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Theorem cdlemg7N 34238
Description: TODO: FIX COMMENT. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg7.b  |-  B  =  ( Base `  K
)
cdlemg7.l  |-  .<_  =  ( le `  K )
cdlemg7.a  |-  A  =  ( Atoms `  K )
cdlemg7.h  |-  H  =  ( LHyp `  K
)
cdlemg7.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemg7N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  ->  ( F `  ( G `  X
) )  =  X )

Proof of Theorem cdlemg7N
StepHypRef Expression
1 simpl1 1017 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simpl31 1095 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  ->  F  e.  T )
3 simpl32 1096 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  ->  G  e.  T )
4 simpl2r 1068 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  ->  X  e.  B )
5 cdlemg7.b . . . . . 6  |-  B  =  ( Base `  K
)
6 cdlemg7.h . . . . . 6  |-  H  =  ( LHyp `  K
)
7 cdlemg7.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
85, 6, 7ltrncl 33735 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  X  e.  B
)  ->  ( G `  X )  e.  B
)
91, 3, 4, 8syl3anc 1276 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  -> 
( G `  X
)  e.  B )
10 simpr 467 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  ->  X  .<_  W )
11 cdlemg7.l . . . . . . 7  |-  .<_  =  ( le `  K )
125, 11, 6, 7ltrnval1 33744 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( G `  X )  =  X )
131, 3, 4, 10, 12syl112anc 1280 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  -> 
( G `  X
)  =  X )
1413, 10eqbrtrd 4437 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  -> 
( G `  X
)  .<_  W )
155, 11, 6, 7ltrnval1 33744 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( G `  X )  e.  B  /\  ( G `  X
)  .<_  W ) )  ->  ( F `  ( G `  X ) )  =  ( G `
 X ) )
161, 2, 9, 14, 15syl112anc 1280 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  -> 
( F `  ( G `  X )
)  =  ( G `
 X ) )
1716, 13eqtrd 2496 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  X  .<_  W )  -> 
( F `  ( G `  X )
)  =  X )
18 simpl1 1017 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  -.  X  .<_  W )  ->  ( K  e.  HL  /\  W  e.  H ) )
19 simpl2l 1067 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  -.  X  .<_  W )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
20 simpl2r 1068 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  -.  X  .<_  W )  ->  X  e.  B
)
21 simpr 467 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  -.  X  .<_  W )  ->  -.  X  .<_  W )
2220, 21jca 539 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  -.  X  .<_  W )  ->  ( X  e.  B  /\  -.  X  .<_  W ) )
23 simpl31 1095 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  -.  X  .<_  W )  ->  F  e.  T
)
24 simpl32 1096 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  -.  X  .<_  W )  ->  G  e.  T
)
25 simpl33 1097 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  -.  X  .<_  W )  ->  ( F `  ( G `  P ) )  =  P )
26 cdlemg7.a . . . 4  |-  A  =  ( Atoms `  K )
275, 11, 26, 6, 7cdlemg7aN 34237 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( F `  ( G `  X )
)  =  X )
2818, 19, 22, 23, 24, 25, 27syl123anc 1293 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B
)  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `
 ( G `  P ) )  =  P ) )  /\  -.  X  .<_  W )  ->  ( F `  ( G `  X ) )  =  X )
2917, 28pm2.61dan 805 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  ->  ( F `  ( G `  X
) )  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   class class class wbr 4416   ` cfv 5601   Basecbs 15170   lecple 15246   Atomscatm 32874   HLchlt 32961   LHypclh 33594   LTrncltrn 33711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-riotaBAD 32570
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-undef 7046  df-map 7500  df-preset 16222  df-poset 16240  df-plt 16253  df-lub 16269  df-glb 16270  df-join 16271  df-meet 16272  df-p0 16334  df-p1 16335  df-lat 16341  df-clat 16403  df-oposet 32787  df-ol 32789  df-oml 32790  df-covers 32877  df-ats 32878  df-atl 32909  df-cvlat 32933  df-hlat 32962  df-llines 33108  df-lplanes 33109  df-lvols 33110  df-lines 33111  df-psubsp 33113  df-pmap 33114  df-padd 33406  df-lhyp 33598  df-laut 33599  df-ldil 33714  df-ltrn 33715  df-trl 33770
This theorem is referenced by: (None)
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