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Theorem cdlemefr27cl 34041
Description: Part of proof of Lemma E in [Crawley] p. 113. Closure of N. (Contributed by NM, 23-Mar-2013.)
Hypotheses
Ref Expression
cdlemefr27.b |- B = ( Base ` K )
cdlemefr27.l |- .<_ = ( le ` K )
cdlemefr27.j |- .\/ = ( join ` K )
cdlemefr27.m |- ./\ = ( meet ` K )
cdlemefr27.a |- A = ( Atoms ` K )
cdlemefr27.h |- H = ( LHyp ` K )
cdlemefr27.u |- U = ( ( P .\/ Q ) ./\ W )
cdlemefr27.c |- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdlemefr27.n |- N = if ( s .<_ ( P .\/ Q ) , I , C )
Assertion
Ref Expression
cdlemefr27cl |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
Proof of Theorem cdlemefr27cl
StepHypRef Expression
1 cdlemefr27.n . . 3 |- N = if ( s .<_ ( P .\/ Q ) , I , C )
2 simpr2 1037 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> -. s .<_ ( P .\/ Q ) )
32iffalsed 3883 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> if ( s .<_ ( P .\/ Q ) , I , C ) = C )
41, 3syl5eq 2517 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N = C )
5 simpl1l 1081 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> K e. HL )
6 simpl1r 1082 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> W e. H )
7 simpl2 1034 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> P e. A )
8 simpl3 1035 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> Q e. A )
9 simpr1 1036 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> s e. A )
10 cdlemefr27.l . . . 4 |- .<_ = ( le ` K )
11 cdlemefr27.j . . . 4 |- .\/ = ( join ` K )
12 cdlemefr27.m . . . 4 |- ./\ = ( meet ` K )
13 cdlemefr27.a . . . 4 |- A = ( Atoms ` K )
14 cdlemefr27.h . . . 4 |- H = ( LHyp ` K )
15 cdlemefr27.u . . . 4 |- U = ( ( P .\/ Q ) ./\ W )
16 cdlemefr27.c . . . 4 |- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
17 cdlemefr27.b . . . 4 |- B = ( Base ` K )
1810, 11, 12, 13, 14, 15, 16, 17cdleme1b 33863 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ s e. A ) ) -> C e. B )
195, 6, 7, 8, 9, 18syl23anc 1299 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> C e. B )
204, 19eqeltrd 2549 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   ifcif 3872   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   Atomscatm 32900   HLchlt 32987   LHypclh 33620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-lat 16370  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-lhyp 33624
This theorem is referenced by:  cdlemefr29bpre0N  34044  cdlemefr29clN  34045  cdlemefr32fvaN  34047  cdlemefr32fva1  34048
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