Theorem cdlemefr27cl 33964

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 1014	. . . 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 3891	. . 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 2496	. 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 1058	. . 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 1059	. . 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 1011	. . 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 1012	. . 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 1013	. . 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 33786	. . 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 1274	. 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 2528	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 371   /\ w3a 984   = wceq 1443   e. wcel 1886   =/= wne 2621   ifcif 3880   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   Basecbs 15114   lecple 15190   joincjn 16182   meetcmee 16183   Atomscatm 32823   HLchlt 32910   LHypclh 33543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-lat 16285  df-ats 32827  df-atl 32858  df-cvlat 32882  df-hlat 32911  df-lhyp 33547

This theorem is referenced by:  cdlemefr29bpre0N  33967  cdlemefr29clN  33968  cdlemefr32fvaN  33970  cdlemefr32fva1  33971