Theorem dihwN 36578

Description: Value of isomorphism H at the fiducial hyperplane W. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihw.b	|- B = ( Base ` K )
dihw.h	|- H = ( LHyp ` K )
dihw.t	|- T = ( ( LTrn ` K ) ` W )
dihw.o	|- .0. = ( f e. T |-> ( _I |` B ) )
dihw.i	|- I = ( ( DIsoH ` K ) ` W )
dihw.k	|- ( ph -> ( K e. HL /\ W e. H ) )

Assertion

Ref	Expression
dihwN	|- ( ph -> ( I ` W ) = ( T X. { .0. } ) )

Distinct variable groups:   f, K   f, W

Allowed substitution hints:   ph( f)   B( f)   T( f)   H( f)   I( f)   .0. ( f)

Proof of Theorem dihwN

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihw.k	. . 3 |- ( ph -> ( K e. HL /\ W e. H ) )
2	1	simprd 479	. . . . 5 |- ( ph -> W e. H )
3		dihw.b	. . . . . 6 |- B = ( Base ` K )
4		dihw.h	. . . . . 6 |- H = ( LHyp ` K )
5	3, 4	lhpbase 35284	. . . . 5 |- ( W e. H -> W e. B )
6	2, 5	syl 17	. . . 4 |- ( ph -> W e. B )
7	1	simpld 475	. . . . . 6 |- ( ph -> K e. HL )
8		hllat 34650	. . . . . 6 |- ( K e. HL -> K e. Lat )
9	7, 8	syl 17	. . . . 5 |- ( ph -> K e. Lat )
10		eqid 2622	. . . . . 6 |- ( le ` K ) = ( le ` K )
11	3, 10	latref 17053	. . . . 5 |- ( ( K e. Lat /\ W e. B ) -> W ( le ` K ) W )
12	9, 6, 11	syl2anc 693	. . . 4 |- ( ph -> W ( le ` K ) W )
13	6, 12	jca 554	. . 3 |- ( ph -> ( W e. B /\ W ( le ` K ) W ) )
14		dihw.i	. . . 4 |- I = ( ( DIsoH ` K ) ` W )
15		eqid 2622	. . . 4 |- ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W )
16	3, 10, 4, 14, 15	dihvalb 36526	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( W e. B /\ W ( le ` K ) W ) ) -> ( I ` W ) = ( ( ( DIsoB ` K ) ` W ) ` W ) )
17	1, 13, 16	syl2anc 693	. 2 |- ( ph -> ( I ` W ) = ( ( ( DIsoB ` K ) ` W ) ` W ) )
18		dihw.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
19		dihw.o	. . . 4 |- .0. = ( f e. T |-> ( _I |` B ) )
20		eqid 2622	. . . 4 |- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
21	3, 10, 4, 18, 19, 20, 15	dibval2 36433	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( W e. B /\ W ( le ` K ) W ) ) -> ( ( ( DIsoB ` K ) ` W ) ` W ) = ( ( ( ( DIsoA ` K ) ` W ) ` W ) X. { .0. } ) )
22	1, 13, 21	syl2anc 693	. 2 |- ( ph -> ( ( ( DIsoB ` K ) ` W ) ` W ) = ( ( ( ( DIsoA ` K ) ` W ) ` W ) X. { .0. } ) )
23		eqid 2622	. . . . . 6 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
24	3, 10, 4, 18, 23, 20	diaval 36321	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( W e. B /\ W ( le ` K ) W ) ) -> ( ( ( DIsoA ` K ) ` W ) ` W ) = { g e. T | ( ( ( trL ` K ) ` W ) ` g ) ( le ` K ) W } )
25	1, 13, 24	syl2anc 693	. . . 4 |- ( ph -> ( ( ( DIsoA ` K ) ` W ) ` W ) = { g e. T | ( ( ( trL ` K ) ` W ) ` g ) ( le ` K ) W } )
26	10, 4, 18, 23	trlle 35471	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ g e. T ) -> ( ( ( trL ` K ) ` W ) ` g ) ( le ` K ) W )
27	1, 26	sylan 488	. . . . . 6 |- ( ( ph /\ g e. T ) -> ( ( ( trL ` K ) ` W ) ` g ) ( le ` K ) W )
28	27	ralrimiva 2966	. . . . 5 |- ( ph -> A. g e. T ( ( ( trL ` K ) ` W ) ` g ) ( le ` K ) W )
29		rabid2 3118	. . . . 5 |- ( T = { g e. T | ( ( ( trL ` K ) ` W ) ` g ) ( le ` K ) W } <-> A. g e. T ( ( ( trL ` K ) ` W ) ` g ) ( le ` K ) W )
30	28, 29	sylibr 224	. . . 4 |- ( ph -> T = { g e. T | ( ( ( trL ` K ) ` W ) ` g ) ( le ` K ) W } )
31	25, 30	eqtr4d 2659	. . 3 |- ( ph -> ( ( ( DIsoA ` K ) ` W ) ` W ) = T )
32	31	xpeq1d 5138	. 2 |- ( ph -> ( ( ( ( DIsoA ` K ) ` W ) ` W ) X. { .0. } ) = ( T X. { .0. } ) )
33	17, 22, 32	3eqtrd 2660	1 |- ( ph -> ( I ` W ) = ( T X. { .0. } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   {csn 4177   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   |` cres 5116   ` cfv 5888   Basecbs 15857   lecple 15948   Latclat 17045   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   DIsoAcdia 36317   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-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-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: (None)