Theorem cdlemk56w 36842

Description: Use a fixed element to eliminate P in cdlemk56 36840. (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 996	. . 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 1022	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> F e. T )
3		simp2r 1023	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> N e. T )
4		simp3 998	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( R ` F ) = ( R ` N ) )
5		eqid 2457	. . . . 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 5873	. . . . . 6 |- ( ._|_ ` W ) = ( ( oc ` K ) ` W )
11	8, 10	eqtri 2486	. . . . 5 |- P = ( ( oc ` K ) ` W )
12	5, 6, 7, 11	lhpocnel2 35886	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( P e. A /\ -. P ( le ` K ) W ) )
13	12	3ad2ant1 1017	. . 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 36840	. . 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 1242	. 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 36841	. 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 973   = wceq 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   ifcif 3944   class class class wbr 4456   |-> cmpt 4515   _I cid 4799   `'ccnv 5007   |` cres 5010   o. ccom 5012   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14644   lecple 14719   occoc 14720   joincjn 15700   meetcmee 15701   Atomscatm 35131   HLchlt 35218   LHypclh 35851   LTrncltrn 35968   trLctrl 36026   TEndoctendo 36621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-llines 35365  df-lplanes 35366  df-lvols 35367  df-lines 35368  df-psubsp 35370  df-pmap 35371  df-padd 35663  df-lhyp 35855  df-laut 35856  df-ldil 35971  df-ltrn 35972  df-trl 36027  df-tendo 36624

This theorem is referenced by:  cdlemk  36843  cdleml6  36850