Theorem dihglblem3N 36584

Description: Isomorphism H of a lattice glb. (Contributed by NM, 20-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 ) }
dihglblem.i	|- J = ( ( DIsoB ` K ) ` W )
dihglblem.ih	|- I = ( ( DIsoH ` K ) ` W )

Assertion

Ref	Expression
dihglblem3N	|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` T ) ) = |^|_ x e. T ( I ` x ) )

Distinct variable groups:   x, u, v, ./\   x, .<_   x, B, u   x, G   x, H   x, K   x, S, u, v   x, T   x, W, u, v   u, .<_ , v   v, B   u, G, v   u, H, v   u, K, v

Allowed substitution hints:   T( v, u)   I( x, v, u)   J( x, v, u)

Proof of Theorem dihglblem3N

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( K e. HL /\ W e. H ) )
2		dihglblem.t	. . . . . 6 |- T = { u e. B | E. v e. S u = ( v ./\ W ) }
3		simp11l 1172	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> K e. HL )
4		hllat 34650	. . . . . . . . . . . 12 |- ( K e. HL -> K e. Lat )
5	3, 4	syl 17	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> K e. Lat )
6		simp12l 1174	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> S C_ B )
7		simp3 1063	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> v e. S )
8	6, 7	sseldd 3604	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> v e. B )
9		simp11r 1173	. . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> W e. H )
10		dihglblem.b	. . . . . . . . . . . . 13 |- B = ( Base ` K )
11		dihglblem.h	. . . . . . . . . . . . 13 |- H = ( LHyp ` K )
12	10, 11	lhpbase 35284	. . . . . . . . . . . 12 |- ( W e. H -> W e. B )
13	9, 12	syl 17	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> W e. B )
14		dihglblem.l	. . . . . . . . . . . 12 |- .<_ = ( le ` K )
15		dihglblem.m	. . . . . . . . . . . 12 |- ./\ = ( meet ` K )
16	10, 14, 15	latmle2 17077	. . . . . . . . . . 11 |- ( ( K e. Lat /\ v e. B /\ W e. B ) -> ( v ./\ W ) .<_ W )
17	5, 8, 13, 16	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> ( v ./\ W ) .<_ W )
18	17	3expia 1267	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B ) -> ( v e. S -> ( v ./\ W ) .<_ W ) )
19		breq1 4656	. . . . . . . . . 10 |- ( u = ( v ./\ W ) -> ( u .<_ W <-> ( v ./\ W ) .<_ W ) )
20	19	biimprcd 240	. . . . . . . . 9 |- ( ( v ./\ W ) .<_ W -> ( u = ( v ./\ W ) -> u .<_ W ) )
21	18, 20	syl6 35	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B ) -> ( v e. S -> ( u = ( v ./\ W ) -> u .<_ W ) ) )
22	21	rexlimdv 3030	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B ) -> ( E. v e. S u = ( v ./\ W ) -> u .<_ W ) )
23	22	ss2rabdv 3683	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> { u e. B | E. v e. S u = ( v ./\ W ) } C_ { u e. B | u .<_ W } )
24	2, 23	syl5eqss 3649	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> T C_ { u e. B | u .<_ W } )
25		dihglblem.i	. . . . . . 7 |- J = ( ( DIsoB ` K ) ` W )
26	10, 14, 11, 25	dibdmN 36446	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> dom J = { u e. B | u .<_ W } )
27	26	3ad2ant1 1082	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> dom J = { u e. B | u .<_ W } )
28	24, 27	sseqtr4d 3642	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> T C_ dom J )
29		dihglblem.g	. . . . . 6 |- G = ( glb ` K )
30	10, 14, 15, 29, 11, 2	dihglblem2aN 36582	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> T =/= (/) )
31	30	3adant3 1081	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> T =/= (/) )
32	29, 11, 25	dibglbN 36455	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( T C_ dom J /\ T =/= (/) ) ) -> ( J ` ( G ` T ) ) = |^|_ x e. T ( J ` x ) )
33	1, 28, 31, 32	syl12anc 1324	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( J ` ( G ` T ) ) = |^|_ x e. T ( J ` x ) )
34	10, 14, 15, 29, 11, 2	dihglblem2N 36583	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ S C_ B /\ ( G ` S ) .<_ W ) -> ( G ` S ) = ( G ` T ) )
35	34	3adant2r 1321	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( G ` S ) = ( G ` T ) )
36	35	fveq2d 6195	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( J ` ( G ` S ) ) = ( J ` ( G ` T ) ) )
37		simpl1 1064	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ x e. T ) -> ( K e. HL /\ W e. H ) )
38	24	sselda 3603	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ x e. T ) -> x e. { u e. B | u .<_ W } )
39		breq1 4656	. . . . . . 7 |- ( u = x -> ( u .<_ W <-> x .<_ W ) )
40	39	elrab 3363	. . . . . 6 |- ( x e. { u e. B | u .<_ W } <-> ( x e. B /\ x .<_ W ) )
41	38, 40	sylib 208	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ x e. T ) -> ( x e. B /\ x .<_ W ) )
42		dihglblem.ih	. . . . . 6 |- I = ( ( DIsoH ` K ) ` W )
43	10, 14, 11, 42, 25	dihvalb 36526	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( x e. B /\ x .<_ W ) ) -> ( I ` x ) = ( J ` x ) )
44	37, 41, 43	syl2anc 693	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ x e. T ) -> ( I ` x ) = ( J ` x ) )
45	44	iineq2dv 4543	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> |^|_ x e. T ( I ` x ) = |^|_ x e. T ( J ` x ) )
46	33, 36, 45	3eqtr4rd 2667	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> |^|_ x e. T ( I ` x ) = ( J ` ( G ` S ) ) )
47		simp1l 1085	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> K e. HL )
48		hlclat 34645	. . . . 5 |- ( K e. HL -> K e. CLat )
49	47, 48	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> K e. CLat )
50		simp2l 1087	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> S C_ B )
51	10, 29	clatglbcl 17114	. . . 4 |- ( ( K e. CLat /\ S C_ B ) -> ( G ` S ) e. B )
52	49, 50, 51	syl2anc 693	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( G ` S ) e. B )
53		simp3 1063	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( G ` S ) .<_ W )
54	10, 14, 11, 42, 25	dihvalb 36526	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( G ` S ) e. B /\ ( G ` S ) .<_ W ) ) -> ( I ` ( G ` S ) ) = ( J ` ( G ` S ) ) )
55	1, 52, 53, 54	syl12anc 1324	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` S ) ) = ( J ` ( G ` S ) ) )
56	35	fveq2d 6195	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` S ) ) = ( I ` ( G ` T ) ) )
57	46, 55, 56	3eqtr2rd 2663	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` T ) ) = |^|_ x e. T ( I ` x ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   |^|_ciin 4521   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   glbcglb 16943   meetcmee 16945   Latclat 17045   CLatccla 17107   HLchlt 34637   LHypclh 35270   DIsoBcdib 36427   DIsoHcdih 36517

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-iin 4523  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-mpt2 6655  df-map 7859  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-p1 17040  df-lat 17046  df-clat 17108  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  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-disoa 36318  df-dib 36428  df-dih 36518

This theorem is referenced by:  dihglblem3aN  36585