Theorem cdlemk56w 34940

Description: Use a fixed element to eliminate P in cdlemk56 34938. (Contributed by NM, 1-Aug-2013.)

Hypotheses

Ref	Expression
cdlemk6.b	|- B = ( Base ` K )
cdlemk6.j	|- .\/ = ( join ` K )
cdlemk6.m	|- ./\ = ( meet ` K )
cdlemk6.o	|- ._|_ = ( oc ` K )
cdlemk6.a	|- A = ( Atoms ` K )
cdlemk6.h	|- H = ( LHyp ` K )
cdlemk6.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk6.r	|- R = ( ( trL ` K ) ` W )
cdlemk6.p	|- P = ( ._|_ ` W )
cdlemk6.z	|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk6.y	|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdlemk6.x	|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
cdlemk6.u	|- U = ( g e. T |-> if ( F = N , g , X ) )
cdlemk6.e	|- E = ( ( TEndo ` K ) ` W )

Assertion

Ref	Expression
cdlemk56w	|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( U e. E /\ ( U ` F ) = N ) )

Distinct variable groups:   g, b, z, ./\   .\/ , b, g, z   A, b, g, z   B, b, g, z   F, b, g, z   H, b, g, z   K, b, g, z   N, b, g, z   P, b, g, z   R, b, g, z   T, b, g, z   W, b, g, z   z, Y   g, Z

Allowed substitution hints:   U( z, g, b)   E( z, g, b)   ._|_ ( z, g, b)   X( z, g, b)   Y( g, b)   Z( z, b)

Proof of Theorem cdlemk56w

Step	Hyp	Ref	Expression
1		simp1 988	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( K e. HL /\ W e. H ) )
2		simp2l 1014	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> F e. T )
3		simp2r 1015	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> N e. T )
4		simp3 990	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( R ` F ) = ( R ` N ) )
5		eqid 2454	. . . . 5 |- ( le ` K ) = ( le ` K )
6		cdlemk6.a	. . . . 5 |- A = ( Atoms ` K )
7		cdlemk6.h	. . . . 5 |- H = ( LHyp ` K )
8		cdlemk6.p	. . . . . 6 |- P = ( ._|_ ` W )
9		cdlemk6.o	. . . . . . 7 |- ._|_ = ( oc ` K )
10	9	fveq1i 5799	. . . . . 6 |- ( ._|_ ` W ) = ( ( oc ` K ) ` W )
11	8, 10	eqtri 2483	. . . . 5 |- P = ( ( oc ` K ) ` W )
12	5, 6, 7, 11	lhpocnel2 33986	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( P e. A /\ -. P ( le ` K ) W ) )
13	12	3ad2ant1 1009	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( P e. A /\ -. P ( le ` K ) W ) )
14		cdlemk6.b	. . . 4 |- B = ( Base ` K )
15		cdlemk6.j	. . . 4 |- .\/ = ( join ` K )
16		cdlemk6.m	. . . 4 |- ./\ = ( meet ` K )
17		cdlemk6.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
18		cdlemk6.r	. . . 4 |- R = ( ( trL ` K ) ` W )
19		cdlemk6.z	. . . 4 |- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
20		cdlemk6.y	. . . 4 |- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
21		cdlemk6.x	. . . 4 |- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
22		cdlemk6.u	. . . 4 |- U = ( g e. T |-> if ( F = N , g , X ) )
23		cdlemk6.e	. . . 4 |- E = ( ( TEndo ` K ) ` W )
24	14, 5, 15, 16, 6, 7, 17, 18, 19, 20, 21, 22, 23	cdlemk56 34938	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P ( le ` K ) W ) ) -> U e. E )
25	1, 2, 3, 4, 13, 24	syl311anc 1233	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> U e. E )
26	14, 15, 16, 9, 6, 7, 17, 18, 8, 19, 20, 21, 22	cdlemk19w 34939	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( U ` F ) = N )
27	25, 26	jca 532	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( U e. E /\ ( U ` F ) = N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2647   A.wral 2798   ifcif 3898   class class class wbr 4399   |-> cmpt 4457   _I cid 4738   `'ccnv 4946   |` cres 4949   o. ccom 4951   ` cfv 5525   iota_crio 6159  (class class class)co 6199   Basecbs 14291   lecple 14363   occoc 14364   joincjn 15232   meetcmee 15233   Atomscatm 33231   HLchlt 33318   LHypclh 33951   LTrncltrn 34068   trLctrl 34125   TEndoctendo 34719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-riotaBAD 32927

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-undef 6901  df-map 7325  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-llines 33465  df-lplanes 33466  df-lvols 33467  df-lines 33468  df-psubsp 33470  df-pmap 33471  df-padd 33763  df-lhyp 33955  df-laut 33956  df-ldil 34071  df-ltrn 34072  df-trl 34126  df-tendo 34722

This theorem is referenced by:  cdlemk  34941  cdleml6  34948