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Theorem cdleml1N 31470
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 957 . . . 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 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 ) ) )  ->  U  e.  E )
3 simp23 992 . . . 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 2412 . . . . 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 31255 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  f  e.  T
)  ->  ( R `  ( U `  f
) ) ( le
`  K ) ( R `  f ) )
101, 2, 3, 9syl3anc 1184 . . 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 981 . . . . 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 29855 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
1311, 12syl 16 . . . 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 31261 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  f  e.  T
)  ->  ( U `  f )  e.  T
)
151, 2, 3, 14syl3anc 1184 . . . . 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 994 . . . . 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 2412 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
1917, 18, 5, 6, 7trlnidat 30667 . . . . 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 1184 . . . 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 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 ) ) )  ->  f  =/=  (  _I  |`  B ) )
2217, 18, 5, 6, 7trlnidat 30667 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  f  =/=  (  _I  |`  B ) )  ->  ( R `  f )  e.  (
Atoms `  K ) )
231, 3, 21, 22syl3anc 1184 . . . 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 29806 . . . 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 1184 . . 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 202 . 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 991 . . . 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 31255 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  f  e.  T
)  ->  ( R `  ( V `  f
) ) ( le
`  K ) ( R `  f ) )
291, 27, 3, 28syl3anc 1184 . . 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 31261 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  f  e.  T
)  ->  ( V `  f )  e.  T
)
311, 27, 3, 30syl3anc 1184 . . . . 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 995 . . . . 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 30667 . . . . 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 1184 . . . 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 29806 . . . 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 1184 . . 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 202 . 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 2447 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180    _I cid 4461    |` cres 4847   ` cfv 5421   Basecbs 13432   lecple 13499   Atomscatm 29758   AtLatcal 29759   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   trLctrl 30652   TEndoctendo 31246
This theorem is referenced by:  cdleml2N  31471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-map 6987  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653  df-tendo 31249
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