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Theorem cdleme20aN 30498
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (Contributed by NM, 14-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
cdleme20aN  |-  ( ( ( 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 ) )

Proof of Theorem cdleme20aN
StepHypRef Expression
1 cdleme20.v . . 3  |-  V  =  ( ( S  .\/  T )  ./\  W )
21oveq1i 5868 . 2  |-  ( V 
.\/  D )  =  ( ( ( S 
.\/  T )  ./\  W )  .\/  D )
3 simp1l 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  HL )
4 simp1r 980 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  H )
5 simp22 989 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  S  e.  A )
6 simp23 990 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  S  .<_  W )
7 simp21 988 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  A )
8 simp33 993 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  .<_  ( P  .\/  Q ) )
9 simp32 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  S  .<_  ( P 
.\/  Q ) )
10 cdleme19.l . . . . . 6  |-  .<_  =  ( le `  K )
11 cdleme19.j . . . . . 6  |-  .\/  =  ( join `  K )
12 cdleme19.m . . . . . 6  |-  ./\  =  ( meet `  K )
13 cdleme19.a . . . . . 6  |-  A  =  ( Atoms `  K )
14 cdleme19.h . . . . . 6  |-  H  =  ( LHyp `  K
)
15 cdleme19.d . . . . . 6  |-  D  =  ( ( R  .\/  S )  ./\  W )
1610, 11, 12, 13, 14, 15cdlemeda 30487 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  D  e.  A )
173, 4, 5, 6, 7, 8, 9, 16syl223anc 1208 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  D  e.  A )
18 simp31 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  T  e.  A )
19 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2019, 11, 13hlatjcl 29556 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
213, 5, 18, 20syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
2219, 14lhpbase 30187 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
234, 22syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  ( Base `  K ) )
24 hllat 29553 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
253, 24syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  Lat )
2619, 11, 13hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
273, 7, 5, 26syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( R  .\/  S
)  e.  ( Base `  K ) )
2819, 10, 12latmle2 14183 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  .<_  W )
2925, 27, 23, 28syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( R  .\/  S )  ./\  W )  .<_  W )
3015, 29syl5eqbr 4056 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  D  .<_  W )
3119, 10, 11, 12, 13atmod4i1 30055 . . . 4  |-  ( ( K  e.  HL  /\  ( D  e.  A  /\  ( S  .\/  T
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  D  .<_  W )  ->  (
( ( S  .\/  T )  ./\  W )  .\/  D )  =  ( ( ( S  .\/  T )  .\/  D ) 
./\  W ) )
323, 17, 21, 23, 30, 31syl131anc 1195 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( S 
.\/  T )  ./\  W )  .\/  D )  =  ( ( ( S  .\/  T ) 
.\/  D )  ./\  W ) )
3310, 11, 12, 13, 14, 15cdleme10 30443 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  D )  =  ( S  .\/  R ) )
343, 4, 7, 5, 6, 33syl212anc 1192 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( S  .\/  D
)  =  ( S 
.\/  R ) )
3534oveq1d 5873 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( S  .\/  D )  .\/  T )  =  ( ( S 
.\/  R )  .\/  T ) )
3611, 13hlatj32 29561 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  D  e.  A  /\  T  e.  A
) )  ->  (
( S  .\/  D
)  .\/  T )  =  ( ( S 
.\/  T )  .\/  D ) )
373, 5, 17, 18, 36syl13anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( S  .\/  D )  .\/  T )  =  ( ( S 
.\/  T )  .\/  D ) )
3835, 37eqtr3d 2317 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( S  .\/  R )  .\/  T )  =  ( ( S 
.\/  T )  .\/  D ) )
3938oveq1d 5873 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( S 
.\/  R )  .\/  T )  ./\  W )  =  ( ( ( S  .\/  T ) 
.\/  D )  ./\  W ) )
4032, 39eqtr4d 2318 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  S  .<_  ( P 
.\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( S 
.\/  T )  ./\  W )  .\/  D )  =  ( ( ( S  .\/  R ) 
.\/  T )  ./\  W ) )
412, 40syl5eq 2327 1  |-  ( ( ( 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 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme20bN  30499
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177
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