Theorem dvhgrp 36396

Description: The full vector space U constructed from a Hilbert lattice K (given a fiducial hyperplane W) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
dvhgrp.b	|- B = ( Base ` K )
dvhgrp.h	|- H = ( LHyp ` K )
dvhgrp.t	|- T = ( ( LTrn ` K ) ` W )
dvhgrp.e	|- E = ( ( TEndo ` K ) ` W )
dvhgrp.u	|- U = ( ( DVecH ` K ) ` W )
dvhgrp.d	|- D = (Scalar ` U )
dvhgrp.p	|- .+^ = ( +g ` D )
dvhgrp.a	|- .+ = ( +g ` U )
dvhgrp.o	|- .0. = ( 0g ` D )
dvhgrp.i	|- I = ( invg ` D )

Assertion

Ref	Expression
dvhgrp	|- ( ( K e. HL /\ W e. H ) -> U e. Grp )

Proof of Theorem dvhgrp

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhgrp.h	. . . 4 |- H = ( LHyp ` K )
2		dvhgrp.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
3		dvhgrp.e	. . . 4 |- E = ( ( TEndo ` K ) ` W )
4		dvhgrp.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
5		eqid 2622	. . . 4 |- ( Base ` U ) = ( Base ` U )
6	1, 2, 3, 4, 5	dvhvbase 36376	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( Base ` U ) = ( T X. E ) )
7	6	eqcomd 2628	. 2 |- ( ( K e. HL /\ W e. H ) -> ( T X. E ) = ( Base ` U ) )
8		dvhgrp.a	. . 3 |- .+ = ( +g ` U )
9	8	a1i 11	. 2 |- ( ( K e. HL /\ W e. H ) -> .+ = ( +g ` U ) )
10		dvhgrp.d	. . . 4 |- D = (Scalar ` U )
11		dvhgrp.p	. . . 4 |- .+^ = ( +g ` D )
12	1, 2, 3, 4, 10, 11, 8	dvhvaddcl 36384	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( T X. E ) /\ g e. ( T X. E ) ) ) -> ( f .+ g ) e. ( T X. E ) )
13	12	3impb 1260	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) /\ g e. ( T X. E ) ) -> ( f .+ g ) e. ( T X. E ) )
14	1, 2, 3, 4, 10, 11, 8	dvhvaddass 36386	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( f e. ( T X. E ) /\ g e. ( T X. E ) /\ h e. ( T X. E ) ) ) -> ( ( f .+ g ) .+ h ) = ( f .+ ( g .+ h ) ) )
15		dvhgrp.b	. . . 4 |- B = ( Base ` K )
16	15, 1, 2	idltrn 35436	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. T )
17		eqid 2622	. . . . . . . 8 |- ( ( EDRing ` K ) ` W ) = ( ( EDRing ` K ) ` W )
18	1, 17, 4, 10	dvhsca 36371	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> D = ( ( EDRing ` K ) ` W ) )
19	1, 17	erngdv 36281	. . . . . . 7 |- ( ( K e. HL /\ W e. H ) -> ( ( EDRing ` K ) ` W ) e. DivRing )
20	18, 19	eqeltrd 2701	. . . . . 6 |- ( ( K e. HL /\ W e. H ) -> D e. DivRing )
21		drnggrp 18755	. . . . . 6 |- ( D e. DivRing -> D e. Grp )
22	20, 21	syl 17	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> D e. Grp )
23		eqid 2622	. . . . . 6 |- ( Base ` D ) = ( Base ` D )
24		dvhgrp.o	. . . . . 6 |- .0. = ( 0g ` D )
25	23, 24	grpidcl 17450	. . . . 5 |- ( D e. Grp -> .0. e. ( Base ` D ) )
26	22, 25	syl 17	. . . 4 |- ( ( K e. HL /\ W e. H ) -> .0. e. ( Base ` D ) )
27	1, 3, 4, 10, 23	dvhbase 36372	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( Base ` D ) = E )
28	26, 27	eleqtrd 2703	. . 3 |- ( ( K e. HL /\ W e. H ) -> .0. e. E )
29		opelxpi 5148	. . 3 |- ( ( ( _I |` B ) e. T /\ .0. e. E ) -> <. ( _I |` B ) , .0. >. e. ( T X. E ) )
30	16, 28, 29	syl2anc 693	. 2 |- ( ( K e. HL /\ W e. H ) -> <. ( _I |` B ) , .0. >. e. ( T X. E ) )
31		simpl 473	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( K e. HL /\ W e. H ) )
32	16	adantr 481	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( _I |` B ) e. T )
33	28	adantr 481	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> .0. e. E )
34		xp1st 7198	. . . . . 6 |- ( f e. ( T X. E ) -> ( 1st ` f ) e. T )
35	34	adantl 482	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( 1st ` f ) e. T )
36		xp2nd 7199	. . . . . 6 |- ( f e. ( T X. E ) -> ( 2nd ` f ) e. E )
37	36	adantl 482	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( 2nd ` f ) e. E )
38	1, 2, 3, 4, 10, 8, 11	dvhopvadd 36382	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( _I |` B ) e. T /\ .0. e. E ) /\ ( ( 1st ` f ) e. T /\ ( 2nd ` f ) e. E ) ) -> ( <. ( _I |` B ) , .0. >. .+ <. ( 1st ` f ) , ( 2nd ` f ) >. ) = <. ( ( _I |` B ) o. ( 1st ` f ) ) , ( .0. .+^ ( 2nd ` f ) ) >. )
