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Theorem cdlemk35u 34284
Description: Substitution version of cdlemk35 34232. (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
cdlemk35u  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  G
)  e.  T )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, G, z    ./\ , b, z    .<_ , b    z,
g,  .<_    .\/ , b, z    A, b, g, z    B, b, z    F, b, g, z   
z, G    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    G, b
Allowed substitution hints:    U( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk35u
StepHypRef Expression
1 simpr 462 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  F  =  N )
2 simpl23 1085 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  G  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, 4cdlemk40t 34238 . . . 4  |-  ( ( F  =  N  /\  G  e.  T )  ->  ( U `  G
)  =  G )
61, 2, 5syl2anc 665 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( U `  G
)  =  G )
76, 2eqeltrd 2508 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( U `  G
)  e.  T )
8 simpr 462 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  ->  F  =/=  N )
9 simpl23 1085 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  ->  G  e.  T )
103, 4cdlemk40f 34239 . . . 4  |-  ( ( F  =/=  N  /\  G  e.  T )  ->  ( U `  G
)  =  [_ G  /  g ]_ X
)
118, 9, 10syl2anc 665 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  -> 
( U `  G
)  =  [_ G  /  g ]_ X
)
12 simpl1l 1056 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  -> 
( K  e.  HL  /\  W  e.  H ) )
13 simpl21 1083 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  ->  F  e.  T )
14 simpl22 1084 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  ->  N  e.  T )
15 simpl1r 1057 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  -> 
( R `  F
)  =  ( R `
 N ) )
16 cdlemk5.b . . . . . . 7  |-  B  =  ( Base `  K
)
17 cdlemk5.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
18 cdlemk5.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
19 cdlemk5.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
2016, 17, 18, 19trlnid 33498 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( F  =/=  N  /\  ( R `  F )  =  ( R `  N ) ) )  ->  F  =/=  (  _I  |`  B ) )
2112, 13, 14, 8, 15, 20syl122anc 1273 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  ->  F  =/=  (  _I  |`  B ) )
2213, 21jca 534 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  -> 
( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )
23 simpl3 1010 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
24 cdlemk5.l . . . . 5  |-  .<_  =  ( le `  K )
25 cdlemk5.j . . . . 5  |-  .\/  =  ( join `  K )
26 cdlemk5.m . . . . 5  |-  ./\  =  ( meet `  K )
27 cdlemk5.a . . . . 5  |-  A  =  ( Atoms `  K )
28 cdlemk5.z . . . . 5  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
29 cdlemk5.y . . . . 5  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
3016, 24, 25, 26, 27, 17, 18, 19, 28, 29, 3cdlemk35s-id 34258 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  [_ G  /  g ]_ X  e.  T )
3112, 22, 9, 14, 23, 15, 30syl132anc 1282 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  ->  [_ G  /  g ]_ X  e.  T
)
3211, 31eqeltrd 2508 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  -> 
( U `  G
)  e.  T )
337, 32pm2.61dane 2740 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  G  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  G
)  e.  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   [_csb 3392   ifcif 3906   class class class wbr 4417    |-> cmpt 4475    _I cid 4755   `'ccnv 4844    |` cres 4847    o. ccom 4849   ` cfv 5592   iota_crio 6257  (class class class)co 6296   Basecbs 15081   lecple 15157   joincjn 16141   meetcmee 16142   Atomscatm 32582   HLchlt 32669   LHypclh 33302   LTrncltrn 33419   trLctrl 33477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-riotaBAD 32278
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-undef 7019  df-map 7473  df-preset 16125  df-poset 16143  df-plt 16156  df-lub 16172  df-glb 16173  df-join 16174  df-meet 16175  df-p0 16237  df-p1 16238  df-lat 16244  df-clat 16306  df-oposet 32495  df-ol 32497  df-oml 32498  df-covers 32585  df-ats 32586  df-atl 32617  df-cvlat 32641  df-hlat 32670  df-llines 32816  df-lplanes 32817  df-lvols 32818  df-lines 32819  df-psubsp 32821  df-pmap 32822  df-padd 33114  df-lhyp 33306  df-laut 33307  df-ldil 33422  df-ltrn 33423  df-trl 33478
This theorem is referenced by:  cdlemk19u  34290  cdlemk56  34291
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