Theorem cdlemefr27cl 33402

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 1004	. . . 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	2	iffalsed 3895	. . 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 )
4	1, 3	syl5eq 2455	. 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 1048	. . 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 1049	. . 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 1001	. . 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 1002	. . 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 1003	. . 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 )
18	10, 11, 12, 13, 14, 15, 16, 17	cdleme1b 33224	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ s e. A ) ) -> C e. B )
19	5, 6, 7, 8, 9, 18	syl23anc 1237	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> C e. B )
20	4, 19	eqeltrd 2490	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 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   ifcif 3884   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   meetcmee 15896   Atomscatm 32261   HLchlt 32348   LHypclh 32981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-lat 15998  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-lhyp 32985

This theorem is referenced by:  cdlemefr29bpre0N  33405  cdlemefr29clN  33406  cdlemefr32fvaN  33408  cdlemefr32fva1  33409