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Theorem cdleme1b 30712
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma showing  F is a lattice element.  F represents their f(r). (Contributed by NM, 6-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l  |-  .<_  =  ( le `  K )
cdleme1.j  |-  .\/  =  ( join `  K )
cdleme1.m  |-  ./\  =  ( meet `  K )
cdleme1.a  |-  A  =  ( Atoms `  K )
cdleme1.h  |-  H  =  ( LHyp `  K
)
cdleme1.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme1.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
cdleme1.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
cdleme1b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  F  e.  B )

Proof of Theorem cdleme1b
StepHypRef Expression
1 cdleme1.f . 2  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
2 hllat 29850 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  K  e.  Lat )
4 simpr3 965 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  R  e.  A )
5 cdleme1.b . . . . . 6  |-  B  =  ( Base `  K
)
6 cdleme1.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 29776 . . . . 5  |-  ( R  e.  A  ->  R  e.  B )
84, 7syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  R  e.  B )
9 cdleme1.l . . . . . 6  |-  .<_  =  ( le `  K )
10 cdleme1.j . . . . . 6  |-  .\/  =  ( join `  K )
11 cdleme1.m . . . . . 6  |-  ./\  =  ( meet `  K )
12 cdleme1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
13 cdleme1.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
149, 10, 11, 6, 12, 13, 5cdleme0aa 30696 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  B )
15143adant3r3 1164 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  U  e.  B )
165, 10latjcl 14438 . . . 4  |-  ( ( K  e.  Lat  /\  R  e.  B  /\  U  e.  B )  ->  ( R  .\/  U
)  e.  B )
173, 8, 15, 16syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( R  .\/  U
)  e.  B )
18 simpr2 964 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  Q  e.  A )
195, 6atbase 29776 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
2018, 19syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  Q  e.  B )
21 simpr1 963 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  P  e.  A )
225, 6atbase 29776 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
2321, 22syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  P  e.  B )
245, 10latjcl 14438 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  R  e.  B )  ->  ( P  .\/  R
)  e.  B )
253, 23, 8, 24syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( P  .\/  R
)  e.  B )
265, 12lhpbase 30484 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2726ad2antlr 708 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  W  e.  B )
285, 11latmcl 14439 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  R
)  ./\  W )  e.  B )
293, 25, 27, 28syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( ( P  .\/  R )  ./\  W )  e.  B )
305, 10latjcl 14438 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  ( ( P  .\/  R )  ./\  W )  e.  B )  ->  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  B
)
313, 20, 29, 30syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( Q  .\/  (
( P  .\/  R
)  ./\  W )
)  e.  B )
325, 11latmcl 14439 . . 3  |-  ( ( K  e.  Lat  /\  ( R  .\/  U )  e.  B  /\  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  B
)  ->  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )  e.  B )
333, 17, 31, 32syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )  e.  B
)
341, 33syl5eqel 2492 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  F  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   Latclat 14433   Atomscatm 29750   HLchlt 29837   LHypclh 30470
This theorem is referenced by:  cdleme3c  30716  cdleme4a  30725  cdleme5  30726  cdleme7e  30733  cdleme11  30756  cdleme15  30764  cdleme22gb  30780  cdleme19b  30790  cdleme19e  30793  cdleme20d  30798  cdleme20j  30804  cdleme20k  30805  cdleme20l2  30807  cdleme20l  30808  cdleme20m  30809  cdleme22e  30830  cdleme22eALTN  30831  cdleme22f  30832  cdleme27cl  30852  cdlemefr27cl  30889  cdleme35fnpq  30935
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-lat 14434  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-lhyp 30474
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