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Theorem cdlemg31a 30704
Description: TODO: fix comment. (Contributed by NM, 29-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 )
cdlemg31.n  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
Assertion
Ref Expression
cdlemg31a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  N  .<_  ( P  .\/  v ) )

Proof of Theorem cdlemg31a
StepHypRef Expression
1 cdlemg31.n . 2  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
2 simp1l 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  K  e.  HL )
3 hllat 29371 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  K  e.  Lat )
5 simp2l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  P  e.  A )
6 simp3l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  v  e.  A )
7 eqid 2316 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 cdlemg12.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdlemg12.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 29374 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  v  e.  A )  ->  ( P  .\/  v
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  ( P  .\/  v )  e.  (
Base `  K )
)
12 simp2r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  Q  e.  A )
137, 9atbase 29297 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1412, 13syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  Q  e.  ( Base `  K )
)
15 simp1 955 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
16 simp3r 984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  F  e.  T )
17 cdlemg12.h . . . . . 6  |-  H  =  ( LHyp `  K
)
18 cdlemg12.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
19 cdlemg12b.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
207, 17, 18, 19trlcl 30171 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
2115, 16, 20syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  ( R `  F )  e.  (
Base `  K )
)
227, 8latjcl 14205 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( R `  F )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( R `  F ) )  e.  ( Base `  K
) )
234, 14, 21, 22syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  ( Q  .\/  ( R `  F
) )  e.  (
Base `  K )
)
24 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
25 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
267, 24, 25latmle1 14231 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  v )  e.  ( Base `  K
)  /\  ( Q  .\/  ( R `  F
) )  e.  (
Base `  K )
)  ->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  .<_  ( P  .\/  v ) )
274, 11, 23, 26syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  .<_  ( P  .\/  v ) )
281, 27syl5eqbr 4093 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  N  .<_  ( P  .\/  v ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   meetcmee 14128   Latclat 14200   Atomscatm 29271   HLchlt 29358   LHypclh 29991   LTrncltrn 30108   trLctrl 30165
This theorem is referenced by:  cdlemg31c  30706  cdlemg33b0  30708  cdlemg33a  30713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-map 6817  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-lhyp 29995  df-laut 29996  df-ldil 30111  df-ltrn 30112  df-trl 30166
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