Theorem cdlemefr29bpre0N 35694

Description: TODO fix comment. (Contributed by NM, 28-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 )

Assertion

Ref	Expression
cdlemefr29bpre0N	|- ( ( ( ( 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 ) ) -> ( A. s e. A ( ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) <-> z = [_ R / s ]_ N ) )

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

Allowed substitution hints:   A( z)   B( z, s)   C( z, s)   P( z)   Q( z)   R( z)   U( z)   H( z)   I( z, s)   .\/ ( z)   K( z)   .<_ ( z)   ./\ ( z)   N( z, s)   W( z)

Proof of Theorem cdlemefr29bpre0N

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 4656	. . 3 |- ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) )
8	7	notbid 308	. 2 |- ( s = R -> ( -. s .<_ ( P .\/ Q ) <-> -. R .<_ ( P .\/ Q ) ) )
9		simp11 1091	. . 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 1174	. . 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 1176	. . 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 1089	. . 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 1135	. . 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 1062	. . 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 35691	. . 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 1341	. 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, 8, 19	cdlemefrs29bpre0 35684	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 ) ) -> ( A. s e. A ( ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) <-> z = [_ R / s ]_ N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274

This theorem is referenced by: (None)