Theorem dihglblem2aN 36582

Description: Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihglblem.b	|- B = ( Base ` K )
dihglblem.l	|- .<_ = ( le ` K )
dihglblem.m	|- ./\ = ( meet ` K )
dihglblem.g	|- G = ( glb ` K )
dihglblem.h	|- H = ( LHyp ` K )
dihglblem.t	|- T = { u e. B | E. v e. S u = ( v ./\ W ) }

Assertion

Ref	Expression
dihglblem2aN	|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> T =/= (/) )

Distinct variable groups:   v, u, ./\   u, B   u, S, v   u, W, v

Allowed substitution hints:   B( v)   T( v, u)   G( v, u)   H( v, u)   K( v, u)   .<_ ( v, u)

Proof of Theorem dihglblem2aN

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihglblem.t	. . 3 |- T = { u e. B | E. v e. S u = ( v ./\ W ) }
2	1	a1i 11	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> T = { u e. B | E. v e. S u = ( v ./\ W ) } )
3		simprr 796	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> S =/= (/) )
4		n0 3931	. . . 4 |- ( S =/= (/) <-> E. z z e. S )
5	3, 4	sylib 208	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> E. z z e. S )
6		hllat 34650	. . . . . . 7 |- ( K e. HL -> K e. Lat )
7	6	ad3antrrr 766	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> K e. Lat )
8		simplrl 800	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> S C_ B )
9		simpr 477	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> z e. S )
10	8, 9	sseldd 3604	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> z e. B )
11		dihglblem.b	. . . . . . . 8 |- B = ( Base ` K )
12		dihglblem.h	. . . . . . . 8 |- H = ( LHyp ` K )
13	11, 12	lhpbase 35284	. . . . . . 7 |- ( W e. H -> W e. B )
14	13	ad3antlr 767	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> W e. B )
15		dihglblem.m	. . . . . . 7 |- ./\ = ( meet ` K )
16	11, 15	latmcl 17052	. . . . . 6 |- ( ( K e. Lat /\ z e. B /\ W e. B ) -> ( z ./\ W ) e. B )
17	7, 10, 14, 16	syl3anc 1326	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> ( z ./\ W ) e. B )
18		eqidd 2623	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> ( z ./\ W ) = ( z ./\ W ) )
19		oveq1 6657	. . . . . . . 8 |- ( v = z -> ( v ./\ W ) = ( z ./\ W ) )
20	19	eqeq2d 2632	. . . . . . 7 |- ( v = z -> ( ( z ./\ W ) = ( v ./\ W ) <-> ( z ./\ W ) = ( z ./\ W ) ) )
21	20	rspcev 3309	. . . . . 6 |- ( ( z e. S /\ ( z ./\ W ) = ( z ./\ W ) ) -> E. v e. S ( z ./\ W ) = ( v ./\ W ) )
22	9, 18, 21	syl2anc 693	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> E. v e. S ( z ./\ W ) = ( v ./\ W ) )
23		ovex 6678	. . . . . 6 |- ( z ./\ W ) e. _V
24		eleq1 2689	. . . . . . 7 |- ( w = ( z ./\ W ) -> ( w e. { u e. B | E. v e. S u = ( v ./\ W ) } <-> ( z ./\ W ) e. { u e. B | E. v e. S u = ( v ./\ W ) } ) )
25		eqeq1 2626	. . . . . . . . 9 |- ( u = ( z ./\ W ) -> ( u = ( v ./\ W ) <-> ( z ./\ W ) = ( v ./\ W ) ) )
26	25	rexbidv 3052	. . . . . . . 8 |- ( u = ( z ./\ W ) -> ( E. v e. S u = ( v ./\ W ) <-> E. v e. S ( z ./\ W ) = ( v ./\ W ) ) )
27	26	elrab 3363	. . . . . . 7 |- ( ( z ./\ W ) e. { u e. B | E. v e. S u = ( v ./\ W ) } <-> ( ( z ./\ W ) e. B /\ E. v e. S ( z ./\ W ) = ( v ./\ W ) ) )
28	24, 27	syl6bb 276	. . . . . 6 |- ( w = ( z ./\ W ) -> ( w e. { u e. B | E. v e. S u = ( v ./\ W ) } <-> ( ( z ./\ W ) e. B /\ E. v e. S ( z ./\ W ) = ( v ./\ W ) ) ) )
29	23, 28	spcev 3300	. . . . 5 |- ( ( ( z ./\ W ) e. B /\ E. v e. S ( z ./\ W ) = ( v ./\ W ) ) -> E. w w e. { u e. B | E. v e. S u = ( v ./\ W ) } )
30	17, 22, 29	syl2anc 693	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> E. w w e. { u e. B | E. v e. S u = ( v ./\ W ) } )
31		n0 3931	. . . 4 |- ( { u e. B | E. v e. S u = ( v ./\ W ) } =/= (/) <-> E. w w e. { u e. B | E. v e. S u = ( v ./\ W ) } )
32	30, 31	sylibr 224	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> { u e. B | E. v e. S u = ( v ./\ W ) } =/= (/) )
33	5, 32	exlimddv 1863	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> { u e. B | E. v e. S u = ( v ./\ W ) } =/= (/) )
34	2, 33	eqnetrd 2861	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> T =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   glbcglb 16943   meetcmee 16945   Latclat 17045   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-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-lat 17046  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274

This theorem is referenced by:  dihglblem3N  36584