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Theorem cdleml5N 35776
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 999 . . . 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 35586 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  E )
81, 7syl 16 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  .0.  e.  E )
9 simpl2l 1049 . . . . 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 35622 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  (  .0.  o.  U )  =  .0.  )
111, 9, 10syl2anc 661 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  (  .0.  o.  U )  =  .0.  )
12 simpr 461 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  V  =  .0.  )
1311, 12eqtr4d 2511 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  (  .0.  o.  U )  =  V )
14 coeq1 5158 . . . . 5  |-  ( s  =  .0.  ->  (
s  o.  U )  =  (  .0.  o.  U ) )
1514eqeq1d 2469 . . . 4  |-  ( s  =  .0.  ->  (
( s  o.  U
)  =  V  <->  (  .0.  o.  U )  =  V ) )
1615rspcev 3214 . . 3  |-  ( (  .0.  e.  E  /\  (  .0.  o.  U )  =  V )  ->  E. s  e.  E  ( s  o.  U
)  =  V )
178, 13, 16syl2anc 661 . 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 999 . . 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 1000 . . 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 1001 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =/=  .0.  )  ->  U  =/=  .0.  )
21 simpr 461 . . 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 35775 . . 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 1232 . 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 2785 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    |-> cmpt 4505    _I cid 4790    |` cres 5001    o. ccom 5003   ` cfv 5586   Basecbs 14486   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   trLctrl 34954   TEndoctendo 35548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-map 7419  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tendo 35551
This theorem is referenced by: (None)
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