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Theorem cdlemg44a 35883
Description: Part of proof of Lemma G of [Crawley] p. 116, fourth line of third paragraph on p. 117: "so fg(p) = gf(p)." (Contributed by NM, 3-Jun-2013.)
Hypotheses
Ref Expression
cdlemg44.h  |-  H  =  ( LHyp `  K
)
cdlemg44.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg44.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg44.l  |-  .<_  =  ( le `  K )
cdlemg44.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdlemg44a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  ( G `  P
) )  =  ( G `  ( F `
 P ) ) )

Proof of Theorem cdlemg44a
StepHypRef Expression
1 simp1l 1020 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  K  e.  HL )
2 hllat 34516 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  K  e.  Lat )
4 simp1 996 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
5 simp22 1030 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  G  e.  T )
6 simp23l 1117 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  P  e.  A )
7 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 cdlemg44.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8atbase 34442 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
106, 9syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  P  e.  ( Base `  K )
)
11 cdlemg44.h . . . . . 6  |-  H  =  ( LHyp `  K
)
12 cdlemg44.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
137, 11, 12ltrncl 35277 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( G `  P )  e.  (
Base `  K )
)
144, 5, 10, 13syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( G `  P )  e.  (
Base `  K )
)
15 simp21 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  F  e.  T )
16 cdlemg44.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
177, 11, 12, 16trlcl 35316 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
184, 15, 17syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( R `  F )  e.  (
Base `  K )
)
19 eqid 2467 . . . . 5  |-  ( join `  K )  =  (
join `  K )
207, 19latjcl 15555 . . . 4  |-  ( ( K  e.  Lat  /\  ( G `  P )  e.  ( Base `  K
)  /\  ( R `  F )  e.  (
Base `  K )
)  ->  ( ( G `  P )
( join `  K )
( R `  F
) )  e.  (
Base `  K )
)
213, 14, 18, 20syl3anc 1228 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( ( G `  P )
( join `  K )
( R `  F
) )  e.  (
Base `  K )
)
227, 11, 12ltrncl 35277 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
234, 15, 10, 22syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  P )  e.  (
Base `  K )
)
247, 11, 12, 16trlcl 35316 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
254, 5, 24syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( R `  G )  e.  (
Base `  K )
)
267, 19latjcl 15555 . . . 4  |-  ( ( K  e.  Lat  /\  ( F `  P )  e.  ( Base `  K
)  /\  ( R `  G )  e.  (
Base `  K )
)  ->  ( ( F `  P )
( join `  K )
( R `  G
) )  e.  (
Base `  K )
)
273, 23, 25, 26syl3anc 1228 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( ( F `  P )
( join `  K )
( R `  G
) )  e.  (
Base `  K )
)
28 eqid 2467 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
297, 28latmcom 15579 . . 3  |-  ( ( K  e.  Lat  /\  ( ( G `  P ) ( join `  K ) ( R `
 F ) )  e.  ( Base `  K
)  /\  ( ( F `  P )
( join `  K )
( R `  G
) )  e.  (
Base `  K )
)  ->  ( (
( G `  P
) ( join `  K
) ( R `  F ) ) (
meet `  K )
( ( F `  P ) ( join `  K ) ( R `
 G ) ) )  =  ( ( ( F `  P
) ( join `  K
) ( R `  G ) ) (
meet `  K )
( ( G `  P ) ( join `  K ) ( R `
 F ) ) ) )
303, 21, 27, 29syl3anc 1228 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( (
( G `  P
) ( join `  K
) ( R `  F ) ) (
meet `  K )
( ( F `  P ) ( join `  K ) ( R `
 G ) ) )  =  ( ( ( F `  P
) ( join `  K
) ( R `  G ) ) (
meet `  K )
( ( G `  P ) ( join `  K ) ( R `
 F ) ) ) )
31 simp23 1031 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
32 simp32 1033 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( G `  P )  =/=  P
)
33 simp33 1034 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( R `  F )  =/=  ( R `  G )
)
34 cdlemg44.l . . . 4  |-  .<_  =  ( le `  K )
3534, 19, 8, 11, 12, 16, 28cdlemg43 35882 . . 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 `  ( G `  P
) )  =  ( ( ( G `  P ) ( join `  K ) ( R `
 F ) ) ( meet `  K
) ( ( F `
 P ) (
join `  K )
( R `  G
) ) ) )
364, 15, 5, 31, 32, 33, 35syl123anc 1245 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  ( G `  P
) )  =  ( ( ( G `  P ) ( join `  K ) ( R `
 F ) ) ( meet `  K
) ( ( F `
 P ) (
join `  K )
( R `  G
) ) ) )
37 simp31 1032 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  P )  =/=  P
)
3833necomd 2738 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( R `  G )  =/=  ( R `  F )
)
3934, 19, 8, 11, 12, 16, 28cdlemg43 35882 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  F  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =/=  P  /\  ( R `  G
)  =/=  ( R `
 F ) ) )  ->  ( G `  ( F `  P
) )  =  ( ( ( F `  P ) ( join `  K ) ( R `
 G ) ) ( meet `  K
) ( ( G `
 P ) (
join `  K )
( R `  F
) ) ) )
404, 5, 15, 31, 37, 38, 39syl123anc 1245 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( G `  ( F `  P
) )  =  ( ( ( F `  P ) ( join `  K ) ( R `
 G ) ) ( meet `  K
) ( ( G `
 P ) (
join `  K )
( R `  F
) ) ) )
4130, 36, 403eqtr4d 2518 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  ( G `  P
) )  =  ( G `  ( F `
 P ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   Latclat 15549   Atomscatm 34416   HLchlt 34503   LHypclh 35136   LTrncltrn 35253   trLctrl 35310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34329  df-ol 34331  df-oml 34332  df-covers 34419  df-ats 34420  df-atl 34451  df-cvlat 34475  df-hlat 34504  df-llines 34650  df-psubsp 34655  df-pmap 34656  df-padd 34948  df-lhyp 35140  df-laut 35141  df-ldil 35256  df-ltrn 35257  df-trl 35311
This theorem is referenced by:  cdlemg44b  35884
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