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Theorem cdleml5N 34618
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 )
cdleml3.o  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
cdleml5N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  ->  E. s  e.  E  ( s  o.  U )  =  V )
Distinct variable groups:    E, s    K, s    R, s    T, s    U, s    V, s    W, s, g    B, g, s   
g, H, s    g, K    .0. , s    T, g    g, W
Allowed substitution hints:    R( g)    U( g)    E( g)    V( g)    .0. ( g)

Proof of Theorem cdleml5N
StepHypRef Expression
1 simpl1 1033 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 cdleml1.b . . . . 5  |-  B  =  ( Base `  K
)
3 cdleml1.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 cdleml1.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
5 cdleml1.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
6 cdleml3.o . . . . 5  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
72, 3, 4, 5, 6tendo0cl 34428 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  E )
81, 7syl 17 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  .0.  e.  E )
9 simpl2l 1083 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  U  e.  E )
102, 3, 4, 5, 6tendo0mul 34464 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  (  .0.  o.  U )  =  .0.  )
111, 9, 10syl2anc 673 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  (  .0.  o.  U )  =  .0.  )
12 simpr 468 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  V  =  .0.  )
1311, 12eqtr4d 2508 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  (  .0.  o.  U )  =  V )
14 coeq1 4997 . . . . 5  |-  ( s  =  .0.  ->  (
s  o.  U )  =  (  .0.  o.  U ) )
1514eqeq1d 2473 . . . 4  |-  ( s  =  .0.  ->  (
( s  o.  U
)  =  V  <->  (  .0.  o.  U )  =  V ) )
1615rspcev 3136 . . 3  |-  ( (  .0.  e.  E  /\  (  .0.  o.  U )  =  V )  ->  E. s  e.  E  ( s  o.  U
)  =  V )
178, 13, 16syl2anc 673 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  E. s  e.  E  ( s  o.  U )  =  V )
18 simpl1 1033 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =/=  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
19 simpl2 1034 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =/=  .0.  )  ->  ( U  e.  E  /\  V  e.  E ) )
20 simpl3 1035 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =/=  .0.  )  ->  U  =/=  .0.  )
21 simpr 468 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =/=  .0.  )  ->  V  =/=  .0.  )
22 cdleml1.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
232, 3, 4, 22, 5, 6cdleml4N 34617 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  ( U  =/=  .0.  /\  V  =/=  .0.  ) )  ->  E. s  e.  E  ( s  o.  U
)  =  V )
2418, 19, 20, 21, 23syl112anc 1296 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =/=  .0.  )  ->  E. s  e.  E  ( s  o.  U
)  =  V )
2517, 24pm2.61dane 2730 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  ->  E. s  e.  E  ( s  o.  U )  =  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757    |-> cmpt 4454    _I cid 4749    |` cres 4841    o. ccom 4843   ` cfv 5589   Basecbs 15199   HLchlt 32987   LHypclh 33620   LTrncltrn 33737   trLctrl 33795   TEndoctendo 34390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-undef 7038  df-map 7492  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796  df-tendo 34393
This theorem is referenced by: (None)
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