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Theorem cdleme20bN 33585
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). We show v  \/ s2 = v  \/ t2. (Contributed by NM, 15-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme19.l  |-  .<_  =  ( le `  K )
cdleme19.j  |-  .\/  =  ( join `  K )
cdleme19.m  |-  ./\  =  ( meet `  K )
cdleme19.a  |-  A  =  ( Atoms `  K )
cdleme19.h  |-  H  =  ( LHyp `  K
)
cdleme19.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme19.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme19.g  |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
cdleme19.d  |-  D  =  ( ( R  .\/  S )  ./\  W )
cdleme19.y  |-  Y  =  ( ( R  .\/  T )  ./\  W )
cdleme20.v  |-  V  =  ( ( S  .\/  T )  ./\  W )
Assertion
Ref Expression
cdleme20bN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( V  .\/  D )  =  ( V  .\/  Y
) )

Proof of Theorem cdleme20bN
StepHypRef Expression
1 simp1l 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  HL )
2 hllat 32637 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  Lat )
4 simp22l 1124 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  S  e.  A )
5 eqid 2429 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme19.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 32563 . . . . 5  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
84, 7syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  S  e.  ( Base `  K
) )
9 simp21 1038 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  A )
105, 6atbase 32563 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
119, 10syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  ( Base `  K
) )
12 simp23l 1126 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  T  e.  A )
135, 6atbase 32563 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1412, 13syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  T  e.  ( Base `  K
) )
15 cdleme19.j . . . . 5  |-  .\/  =  ( join `  K )
165, 15latj31 16296 . . . 4  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  T  e.  ( Base `  K
) ) )  -> 
( ( S  .\/  R )  .\/  T )  =  ( ( T 
.\/  R )  .\/  S ) )
173, 8, 11, 14, 16syl13anc 1266 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( S  .\/  R
)  .\/  T )  =  ( ( T 
.\/  R )  .\/  S ) )
1817oveq1d 6320 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( ( S  .\/  R )  .\/  T ) 
./\  W )  =  ( ( ( T 
.\/  R )  .\/  S )  ./\  W )
)
19 simp1r 1030 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  H )
20 simp22r 1125 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  S  .<_  W )
21 simp31 1041 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
22 simp33 1043 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  .<_  ( P  .\/  Q
) )
23 cdleme19.l . . . 4  |-  .<_  =  ( le `  K )
24 cdleme19.m . . . 4  |-  ./\  =  ( meet `  K )
25 cdleme19.h . . . 4  |-  H  =  ( LHyp `  K
)
26 cdleme19.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
27 cdleme19.f . . . 4  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
28 cdleme19.g . . . 4  |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
29 cdleme19.d . . . 4  |-  D  =  ( ( R  .\/  S )  ./\  W )
30 cdleme19.y . . . 4  |-  Y  =  ( ( R  .\/  T )  ./\  W )
31 cdleme20.v . . . 4  |-  V  =  ( ( S  .\/  T )  ./\  W )
3223, 15, 24, 6, 25, 26, 27, 28, 29, 30, 31cdleme20aN 33584 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( V  .\/  D
)  =  ( ( ( S  .\/  R
)  .\/  T )  ./\  W ) )
331, 19, 9, 4, 20, 12, 21, 22, 32syl233anc 1293 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( V  .\/  D )  =  ( ( ( S 
.\/  R )  .\/  T )  ./\  W )
)
3415, 6hlatjcom 32641 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  =  ( T 
.\/  S ) )
351, 4, 12, 34syl3anc 1264 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( S  .\/  T )  =  ( T  .\/  S
) )
3635oveq1d 6320 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( S  .\/  T
)  ./\  W )  =  ( ( T 
.\/  S )  ./\  W ) )
3731, 36syl5eq 2482 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  V  =  ( ( T 
.\/  S )  ./\  W ) )
3837oveq1d 6320 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( V  .\/  Y )  =  ( ( ( T 
.\/  S )  ./\  W )  .\/  Y ) )
39 simp23r 1127 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  T  .<_  W )
40 simp32 1042 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  T  .<_  ( P  .\/  Q ) )
41 eqid 2429 . . . . 5  |-  ( ( T  .\/  S ) 
./\  W )  =  ( ( T  .\/  S )  ./\  W )
4223, 15, 24, 6, 25, 26, 28, 27, 30, 29, 41cdleme20aN 33584 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  T  e.  A  /\  -.  T  .<_  W )  /\  ( S  e.  A  /\  -.  T  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( T 
.\/  S )  ./\  W )  .\/  Y )  =  ( ( ( T  .\/  R ) 
.\/  S )  ./\  W ) )
431, 19, 9, 12, 39, 4, 40, 22, 42syl233anc 1293 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( ( T  .\/  S )  ./\  W )  .\/  Y )  =  ( ( ( T  .\/  R )  .\/  S ) 
./\  W ) )
4438, 43eqtrd 2470 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( V  .\/  Y )  =  ( ( ( T 
.\/  R )  .\/  S )  ./\  W )
)
4518, 33, 443eqtr4d 2480 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( V  .\/  D )  =  ( V  .\/  Y
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   Latclat 16242   Atomscatm 32537   HLchlt 32624   LHypclh 33257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32450  df-ol 32452  df-oml 32453  df-covers 32540  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625  df-psubsp 32776  df-pmap 32777  df-padd 33069  df-lhyp 33261
This theorem is referenced by: (None)
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