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Theorem cdlemg44 30052
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
StepHypRef Expression
1 eqid 2256 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2256 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 cdlemg44.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 29325 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p ( le `  K ) W )
543ad2ant1 981 . 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 990 . . . 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 1073 . . . . 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 1074 . . . . 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 30038 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( F  o.  G )  e.  T
)
116, 7, 8, 10syl3anc 1187 . . . 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 30038 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  F  e.  T
)  ->  ( G  o.  F )  e.  T
)
136, 8, 7, 12syl3anc 1187 . . . 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 959 . . . 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 992 . . . . . 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 30051 . . . . . 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 1200 . . . . 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 991 . . . . . 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 961 . . . . . 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 29464 . . . . . 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 1187 . . . . 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 29464 . . . . . 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 1192 . . . . 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 2298 . . . 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 29526 . . . 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 1201 . . 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 2640 . 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 16 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 set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517   class class class wbr 3963    o. ccom 4630   ` cfv 4638   lecple 13142   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477
This theorem is referenced by:  cdlemg47  30055  ltrncom  30057
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478
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