Theorem cdlemefr29clN 34045

Description: Show closure of the unique element in cdleme29c 34014. TODO fix comment. TODO Not needed? (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)

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 )
cdlemefr29cl.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) )

Assertion

Ref	Expression
cdlemefr29clN	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> O e. B )

Distinct variable groups:   A, s   .\/ , s   .<_ , s   ./\ , s   P, s   Q, s   R, s   U, s   W, s, z   H, s   K, s   z, A   B, s, z   z, H   z, .\/   z, K   z, .<_   z, ./\   z, N   z, P   z, Q   z, R   z, W

Allowed substitution hints:   C( z, s)   U( z)   I( z, s)   N( s)   O( z, s)

Proof of Theorem cdlemefr29clN

Step	Hyp	Ref	Expression
1		cdlemefr27.b	. 2 |- B = ( Base ` K )
2		cdlemefr27.l	. 2 |- .<_ = ( le ` K )
3		cdlemefr27.j	. 2 |- .\/ = ( join ` K )
4		cdlemefr27.m	. 2 |- ./\ = ( meet ` K )
5		cdlemefr27.a	. 2 |- A = ( Atoms ` K )
6		cdlemefr27.h	. 2 |- H = ( LHyp ` K )
7		breq1 4398	. . 3 |- ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) )
8	7	notbid 301	. 2 |- ( s = R -> ( -. s .<_ ( P .\/ Q ) <-> -. R .<_ ( P .\/ Q ) ) )
9		simp11 1060	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> ( K e. HL /\ W e. H ) )
10		simp12l 1143	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> P e. A )
11		simp13l 1145	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> Q e. A )
12		simp3l 1058	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> s e. A )
13		simp3rr 1104	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> -. s .<_ ( P .\/ Q ) )
14		simp2 1031	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> P =/= Q )
15		cdlemefr27.u	. . . 4 |- U = ( ( P .\/ Q ) ./\ W )
16		cdlemefr27.c	. . . 4 |- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
17		cdlemefr27.n	. . . 4 |- N = if ( s .<_ ( P .\/ Q ) , I , C )
18	1, 2, 3, 4, 5, 6, 15, 16, 17	cdlemefr27cl 34041	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
19	9, 10, 11, 12, 13, 14, 18	syl33anc 1307	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) ) ) -> N e. B )
20	1, 2, 3, 4, 5, 6, 15, 16, 17	cdlemefr32snb 34043	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N e. B )
21		cdlemefr29cl.o	. 2 |- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) )
22	1, 2, 3, 4, 5, 6, 8, 19, 20, 21	cdlemefrs29clN 34037	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> O e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   ifcif 3872   class class class wbr 4395   ` cfv 5589   iota_crio 6269  (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-rmo 2764  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-iin 4272  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-mpt2 6313  df-1st 6812  df-2nd 6813  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624

This theorem is referenced by: (None)