Theorem cdlemefr27cl 35074

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

Step	Hyp	Ref	Expression
1		cdlemefr27.n	. . 3 |- N = if ( s .<_ ( P .\/ Q ) , I , C )
2		simpr2 998	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> -. s .<_ ( P .\/ Q ) )
3		iffalse 3941	. . . 4 |- ( -. s .<_ ( P .\/ Q ) -> if ( s .<_ ( P .\/ Q ) , I , C ) = C )
4	2, 3	syl 16	. . 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 )
5	1, 4	syl5eq 2513	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N = C )
6		simpl1l 1042	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> K e. HL )
7		simpl1r 1043	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> W e. H )
8		simpl2 995	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> P e. A )
9		simpl3 996	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> Q e. A )
10		simpr1 997	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> s e. A )
11		cdlemefr27.l	. . . 4 |- .<_ = ( le ` K )
12		cdlemefr27.j	. . . 4 |- .\/ = ( join ` K )
13		cdlemefr27.m	. . . 4 |- ./\ = ( meet ` K )
14		cdlemefr27.a	. . . 4 |- A = ( Atoms ` K )
15		cdlemefr27.h	. . . 4 |- H = ( LHyp ` K )
16		cdlemefr27.u	. . . 4 |- U = ( ( P .\/ Q ) ./\ W )
17		cdlemefr27.c	. . . 4 |- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
18		cdlemefr27.b	. . . 4 |- B = ( Base ` K )
19	11, 12, 13, 14, 15, 16, 17, 18	cdleme1b 34897	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ s e. A ) ) -> C e. B )
20	6, 7, 8, 9, 10, 19	syl23anc 1230	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> C e. B )
21	5, 20	eqeltrd 2548	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   =/= wne 2655   ifcif 3932   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Atomscatm 33935   HLchlt 34022   LHypclh 34655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-lat 15522  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-lhyp 34659

This theorem is referenced by:  cdlemefr29bpre0N  35077  cdlemefr29clN  35078  cdlemefr32fvaN  35080  cdlemefr32fva1  35081