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Theorem cdleme3e 35519
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 35523 and cdleme3 35524. (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 ) ) )
cdleme3.3 |- V = ( ( P .\/ R ) ./\ W )
Assertion
Ref Expression
cdleme3e |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> V e. A )
Proof of Theorem cdleme3e
StepHypRef Expression
1 cdleme3.3 . 2 |- V = ( ( P .\/ R ) ./\ W )
2 simpl 473 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( K e. HL /\ W e. H ) )
3 simpr1 1067 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
4 simpr3l 1122 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R e. A )
5 hllat 34650 . . . . . 6 |- ( K e. HL -> K e. Lat )
65ad2antrr 762 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> K e. Lat )
7 eqid 2622 . . . . . . 7 |- ( Base ` K ) = ( Base ` K )
8 cdleme1.a . . . . . . 7 |- A = ( Atoms ` K )
97, 8atbase 34576 . . . . . 6 |- ( R e. A -> R e. ( Base ` K ) )
104, 9syl 17 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R e. ( Base ` K ) )
11 simpr1l 1118 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> P e. A )
127, 8atbase 34576 . . . . . 6 |- ( P e. A -> P e. ( Base ` K ) )
1311, 12syl 17 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> P e. ( Base ` K ) )
14 simpr2 1068 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> Q e. A )
157, 8atbase 34576 . . . . . 6 |- ( Q e. A -> Q e. ( Base ` K ) )
1614, 15syl 17 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> Q e. ( Base ` K ) )
17 simpr3r 1123 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> -. R .<_ ( P .\/ Q ) )
18 cdleme1.l . . . . . 6 |- .<_ = ( le ` K )
19 cdleme1.j . . . . . 6 |- .\/ = ( join ` K )
207, 18, 19latnlej1l 17069 . . . . 5 |- ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P )
216, 10, 13, 16, 17, 20syl131anc 1339 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R =/= P )
2221necomd 2849 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> P =/= R )
23 cdleme1.m . . . 4 |- ./\ = ( meet ` K )
24 cdleme1.h . . . 4 |- H = ( LHyp ` K )
2518, 19, 23, 8, 24lhpat 35329 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ P =/= R ) ) -> ( ( P .\/ R ) ./\ W ) e. A )
262, 3, 4, 22, 25syl112anc 1330 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( ( P .\/ R ) ./\ W ) e. A )
271, 26syl5eqel 2705 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ ( P .\/ Q ) ) ) ) -> V e. A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Latclat 17045   Atomscatm 34550   HLchlt 34637   LHypclh 35270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274
This theorem is referenced by:  cdleme3g  35521  cdleme3h  35522
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