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Theorem cdleml2N 30434
Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleml1.b  |-  B  =  ( Base `  K
)
cdleml1.h  |-  H  =  ( LHyp `  K
)
cdleml1.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdleml1.r  |-  R  =  ( ( trL `  K
) `  W )
cdleml1.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
cdleml2N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  E. s  e.  E  ( s `  ( U `  f
) )  =  ( V `  f ) )
Distinct variable groups:    E, s    K, s    R, s    T, s    U, s    V, s    W, s    f, s
Allowed substitution hints:    B( f, s)    R( f)    T( f)    U( f)    E( f)    H( f, s)    K( f)    V( f)    W( f)

Proof of Theorem cdleml2N
StepHypRef Expression
1 simp1 957 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp21 990 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  U  e.  E )
3 simp23 992 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  f  e.  T )
4 cdleml1.h . . . 4  |-  H  =  ( LHyp `  K
)
5 cdleml1.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
6 cdleml1.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
74, 5, 6tendocl 30224 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  f  e.  T
)  ->  ( U `  f )  e.  T
)
81, 2, 3, 7syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  ( U `  f )  e.  T
)
9 simp22 991 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  V  e.  E )
104, 5, 6tendocl 30224 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  f  e.  T
)  ->  ( V `  f )  e.  T
)
111, 9, 3, 10syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  ( V `  f )  e.  T
)
12 cdleml1.b . . 3  |-  B  =  ( Base `  K
)
13 cdleml1.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
1412, 4, 5, 13, 6cdleml1N 30433 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  ( R `  ( U `  f
) )  =  ( R `  ( V `
 f ) ) )
154, 5, 13, 6cdlemk 30431 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( U `
 f )  e.  T  /\  ( V `
 f )  e.  T )  /\  ( R `  ( U `  f ) )  =  ( R `  ( V `  f )
) )  ->  E. s  e.  E  ( s `  ( U `  f
) )  =  ( V `  f ) )
161, 8, 11, 14, 15syl121anc 1189 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  f  e.  T )  /\  (
f  =/=  (  _I  |`  B )  /\  ( U `  f )  =/=  (  _I  |`  B )  /\  ( V `  f )  =/=  (  _I  |`  B ) ) )  ->  E. s  e.  E  ( s `  ( U `  f
) )  =  ( V `  f ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2448   E.wrex 2546    _I cid 4304    |` cres 4691   ` cfv 5222   Basecbs 13143   HLchlt 28808   LHypclh 29441   LTrncltrn 29558   trLctrl 29615   TEndoctendo 30209
This theorem is referenced by:  cdleml3N  30435
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-llines 28955  df-lplanes 28956  df-lvols 28957  df-lines 28958  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616  df-tendo 30212
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