Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleml5N Structured version   Unicode version

Theorem cdleml5N 34933
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 991 . . . 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 34743 . . . 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 1041 . . . . 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 34779 . . . . 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 2495 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  U  =/=  .0.  )  /\  V  =  .0.  )  ->  (  .0.  o.  U )  =  V )
14 coeq1 5098 . . . . 5  |-  ( s  =  .0.  ->  (
s  o.  U )  =  (  .0.  o.  U ) )
1514eqeq1d 2453 . . . 4  |-  ( s  =  .0.  ->  (
( s  o.  U
)  =  V  <->  (  .0.  o.  U )  =  V ) )
1615rspcev 3172 . . 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 991 . . 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 992 . . 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 993 . . 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 34932 . . 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 1223 . 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 2766 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796    |-> cmpt 4451    _I cid 4732    |` cres 4943    o. ccom 4945   ` cfv 5519   Basecbs 14285   HLchlt 33304   LHypclh 33937   LTrncltrn 34054   trLctrl 34111   TEndoctendo 34705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-riotaBAD 32913
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-undef 6895  df-map 7319  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451  df-lplanes 33452  df-lvols 33453  df-lines 33454  df-psubsp 33456  df-pmap 33457  df-padd 33749  df-lhyp 33941  df-laut 33942  df-ldil 34057  df-ltrn 34058  df-trl 34112  df-tendo 34708
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator