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Theorem cdleml1N 31787
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
cdleml1N  |-  ( ( ( 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 ) ) )

Proof of Theorem cdleml1N
StepHypRef Expression
1 simp1 955 . . . 4  |-  ( ( ( 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 988 . . . 4  |-  ( ( ( 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 990 . . . 4  |-  ( ( ( 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 eqid 2296 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
5 cdleml1.h . . . . 5  |-  H  =  ( LHyp `  K
)
6 cdleml1.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
7 cdleml1.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
8 cdleml1.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
94, 5, 6, 7, 8tendotp 31572 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  f  e.  T
)  ->  ( R `  ( U `  f
) ) ( le
`  K ) ( R `  f ) )
101, 2, 3, 9syl3anc 1182 . . 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 ) ) )  ->  ( R `  ( U `  f
) ) ( le
`  K ) ( R `  f ) )
11 simp1l 979 . . . . 5  |-  ( ( ( 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 )
12 hlatl 30172 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
1311, 12syl 15 . . . 4  |-  ( ( ( 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.  AtLat
)
145, 6, 8tendocl 31578 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  f  e.  T
)  ->  ( U `  f )  e.  T
)
151, 2, 3, 14syl3anc 1182 . . . . 5  |-  ( ( ( 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
)
16 simp32 992 . . . . 5  |-  ( ( ( 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 )  =/=  (  _I  |`  B ) )
17 cdleml1.b . . . . . 6  |-  B  =  ( Base `  K
)
18 eqid 2296 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
1917, 18, 5, 6, 7trlnidat 30984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  f )  e.  T  /\  ( U `  f
)  =/=  (  _I  |`  B ) )  -> 
( R `  ( U `  f )
)  e.  ( Atoms `  K ) )
201, 15, 16, 19syl3anc 1182 . . . 4  |-  ( ( ( 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
) )  e.  (
Atoms `  K ) )
21 simp31 991 . . . . 5  |-  ( ( ( 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  =/=  (  _I  |`  B ) )
2217, 18, 5, 6, 7trlnidat 30984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  ( R `  f )  e.  (
Atoms `  K ) )
231, 3, 21, 22syl3anc 1182 . . . 4  |-  ( ( ( 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 `  f )  e.  (
Atoms `  K ) )
244, 18atcmp 30123 . . . 4  |-  ( ( K  e.  AtLat  /\  ( R `  ( U `  f ) )  e.  ( Atoms `  K )  /\  ( R `  f
)  e.  ( Atoms `  K ) )  -> 
( ( R `  ( U `  f ) ) ( le `  K ) ( R `
 f )  <->  ( R `  ( U `  f
) )  =  ( R `  f ) ) )
2513, 20, 23, 24syl3anc 1182 . . 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 ) ) )  ->  ( ( R `  ( U `  f ) ) ( le `  K ) ( R `  f
)  <->  ( R `  ( U `  f ) )  =  ( R `
 f ) ) )
2610, 25mpbid 201 . 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 `  f ) )
27 simp22 989 . . . 4  |-  ( ( ( 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 )
284, 5, 6, 7, 8tendotp 31572 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  f  e.  T
)  ->  ( R `  ( V `  f
) ) ( le
`  K ) ( R `  f ) )
291, 27, 3, 28syl3anc 1182 . . 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 ) ) )  ->  ( R `  ( V `  f
) ) ( le
`  K ) ( R `  f ) )
305, 6, 8tendocl 31578 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  f  e.  T
)  ->  ( V `  f )  e.  T
)
311, 27, 3, 30syl3anc 1182 . . . . 5  |-  ( ( ( 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
)
32 simp33 993 . . . . 5  |-  ( ( ( 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 )  =/=  (  _I  |`  B ) )
3317, 18, 5, 6, 7trlnidat 30984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( V `  f )  e.  T  /\  ( V `  f
)  =/=  (  _I  |`  B ) )  -> 
( R `  ( V `  f )
)  e.  ( Atoms `  K ) )
341, 31, 32, 33syl3anc 1182 . . . 4  |-  ( ( ( 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 `  ( V `  f
) )  e.  (
Atoms `  K ) )
354, 18atcmp 30123 . . . 4  |-  ( ( K  e.  AtLat  /\  ( R `  ( V `  f ) )  e.  ( Atoms `  K )  /\  ( R `  f
)  e.  ( Atoms `  K ) )  -> 
( ( R `  ( V `  f ) ) ( le `  K ) ( R `
 f )  <->  ( R `  ( V `  f
) )  =  ( R `  f ) ) )
3613, 34, 23, 35syl3anc 1182 . . 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 ) ) )  ->  ( ( R `  ( V `  f ) ) ( le `  K ) ( R `  f
)  <->  ( R `  ( V `  f ) )  =  ( R `
 f ) ) )
3729, 36mpbid 201 . 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 `  ( V `  f
) )  =  ( R `  f ) )
3826, 37eqtr4d 2331 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 ) ) )  ->  ( R `  ( U `  f
) )  =  ( R `  ( V `
 f ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039    _I cid 4320    |` cres 4707   ` cfv 5271   Basecbs 13164   lecple 13231   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  cdleml2N  31788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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