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Theorem cdlemg27a 36834
Description: For use with case when  ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) or  ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) is zero, letting us establish  -.  z  .<_  W  /\  z  .<_  ( P 
.\/  v ) via 4atex 36216. TODO: Fix comment. (Contributed by NM, 28-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg27a  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  -.  ( R `  F ) 
.<_  ( P  .\/  z
) )

Proof of Theorem cdlemg27a
StepHypRef Expression
1 simp11 1024 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 1025 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp31 1030 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  v  =/=  ( R `  F
) )
4 simp13 1026 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  (
v  e.  A  /\  v  .<_  W ) )
5 simp2r 1021 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  F  e.  T )
6 simp33 1032 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  ( F `  P )  =/=  P )
7 cdlemg12.l . . . . 5  |-  .<_  =  ( le `  K )
8 cdlemg12.a . . . . 5  |-  A  =  ( Atoms `  K )
9 cdlemg12.h . . . . 5  |-  H  =  ( LHyp `  K
)
10 cdlemg12.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
11 cdlemg12b.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
127, 8, 9, 10, 11trlat 36310 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
131, 2, 5, 6, 12syl112anc 1230 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  ( R `  F )  e.  A )
147, 9, 10, 11trlle 36325 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
151, 5, 14syl2anc 659 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  ( R `  F )  .<_  W )
16 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
177, 16, 8, 9lhp2atnle 36173 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  v  =/=  ( R `  F )
)  /\  ( v  e.  A  /\  v  .<_  W )  /\  (
( R `  F
)  e.  A  /\  ( R `  F ) 
.<_  W ) )  ->  -.  ( R `  F
)  .<_  ( P  .\/  v ) )
181, 2, 3, 4, 13, 15, 17syl312anc 1247 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  -.  ( R `  F ) 
.<_  ( P  .\/  v
) )
19 simp11l 1105 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  K  e.  HL )
20 simp12l 1107 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  P  e.  A )
21 simp13l 1109 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  v  e.  A )
227, 16, 8hlatlej1 35515 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  v  e.  A )  ->  P  .<_  ( P  .\/  v ) )
2319, 20, 21, 22syl3anc 1226 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  P  .<_  ( P  .\/  v
) )
24 simp32 1031 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  z  .<_  ( P  .\/  v
) )
25 hllat 35504 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2619, 25syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  K  e.  Lat )
27 eqid 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2827, 8atbase 35430 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2920, 28syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  P  e.  ( Base `  K
) )
30 simp2l 1020 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  z  e.  A )
3127, 8atbase 35430 . . . . . 6  |-  ( z  e.  A  ->  z  e.  ( Base `  K
) )
3230, 31syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  z  e.  ( Base `  K
) )
3327, 16, 8hlatjcl 35507 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  v  e.  A )  ->  ( P  .\/  v
)  e.  ( Base `  K ) )
3419, 20, 21, 33syl3anc 1226 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  ( P  .\/  v )  e.  ( Base `  K
) )
3527, 7, 16latjle12 15894 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  z  e.  ( Base `  K )  /\  ( P  .\/  v )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( P  .\/  v )  /\  z  .<_  ( P 
.\/  v ) )  <-> 
( P  .\/  z
)  .<_  ( P  .\/  v ) ) )
3626, 29, 32, 34, 35syl13anc 1228 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  (
( P  .<_  ( P 
.\/  v )  /\  z  .<_  ( P  .\/  v ) )  <->  ( P  .\/  z )  .<_  ( P 
.\/  v ) ) )
3723, 24, 36mpbi2and 919 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  ( P  .\/  z )  .<_  ( P  .\/  v ) )
3827, 8atbase 35430 . . . . 5  |-  ( ( R `  F )  e.  A  ->  ( R `  F )  e.  ( Base `  K
) )
3913, 38syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  ( R `  F )  e.  ( Base `  K
) )
4027, 16, 8hlatjcl 35507 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  z  e.  A )  ->  ( P  .\/  z
)  e.  ( Base `  K ) )
4119, 20, 30, 40syl3anc 1226 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  ( P  .\/  z )  e.  ( Base `  K
) )
4227, 7lattr 15888 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( R `  F )  e.  (
Base `  K )  /\  ( P  .\/  z
)  e.  ( Base `  K )  /\  ( P  .\/  v )  e.  ( Base `  K
) ) )  -> 
( ( ( R `
 F )  .<_  ( P  .\/  z )  /\  ( P  .\/  z )  .<_  ( P 
.\/  v ) )  ->  ( R `  F )  .<_  ( P 
.\/  v ) ) )
4326, 39, 41, 34, 42syl13anc 1228 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  (
( ( R `  F )  .<_  ( P 
.\/  z )  /\  ( P  .\/  z ) 
.<_  ( P  .\/  v
) )  ->  ( R `  F )  .<_  ( P  .\/  v
) ) )
4437, 43mpan2d 672 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  (
( R `  F
)  .<_  ( P  .\/  z )  ->  ( R `  F )  .<_  ( P  .\/  v
) ) )
4518, 44mtod 177 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( z  e.  A  /\  F  e.  T
)  /\  ( v  =/=  ( R `  F
)  /\  z  .<_  ( P  .\/  v )  /\  ( F `  P )  =/=  P
) )  ->  -.  ( R `  F ) 
.<_  ( P  .\/  z
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776   Latclat 15877   Atomscatm 35404   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   trLctrl 36299
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-map 7414  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-psubsp 35643  df-pmap 35644  df-padd 35936  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300
This theorem is referenced by:  cdlemg28a  36835
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