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Theorem cdlemk19x 31802
Description: cdlemk19 31728 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 ) ) ) )
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 ) )
Assertion
Ref Expression
cdlemk19x  |-  ( ( ( ( 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 ) )
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:    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk19x
StepHypRef Expression
1 simp1l 982 . . 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 ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 cdlemk5.b . . . 4  |-  B  =  ( Base `  K
)
3 cdlemk5.h . . . 4  |-  H  =  ( LHyp `  K
)
4 cdlemk5.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 cdlemk5.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
62, 3, 4, 5cdlemftr1 31426 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F ) ) )
71, 6syl 16 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )
) )
8 nfv 1630 . . 3  |-  F/ b ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )
9 nfcv 2574 . . . . . 6  |-  F/_ b F
10 cdlemk5.x . . . . . . 7  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
11 nfra1 2758 . . . . . . . 8  |-  F/ b A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )
12 nfcv 2574 . . . . . . . 8  |-  F/_ b T
1311, 12nfriota 6561 . . . . . . 7  |-  F/_ b
( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
1410, 13nfcxfr 2571 . . . . . 6  |-  F/_ b X
159, 14nfcsb 3287 . . . . 5  |-  F/_ b [_ F  /  g ]_ X
16 nfcv 2574 . . . . 5  |-  F/_ b P
1715, 16nffv 5737 . . . 4  |-  F/_ b
( [_ F  /  g ]_ X `  P )
1817nfeq1 2583 . . 3  |-  F/ b ( [_ F  / 
g ]_ X `  P
)  =  ( N `
 P )
19 simpl1 961 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
) ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) ) )
20 simpl2 962 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
) ) ) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
) )
21 simpl3 963 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
22 simpr 449 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
) ) ) )  ->  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )
) ) )
23 cdlemk5.l . . . . . 6  |-  .<_  =  ( le `  K )
24 cdlemk5.j . . . . . 6  |-  .\/  =  ( join `  K )
25 cdlemk5.m . . . . . 6  |-  ./\  =  ( meet `  K )
26 cdlemk5.a . . . . . 6  |-  A  =  ( Atoms `  K )
27 cdlemk5.z . . . . . 6  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
28 cdlemk5.y . . . . . 6  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
292, 23, 24, 25, 26, 3, 4, 5, 27, 28, 10cdlemk19xlem 31801 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F ) ) ) )  ->  ( [_ F  /  g ]_ X `  P )  =  ( N `  P ) )
3019, 20, 21, 22, 29syl121anc 1190 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
) ) ) )  ->  ( [_ F  /  g ]_ X `  P )  =  ( N `  P ) )
3130exp32 590 . . 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 ) )  ->  ( b  e.  T  ->  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
) )  ->  ( [_ F  /  g ]_ X `  P )  =  ( N `  P ) ) ) )
328, 18, 31rexlimd 2829 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( E. b  e.  T  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
) )  ->  ( [_ F  /  g ]_ X `  P )  =  ( N `  P ) ) )
337, 32mpd 15 1  |-  ( ( ( ( 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 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   [_csb 3253   class class class wbr 4214    _I cid 4495   `'ccnv 4879    |` cres 4882    o. ccom 4884   ` cfv 5456  (class class class)co 6083   iota_crio 6544   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem is referenced by:  cdlemk19u1  31828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359  df-lines 30360  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
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