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Theorem cdlemg44a 34714
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 1012 . . . 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 33347 . . . 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 988 . . . . 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 1022 . . . . 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 1109 . . . . . 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 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 cdlemg44.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8atbase 33273 . . . . . 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 34108 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( G `  P )  e.  (
Base `  K )
)
144, 5, 10, 13syl3anc 1219 . . . 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 1021 . . . . 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 34147 . . . . 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 2454 . . . . 5  |-  ( join `  K )  =  (
join `  K )
207, 19latjcl 15341 . . . 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 1219 . . 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 34108 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
234, 15, 10, 22syl3anc 1219 . . . 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 34147 . . . . 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 15341 . . . 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 1219 . . 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 2454 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
297, 28latmcom 15365 . . 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 1219 . 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 1023 . . 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 1025 . . 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 1026 . . 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 34713 . . 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 1236 . 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 1024 . . 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 2723 . . 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 34713 . . 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 1236 . 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 2505 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   Latclat 15335   Atomscatm 33247   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   trLctrl 34141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142
This theorem is referenced by:  cdlemg44b  34715
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