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Theorem cdleme20bN 35124
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 1020 . . . . 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 34178 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . 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 1115 . . . . 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 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme19.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 34104 . . . . 5  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
84, 7syl 16 . . . 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 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 ) ) )  ->  R  e.  A )
105, 6atbase 34104 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
119, 10syl 16 . . . 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 1117 . . . . 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 34104 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1412, 13syl 16 . . . 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 15586 . . . 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 1230 . . 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 6299 . 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 1021 . . 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 1116 . . 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 1032 . . 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 1034 . . 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 35123 . . 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 1257 . 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 34182 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  =  ( T 
.\/  S ) )
351, 4, 12, 34syl3anc 1228 . . . . . 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 6299 . . . . 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 2520 . . . 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 6299 . . 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 1118 . . . 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 1033 . . . 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 2467 . . . . 5  |-  ( ( T  .\/  S ) 
./\  W )  =  ( ( T  .\/  S )  ./\  W )
4223, 15, 24, 6, 25, 26, 28, 27, 30, 29, 41cdleme20aN 35123 . . . 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 1257 . . 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 2508 . 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 2518 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   lecple 14562   joincjn 15431   meetcmee 15432   Latclat 15532   Atomscatm 34078   HLchlt 34165   LHypclh 34798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-psubsp 34317  df-pmap 34318  df-padd 34610  df-lhyp 34802
This theorem is referenced by: (None)
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