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Theorem cdleml5N 33979
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 1000 . . . 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 33789 . . . 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 1050 . . . . 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 33825 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  (  .0.  o.  U )  =  .0.  )
111, 9, 10syl2anc 659 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  (  .0.  o.  U )  =  .0.  )
12 simpr 459 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  V  =  .0.  )
1311, 12eqtr4d 2446 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  (  .0.  o.  U )  =  V )
14 coeq1 4980 . . . . 5  |-  ( s  =  .0.  ->  (
s  o.  U )  =  (  .0.  o.  U ) )
1514eqeq1d 2404 . . . 4  |-  ( s  =  .0.  ->  (
( s  o.  U
)  =  V  <->  (  .0.  o.  U )  =  V ) )
1615rspcev 3159 . . 3  |-  ( (  .0.  e.  E  /\  (  .0.  o.  U )  =  V )  ->  E. s  e.  E  ( s  o.  U
)  =  V )
178, 13, 16syl2anc 659 . 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 1000 . . 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 1001 . . 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 1002 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =/=  .0.  )  ->  U  =/=  .0.  )
21 simpr 459 . . 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 33978 . . 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 1234 . 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 2721 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754    |-> cmpt 4452    _I cid 4732    |` cres 4824    o. ccom 4826   ` cfv 5568   Basecbs 14839   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   trLctrl 33156   TEndoctendo 33751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-map 7458  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157  df-tendo 33754
This theorem is referenced by: (None)
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