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Theorem cdlemg43 31216
Description: Part of proof of Lemma G of [Crawley] p. 116, third line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)
Hypotheses
Ref Expression
cdlemg42.l  |-  .<_  =  ( le `  K )
cdlemg42.j  |-  .\/  =  ( join `  K )
cdlemg42.a  |-  A  =  ( Atoms `  K )
cdlemg42.h  |-  H  =  ( LHyp `  K
)
cdlemg42.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg42.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg42.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
cdlemg43  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  ( G `  P
) )  =  ( ( ( G `  P )  .\/  ( R `  F )
)  ./\  ( ( F `  P )  .\/  ( R `  G
) ) ) )

Proof of Theorem cdlemg43
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2l 983 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  F  e.  T )
3 simp31 993 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp2r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  G  e.  T )
5 cdlemg42.l . . . . 5  |-  .<_  =  ( le `  K )
6 cdlemg42.a . . . . 5  |-  A  =  ( Atoms `  K )
7 cdlemg42.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 cdlemg42.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
95, 6, 7, 8ltrnel 30625 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
101, 4, 3, 9syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
11 cdlemg42.j . . . 4  |-  .\/  =  ( join `  K )
12 cdlemg42.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
135, 11, 6, 7, 8, 12cdlemg42 31215 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  -.  ( G `  P )  .<_  ( P  .\/  ( F `  P )
) )
14 cdlemg42.m . . . 4  |-  ./\  =  ( meet `  K )
155, 11, 14, 6, 7, 8, 12cdlemc 30683 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )  /\  -.  ( G `  P )  .<_  ( P  .\/  ( F `  P )
) )  ->  ( F `  ( G `  P ) )  =  ( ( ( G `
 P )  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( ( P  .\/  ( G `  P ) )  ./\  W )
) ) )
161, 2, 3, 10, 13, 15syl131anc 1197 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  ( G `  P
) )  =  ( ( ( G `  P )  .\/  ( R `  F )
)  ./\  ( ( F `  P )  .\/  ( ( P  .\/  ( G `  P ) )  ./\  W )
) ) )
175, 11, 14, 6, 7, 8, 12trlval2 30649 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
)  ./\  W )
)
181, 4, 3, 17syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
)  ./\  W )
)
1918oveq2d 6060 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( ( F `  P )  .\/  ( R `  G
) )  =  ( ( F `  P
)  .\/  ( ( P  .\/  ( G `  P ) )  ./\  W ) ) )
2019oveq2d 6060 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( (
( G `  P
)  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( R `  G )
) )  =  ( ( ( G `  P )  .\/  ( R `  F )
)  ./\  ( ( F `  P )  .\/  ( ( P  .\/  ( G `  P ) )  ./\  W )
) ) )
2116, 20eqtr4d 2443 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  ( G `  P
) )  =  ( ( ( G `  P )  .\/  ( R `  F )
)  ./\  ( ( F `  P )  .\/  ( R `  G
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   meetcmee 14361   Atomscatm 29750   HLchlt 29837   LHypclh 30470   LTrncltrn 30587   trLctrl 30644
This theorem is referenced by:  cdlemg44a  31217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-map 6983  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645
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