 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 31-Jul-2013.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk5.x  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
cdlemk5.u  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
Assertion
Ref Expression
cdlemk19u1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( ( U `  F ) `  P
)  =  ( N `
 P ) )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, z,  ./\    .<_ , b   
z, g,  .<_    .\/ , b,
z    A, b, g, z    B, b, z    F, b, g, z    H, b, g, z    K, b, g, z    N, b, g, z    P, b, z    R, b, z    T, b, z    W, b, g, z    z, Y
Allowed substitution hints:    U( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk19u1
StepHypRef Expression
1 simp22 1031 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  =/=  N )
2 simp21 1030 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
3 cdlemk5.x . . . . 5  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
4 cdlemk5.u . . . . 5  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
53, 4cdlemk40f 33938 . . . 4  |-  ( ( F  =/=  N  /\  F  e.  T )  ->  ( U `  F
)  =  [_ F  /  g ]_ X
)
61, 2, 5syl2anc 659 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  F
)  =  [_ F  /  g ]_ X
)
76fveq1d 5851 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( ( U `  F ) `  P
)  =  ( [_ F  /  g ]_ X `  P ) )
8 simp1l 1021 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
9 simp23 1032 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  N  e.  T )
10 simp1r 1022 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( R `  F
)  =  ( R `
 N ) )
11 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
12 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
13 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemk5.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
1511, 12, 13, 14trlnid 33197 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( F  =/=  N  /\  ( R `  F )  =  ( R `  N ) ) )  ->  F  =/=  (  _I  |`  B ) )
168, 2, 9, 1, 10, 15syl122anc 1239 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  =/=  (  _I  |`  B ) )
172, 16, 93jca 1177 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )
)
18 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
19 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
20 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
21 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
22 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
23 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
2411, 18, 19, 20, 21, 12, 13, 14, 22, 23, 3cdlemk19x 33962 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( [_ F  /  g ]_ X `  P )  =  ( N `  P ) )
2517, 24syld3an2 1277 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( [_ F  /  g ]_ X `  P )  =  ( N `  P ) )
267, 25eqtrd 2443 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( ( U `  F ) `  P
)  =  ( N `
 P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   [_csb 3373   ifcif 3885   class class class wbr 4395    |-> cmpt 4453    _I cid 4733   `'ccnv 4822    |` cres 4825    o. ccom 4827   ` cfv 5569   iota_crio 6239  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Atomscatm 32281   HLchlt 32368   LHypclh 33001   LTrncltrn 33118   trLctrl 33176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-riotaBAD 31977
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-undef 7005  df-map 7459  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516  df-lvols 32517  df-lines 32518  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177
This theorem is referenced by:  cdlemk19u  33989
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