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Theorem cdleme35c 29909
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT (Contributed by NM, 10-Mar-2013.)
Hypotheses
Ref Expression
cdleme35.l  |-  .<_  =  ( le `  K )
cdleme35.j  |-  .\/  =  ( join `  K )
cdleme35.m  |-  ./\  =  ( meet `  K )
cdleme35.a  |-  A  =  ( Atoms `  K )
cdleme35.h  |-  H  =  ( LHyp `  K
)
cdleme35.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme35.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
Assertion
Ref Expression
cdleme35c  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  F )  =  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
) )

Proof of Theorem cdleme35c
StepHypRef Expression
1 cdleme35.f . . 3  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
21oveq2i 5832 . 2  |-  ( Q 
.\/  F )  =  ( Q  .\/  (
( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
3 simp11l 1068 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  K  e.  HL )
4 simp13l 1072 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  Q  e.  A )
5 simp2rl 1026 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  R  e.  A )
6 simp11 987 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 simp12 988 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp2l 983 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  P  =/=  Q )
9 cdleme35.l . . . . . . 7  |-  .<_  =  ( le `  K )
10 cdleme35.j . . . . . . 7  |-  .\/  =  ( join `  K )
11 cdleme35.m . . . . . . 7  |-  ./\  =  ( meet `  K )
12 cdleme35.a . . . . . . 7  |-  A  =  ( Atoms `  K )
13 cdleme35.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
14 cdleme35.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
159, 10, 11, 12, 13, 14cdleme0a 29669 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
166, 7, 4, 8, 15syl112anc 1188 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  U  e.  A )
17 eqid 2286 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1817, 10, 12hlatjcl 28825 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
193, 5, 16, 18syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( R  .\/  U )  e.  ( Base `  K
) )
20 hllat 28822 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213, 20syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  K  e.  Lat )
2217, 12atbase 28748 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
234, 22syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  Q  e.  ( Base `  K
) )
24 simp12l 1070 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  P  e.  A )
2517, 10, 12hlatjcl 28825 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
263, 24, 5, 25syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
27 simp11r 1069 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  W  e.  H )
2817, 13lhpbase 29456 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2927, 28syl 17 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  W  e.  ( Base `  K
) )
3017, 11latmcl 14153 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R )  ./\  W )  e.  ( Base `  K ) )
3121, 26, 29, 30syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  R
)  ./\  W )  e.  ( Base `  K
) )
3217, 10latjcl 14152 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  R
)  ./\  W )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  (
Base `  K )
)
3321, 23, 31, 32syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  (
Base `  K )
)
3417, 9, 10latlej1 14162 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  R
)  ./\  W )  e.  ( Base `  K
) )  ->  Q  .<_  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
) )
3521, 23, 31, 34syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  Q  .<_  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
) )
3617, 9, 10, 11, 12atmod1i1 29315 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( R  .\/  U
)  e.  ( Base `  K )  /\  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  (
Base `  K )
)  /\  Q  .<_  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )  ->  ( Q  .\/  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )  =  ( ( Q  .\/  ( R  .\/  U ) )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
373, 4, 19, 33, 35, 36syl131anc 1197 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( ( R 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) )  =  ( ( Q  .\/  ( R  .\/  U ) ) 
./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
389, 10, 11, 12, 13, 14, 1cdleme35b 29908 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  .<_  ( Q 
.\/  ( R  .\/  U ) ) )
3917, 10latjcl 14152 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( R  .\/  U )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( R  .\/  U ) )  e.  (
Base `  K )
)
4021, 23, 19, 39syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( R  .\/  U ) )  e.  (
Base `  K )
)
4117, 9, 11latleeqm2 14182 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) )  e.  ( Base `  K
)  /\  ( Q  .\/  ( R  .\/  U
) )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  .<_  ( Q 
.\/  ( R  .\/  U ) )  <->  ( ( Q  .\/  ( R  .\/  U ) )  ./\  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
) ) )
4221, 33, 40, 41syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( Q  .\/  (
( P  .\/  R
)  ./\  W )
)  .<_  ( Q  .\/  ( R  .\/  U ) )  <->  ( ( Q 
.\/  ( R  .\/  U ) )  ./\  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
) ) )
4338, 42mpbid 203 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( Q  .\/  ( R  .\/  U ) ) 
./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )  =  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )
4437, 43eqtrd 2318 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( ( R 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) )  =  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )
452, 44syl5eq 2330 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  F )  =  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1625    e. wcel 1687    =/= wne 2449   class class class wbr 4026   ` cfv 5223  (class class class)co 5821   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   Latclat 14147   Atomscatm 28722   HLchlt 28809   LHypclh 29442
This theorem is referenced by:  cdleme35d  29910
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-iota 6254  df-undef 6293  df-riota 6301  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28635  df-ol 28637  df-oml 28638  df-covers 28725  df-ats 28726  df-atl 28757  df-cvlat 28781  df-hlat 28810  df-psubsp 28961  df-pmap 28962  df-padd 29254  df-lhyp 29446
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