Theorem cdlemefs29cpre1N 30900

Description: TODO FIX COMMENT (Contributed by NM, 26-Mar-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemefs32.b	|- B = ( Base ` K )
cdlemefs32.l	|- .<_ = ( le ` K )
cdlemefs32.j	|- .\/ = ( join ` K )
cdlemefs32.m	|- ./\ = ( meet ` K )
cdlemefs32.a	|- A = ( Atoms ` K )
cdlemefs32.h	|- H = ( LHyp ` K )
cdlemefs32.u	|- U = ( ( P .\/ Q ) ./\ W )
cdlemefs32.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs32.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemefs32.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
cdlemefs32.n	|- N = if ( s .<_ ( P .\/ Q ) , I , C )

Assertion

Ref	Expression
cdlemefs29cpre1N	|- ( ( ( ( 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 ) ) -> E! z e. B A. s e. A ( ( ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) )

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

Allowed substitution hints:   C( y, z, t, s)   D( z, t)   U( z, s)   E( z, t, s)   I( y, z, t, s)   N( y, t, s)

Proof of Theorem cdlemefs29cpre1N

Step	Hyp	Ref	Expression
1		cdlemefs32.b	. 2 |- B = ( Base ` K )
2		cdlemefs32.l	. 2 |- .<_ = ( le ` K )
3		cdlemefs32.j	. 2 |- .\/ = ( join ` K )
4		cdlemefs32.m	. 2 |- ./\ = ( meet ` K )
5		cdlemefs32.a	. 2 |- A = ( Atoms ` K )
6		cdlemefs32.h	. 2 |- H = ( LHyp ` K )
7		breq1 4175	. 2 |- ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) )
8		simp1 957	. . 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 ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
9		simp3l 985	. . . 4 |- ( ( ( ( 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 )
10		simp3rl 1030	. . . 4 |- ( ( ( ( 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 .<_ W )
11	9, 10	jca 519	. . 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 /\ -. s .<_ W ) )
12		simp3rr 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 ) ) ) ) -> s .<_ ( P .\/ Q ) )
13		simp2 958	. . 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 )
14		cdlemefs32.u	. . . 4 |- U = ( ( P .\/ Q ) ./\ W )
15		cdlemefs32.d	. . . 4 |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
16		cdlemefs32.e	. . . 4 |- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
17		cdlemefs32.i	. . . 4 |- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
18		cdlemefs32.n	. . . 4 |- N = if ( s .<_ ( P .\/ Q ) , I , C )
19	1, 2, 3, 4, 5, 6, 14, 15, 16, 17, 18	cdlemefs27cl 30895	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
20	8, 11, 12, 13, 19	syl13anc 1186	. 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 )
21	1, 2, 3, 4, 5, 6, 14, 15, 16, 17, 18	cdlemefs32snb 30897	. 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 )
22	1, 2, 3, 4, 5, 6, 7, 20, 21	cdlemefrs29cpre1 30880	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 ) ) -> E! z e. B A. s e. A ( ( ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E!wreu 2668   ifcif 3699   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470