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Theorem cdlemk38 31443
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. TODO: derive more directly with r19.23 2808? (Contributed by NM, 19-Jul-2013.)
Hypotheses
Ref Expression
cdlemk4.b  |-  B  =  ( Base `  K
)
cdlemk4.l  |-  .<_  =  ( le `  K )
cdlemk4.j  |-  .\/  =  ( join `  K )
cdlemk4.m  |-  ./\  =  ( meet `  K )
cdlemk4.a  |-  A  =  ( Atoms `  K )
cdlemk4.h  |-  H  =  ( LHyp `  K
)
cdlemk4.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk4.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk4.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk4.y  |-  Y  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `  ( G  o.  `' b
) ) ) )
cdlemk4.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
cdlemk38  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  ( X `  P )  .<_  ( P  .\/  ( R `  G )
) )
Distinct variable groups:    z, b,  ./\    .<_ , b, z    .\/ , b, z    A, b, z    B, b, z    F, b, z    G, b, z    H, b, z    K, b, z    N, b, z    P, b, z    R, b, z    T, b, z    W, b, z    z, Y
Allowed substitution hints:    X( z, b)    Y( b)    Z( z, b)

Proof of Theorem cdlemk38
StepHypRef Expression
1 cdlemk4.b . . . 4  |-  B  =  ( Base `  K
)
2 cdlemk4.h . . . 4  |-  H  =  ( LHyp `  K
)
3 cdlemk4.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
4 cdlemk4.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
51, 2, 3, 4cdlemftr2 31094 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) )
653ad2ant1 978 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) )
7 nfv 1629 . . 3  |-  F/ b ( ( K  e.  HL  /\  W  e.  H )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
8 cdlemk4.x . . . . . 6  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  Y ) )
9 nfra1 2743 . . . . . . 7  |-  F/ b A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  Y )
10 nfcv 2566 . . . . . . 7  |-  F/_ b T
119, 10nfriota 6545 . . . . . 6  |-  F/_ b
( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  Y ) )
128, 11nfcxfr 2563 . . . . 5  |-  F/_ b X
13 nfcv 2566 . . . . 5  |-  F/_ b P
1412, 13nffv 5721 . . . 4  |-  F/_ b
( X `  P
)
15 nfcv 2566 . . . 4  |-  F/_ b  .<_
16 nfcv 2566 . . . 4  |-  F/_ b
( P  .\/  ( R `  G )
)
1714, 15, 16nfbr 4243 . . 3  |-  F/ b ( X `  P
)  .<_  ( P  .\/  ( R `  G ) )
18 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  (
b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
19 simpl21 1035 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  (
b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )
20 simpl22 1036 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  (
b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  ->  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )
21 simpl23 1037 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  (
b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  ->  N  e.  T
)
22 simpl3l 1012 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  (
b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
23 simpl3r 1013 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  (
b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
24 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  (
b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  ->  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )
25 cdlemk4.l . . . . . 6  |-  .<_  =  ( le `  K )
26 cdlemk4.j . . . . . 6  |-  .\/  =  ( join `  K )
27 cdlemk4.m . . . . . 6  |-  ./\  =  ( meet `  K )
28 cdlemk4.a . . . . . 6  |-  A  =  ( Atoms `  K )
29 cdlemk4.z . . . . . 6  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
30 cdlemk4.y . . . . . 6  |-  Y  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `  ( G  o.  `' b
) ) ) )
311, 25, 26, 27, 28, 2, 3, 4, 29, 30, 8cdlemk37 31442 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) )  -> 
( X `  P
)  .<_  ( P  .\/  ( R `  G ) ) )
3218, 19, 20, 21, 22, 23, 24, 31syl331anc 1209 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  (
b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  ->  ( X `  P )  .<_  ( P 
.\/  ( R `  G ) ) )
3332exp32 589 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  (
b  e.  T  -> 
( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( X `  P )  .<_  ( P 
.\/  ( R `  G ) ) ) ) )
347, 17, 33rexlimd 2814 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  ( E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  ( X `  P )  .<_  ( P  .\/  ( R `  G )
) ) )
356, 34mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  ( X `  P )  .<_  ( P  .\/  ( R `  G )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   A.wral 2692   E.wrex 2693   class class class wbr 4199    _I cid 4480   `'ccnv 4863    |` cres 4866    o. ccom 4868   ` cfv 5440  (class class class)co 6067   iota_crio 6528   Basecbs 13452   lecple 13519   joincjn 14384   meetcmee 14385   Atomscatm 29792   HLchlt 29879   LHypclh 30512   LTrncltrn 30629   trLctrl 30686
This theorem is referenced by:  cdlemk39  31444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-map 7006  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-p1 14452  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026  df-lplanes 30027  df-lvols 30028  df-lines 30029  df-psubsp 30031  df-pmap 30032  df-padd 30324  df-lhyp 30516  df-laut 30517  df-ldil 30632  df-ltrn 30633  df-trl 30687
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