Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme0cp Structured version   Unicode version

Theorem cdleme0cp 36336
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 36720- swap consequent equality; make antecedent use df-3an 973. (Contributed by NM, 13-Jun-2012.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme0cp  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )

Proof of Theorem cdleme0cp
StepHypRef Expression
1 cdleme0.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
21oveq2i 6281 . 2  |-  ( P 
.\/  U )  =  ( P  .\/  (
( P  .\/  Q
)  ./\  W )
)
3 simpll 751 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  K  e.  HL )
4 simprll 761 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  P  e.  A )
5 hllat 35485 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
65ad2antrr 723 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  K  e.  Lat )
7 eqid 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8atbase 35411 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
104, 9syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  P  e.  ( Base `  K )
)
11 simprr 755 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  Q  e.  A )
127, 8atbase 35411 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1311, 12syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  Q  e.  ( Base `  K )
)
14 cdleme0.j . . . . . 6  |-  .\/  =  ( join `  K )
157, 14latjcl 15880 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
166, 10, 13, 15syl3anc 1226 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
17 cdleme0.h . . . . . 6  |-  H  =  ( LHyp `  K
)
187, 17lhpbase 36119 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1918ad2antlr 724 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  W  e.  ( Base `  K )
)
20 cdleme0.l . . . . . 6  |-  .<_  =  ( le `  K )
2120, 14, 8hlatlej1 35496 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
223, 4, 11, 21syl3anc 1226 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  P  .<_  ( P  .\/  Q ) )
23 cdleme0.m . . . . 5  |-  ./\  =  ( meet `  K )
247, 20, 14, 23, 8atmod3i1 35985 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  ( P  .\/  W ) ) )
253, 4, 16, 19, 22, 24syl131anc 1239 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  =  ( ( P  .\/  Q ) 
./\  ( P  .\/  W ) ) )
26 eqid 2454 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
2720, 14, 26, 8, 17lhpjat2 36142 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
2827adantrr 714 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  W )  =  ( 1. `  K ) )
2928oveq2d 6286 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( P  .\/  W ) )  =  ( ( P  .\/  Q ) 
./\  ( 1. `  K ) ) )
30 hlol 35483 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
3130ad2antrr 723 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  K  e.  OL )
327, 23, 26olm11 35349 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
3331, 16, 32syl2anc 659 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( 1. `  K ) )  =  ( P 
.\/  Q ) )
3425, 29, 333eqtrd 2499 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  =  ( P 
.\/  Q ) )
352, 34syl5eq 2507 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   1.cp1 15867   Latclat 15874   OLcol 35296   Atomscatm 35385   HLchlt 35472   LHypclh 36105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109
This theorem is referenced by:  cdleme11c  36383  cdlemg4b1  36732  cdlemg4g  36739  cdlemg13a  36774  cdlemg17a  36784  cdlemg17f  36789  cdlemg18b  36802  cdlemg18c  36803
  Copyright terms: Public domain W3C validator