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Theorem cdlemk5a 29713
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
Hypotheses
Ref Expression
cdlemk.b  |-  B  =  ( Base `  K
)
cdlemk.l  |-  .<_  =  ( le `  K )
cdlemk.j  |-  .\/  =  ( join `  K )
cdlemk.a  |-  A  =  ( Atoms `  K )
cdlemk.h  |-  H  =  ( LHyp `  K
)
cdlemk.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
cdlemk5a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( F `  P
)  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  .<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F
) ) ) )

Proof of Theorem cdlemk5a
StepHypRef Expression
1 simp1l 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
2 simp1r 985 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H )
3 simp21 993 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T )
4 simp22 994 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  G  e.  T )
5 simp3 962 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( R `  G )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )
6 cdlemk.b . . . 4  |-  B  =  ( Base `  K
)
7 cdlemk.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemk.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemk.a . . . 4  |-  A  =  ( Atoms `  K )
10 cdlemk.h . . . 4  |-  H  =  ( LHyp `  K
)
11 cdlemk.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
12 cdlemk.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
13 cdlemk.m . . . 4  |-  ./\  =  ( meet `  K )
146, 7, 8, 9, 10, 11, 12, 13cdlemk3 29711 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( F `  P
)  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  =  ( F `
 P ) )
151, 2, 3, 4, 5, 14syl221anc 1198 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( F `  P
)  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  =  ( F `
 P ) )
16 simp23 995 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  X  e.  T )
17 simp33l 1087 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  P  e.  A )
18 simp33r 1088 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  -.  P  .<_  W )
196, 7, 8, 9, 10, 11, 12, 13cdlemk4 29712 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  .<_  ( ( X `  P ) 
.\/  ( R `  ( X  o.  `' F ) ) ) )
201, 2, 3, 16, 17, 18, 19syl222anc 1203 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( F `  P )  .<_  ( ( X `  P ) 
.\/  ( R `  ( X  o.  `' F ) ) ) )
2115, 20eqbrtrd 3940 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( F `  P
)  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  .<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920    _I cid 4197   `'ccnv 4579    |` cres 4582    o. ccom 4584   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28142   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036
This theorem is referenced by:  cdlemk5  29714  cdlemk5auN  29738  cdlemk5u  29739
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037
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