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Theorem cdlemk19y 33931
Description: cdlemk19 33868 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 30-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 ) ) ) )
Assertion
Ref Expression
cdlemk19y  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
) ) ) )  ->  [_ F  /  g ]_ Y  =  ( N `  P )
)
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b,  ./\    .\/ , b    F, b    N, b    P, b    R, b    T, b    g, F
Allowed substitution hints:    A( g, b)    B( b)    H( g, b)    K( g, b)    .<_ ( g, b)    N( g)    W( g, b)    Y( g, b)    Z( b)

Proof of Theorem cdlemk19y
Dummy variables  e 
f  i  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemk5.b . 2  |-  B  =  ( Base `  K
)
2 cdlemk5.l . 2  |-  .<_  =  ( le `  K )
3 cdlemk5.j . 2  |-  .\/  =  ( join `  K )
4 cdlemk5.m . 2  |-  ./\  =  ( meet `  K )
5 cdlemk5.a . 2  |-  A  =  ( Atoms `  K )
6 cdlemk5.h . 2  |-  H  =  ( LHyp `  K
)
7 cdlemk5.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemk5.r . 2  |-  R  =  ( ( trL `  K
) `  W )
9 cdlemk5.z . 2  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
10 cdlemk5.y . 2  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
11 eqid 2402 . 2  |-  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `
 P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) ) ) )  =  ( f  e.  T  |->  (
iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
12 eqid 2402 . 2  |-  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `
 P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( (
( ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P )  =  ( ( P  .\/  ( R `  f )
)  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) ) ) ) `  b
) `  P )  .\/  ( R `  (
e  o.  `' b ) ) ) ) ) )  =  ( e  e.  T  |->  (
iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) ) `  b ) `
 P )  .\/  ( R `  ( e  o.  `' b ) ) ) ) ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemk19ylem 33929 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
) ) ) )  ->  [_ F  /  g ]_ Y  =  ( 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   [_csb 3372   class class class wbr 4394    |-> cmpt 4452    _I cid 4732   `'ccnv 4821    |` cres 4824    o. ccom 4826   ` cfv 5568   iota_crio 6238  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   meetcmee 15896   Atomscatm 32261   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   trLctrl 33156
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 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  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 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-map 7458  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157
This theorem is referenced by:  cdlemk19xlem  33941
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