Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg44 Structured version   Unicode version

Theorem cdlemg44 34700
Description: Part of proof of Lemma G of [Crawley] p. 116, fifth line of third paragraph on p. 117: "and hence fg = gf." (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 )
Assertion
Ref Expression
cdlemg44  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F )  =/=  ( R `  G
) )  ->  ( F  o.  G )  =  ( G  o.  F ) )

Proof of Theorem cdlemg44
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2454 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 cdlemg44.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 33973 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p ( le `  K ) W )
543ad2ant1 1009 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F )  =/=  ( R `  G
) )  ->  E. p  e.  ( Atoms `  K )  -.  p ( le `  K ) W )
6 simp11 1018 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 simp12l 1101 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  F  e.  T
)
8 simp12r 1102 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  G  e.  T
)
9 cdlemg44.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
103, 9ltrnco 34686 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( F  o.  G )  e.  T
)
116, 7, 8, 10syl3anc 1219 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( F  o.  G )  e.  T
)
123, 9ltrnco 34686 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  F  e.  T
)  ->  ( G  o.  F )  e.  T
)
136, 8, 7, 12syl3anc 1219 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( G  o.  F )  e.  T
)
14 3simpc 987 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )
15 simp13 1020 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( R `  F )  =/=  ( R `  G )
)
16 cdlemg44.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
173, 9, 16, 1, 2cdlemg44b 34699 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( p  e.  ( Atoms `  K
)  /\  -.  p
( le `  K
) W ) )  /\  ( R `  F )  =/=  ( R `  G )
)  ->  ( F `  ( G `  p
) )  =  ( G `  ( F `
 p ) ) )
186, 7, 8, 14, 15, 17syl131anc 1232 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( F `  ( G `  p ) )  =  ( G `
 ( F `  p ) ) )
19 simp12 1019 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( F  e.  T  /\  G  e.  T ) )
20 simp2 989 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  p  e.  (
Atoms `  K ) )
211, 2, 3, 9ltrncoval 34112 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  p  e.  ( Atoms `  K )
)  ->  ( ( F  o.  G ) `  p )  =  ( F `  ( G `
 p ) ) )
226, 19, 20, 21syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( ( F  o.  G ) `  p )  =  ( F `  ( G `
 p ) ) )
231, 2, 3, 9ltrncoval 34112 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  F  e.  T )  /\  p  e.  ( Atoms `  K )
)  ->  ( ( G  o.  F ) `  p )  =  ( G `  ( F `
 p ) ) )
246, 8, 7, 20, 23syl121anc 1224 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( ( G  o.  F ) `  p )  =  ( G `  ( F `
 p ) ) )
2518, 22, 243eqtr4d 2505 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( ( F  o.  G ) `  p )  =  ( ( G  o.  F
) `  p )
)
261, 2, 3, 9cdlemd 34174 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  o.  G )  e.  T  /\  ( G  o.  F )  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W )  /\  ( ( F  o.  G ) `  p )  =  ( ( G  o.  F
) `  p )
)  ->  ( F  o.  G )  =  ( G  o.  F ) )
276, 11, 13, 14, 25, 26syl311anc 1233 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F
)  =/=  ( R `
 G ) )  /\  p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W )  ->  ( F  o.  G )  =  ( G  o.  F ) )
2827rexlimdv3a 2947 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F )  =/=  ( R `  G
) )  ->  ( E. p  e.  ( Atoms `  K )  -.  p ( le `  K ) W  -> 
( F  o.  G
)  =  ( G  o.  F ) ) )
295, 28mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( R `  F )  =/=  ( R `  G
) )  ->  ( F  o.  G )  =  ( G  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799   class class class wbr 4399    o. ccom 4951   ` cfv 5525   lecple 14363   Atomscatm 33231   HLchlt 33318   LHypclh 33951   LTrncltrn 34068   trLctrl 34125
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 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-riotaBAD 32927
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-undef 6901  df-map 7325  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-llines 33465  df-lplanes 33466  df-lvols 33467  df-lines 33468  df-psubsp 33470  df-pmap 33471  df-padd 33763  df-lhyp 33955  df-laut 33956  df-ldil 34071  df-ltrn 34072  df-trl 34126
This theorem is referenced by:  cdlemg47  34703  ltrncom  34705
  Copyright terms: Public domain W3C validator