39	31, 32, 33, 35, 37, 38	syl122anc 1335	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( <. ( _I |` B ) , .0. >. .+ <. ( 1st ` f ) , ( 2nd ` f ) >. ) = <. ( ( _I |` B ) o. ( 1st ` f ) ) , ( .0. .+^ ( 2nd ` f ) ) >. )
40	15, 1, 2	ltrn1o 35410	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( 1st ` f ) e. T ) -> ( 1st ` f ) : B -1-1-onto-> B )
41	35, 40	syldan 487	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( 1st ` f ) : B -1-1-onto-> B )
42		f1of 6137	. . . . . 6 |- ( ( 1st ` f ) : B -1-1-onto-> B -> ( 1st ` f ) : B --> B )
43		fcoi2 6079	. . . . . 6 |- ( ( 1st ` f ) : B --> B -> ( ( _I |` B ) o. ( 1st ` f ) ) = ( 1st ` f ) )
44	41, 42, 43	3syl 18	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( ( _I |` B ) o. ( 1st ` f ) ) = ( 1st ` f ) )
45	22	adantr 481	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> D e. Grp )
46	27	adantr 481	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( Base ` D ) = E )
47	37, 46	eleqtrrd 2704	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( 2nd ` f ) e. ( Base ` D ) )
48	23, 11, 24	grplid 17452	. . . . . 6 |- ( ( D e. Grp /\ ( 2nd ` f ) e. ( Base ` D ) ) -> ( .0. .+^ ( 2nd ` f ) ) = ( 2nd ` f ) )
49	45, 47, 48	syl2anc 693	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( .0. .+^ ( 2nd ` f ) ) = ( 2nd ` f ) )
50	44, 49	opeq12d 4410	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> <. ( ( _I |` B ) o. ( 1st ` f ) ) , ( .0. .+^ ( 2nd ` f ) ) >. = <. ( 1st ` f ) , ( 2nd ` f ) >. )
51	39, 50	eqtrd 2656	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( <. ( _I |` B ) , .0. >. .+ <. ( 1st ` f ) , ( 2nd ` f ) >. ) = <. ( 1st ` f ) , ( 2nd ` f ) >. )
52		1st2nd2 7205	. . . . 5 |- ( f e. ( T X. E ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
53	52	adantl 482	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
54	53	oveq2d 6666	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( <. ( _I |` B ) , .0. >. .+ f ) = ( <. ( _I |` B ) , .0. >. .+ <. ( 1st ` f ) , ( 2nd ` f ) >. ) )
55	51, 54, 53	3eqtr4d 2666	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( <. ( _I |` B ) , .0. >. .+ f ) = f )
56	1, 2	ltrncnv 35432	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( 1st ` f ) e. T ) -> `' ( 1st ` f ) e. T )
57	35, 56	syldan 487	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> `' ( 1st ` f ) e. T )
58		dvhgrp.i	. . . . . 6 |- I = ( invg ` D )
59	23, 58	grpinvcl 17467	. . . . 5 |- ( ( D e. Grp /\ ( 2nd ` f ) e. ( Base ` D ) ) -> ( I ` ( 2nd ` f ) ) e. ( Base ` D ) )
60	45, 47, 59	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( I ` ( 2nd ` f ) ) e. ( Base ` D ) )
61	60, 46	eleqtrd 2703	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( I ` ( 2nd ` f ) ) e. E )
62		opelxpi 5148	. . 3 |- ( ( `' ( 1st ` f ) e. T /\ ( I ` ( 2nd ` f ) ) e. E ) -> <. `' ( 1st ` f ) , ( I ` ( 2nd ` f ) ) >. e. ( T X. E ) )
63	57, 61, 62	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> <. `' ( 1st ` f ) , ( I ` ( 2nd ` f ) ) >. e. ( T X. E ) )
64	53	oveq2d 6666	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( <. `' ( 1st ` f ) , ( I ` ( 2nd ` f ) ) >. .+ f ) = ( <. `' ( 1st ` f ) , ( I ` ( 2nd ` f ) ) >. .+ <. ( 1st ` f ) , ( 2nd ` f ) >. ) )
65	1, 2, 3, 4, 10, 8, 11	dvhopvadd 36382	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( `' ( 1st ` f ) e. T /\ ( I ` ( 2nd ` f ) ) e. E ) /\ ( ( 1st ` f ) e. T /\ ( 2nd ` f ) e. E ) ) -> ( <. `' ( 1st ` f ) , ( I ` ( 2nd ` f ) ) >. .+ <. ( 1st ` f ) , ( 2nd ` f ) >. ) = <. ( `' ( 1st ` f ) o. ( 1st ` f ) ) , ( ( I ` ( 2nd ` f ) ) .+^ ( 2nd ` f ) ) >. )
66	31, 57, 61, 35, 37, 65	syl122anc 1335	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( <. `' ( 1st ` f ) , ( I ` ( 2nd ` f ) ) >. .+ <. ( 1st ` f ) , ( 2nd ` f ) >. ) = <. ( `' ( 1st ` f ) o. ( 1st ` f ) ) , ( ( I ` ( 2nd ` f ) ) .+^ ( 2nd ` f ) ) >. )
67		f1ococnv1 6165	. . . . . 6 |- ( ( 1st ` f ) : B -1-1-onto-> B -> ( `' ( 1st ` f ) o. ( 1st ` f ) ) = ( _I |` B ) )
68	41, 67	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( `' ( 1st ` f ) o. ( 1st ` f ) ) = ( _I |` B ) )
69	23, 11, 24, 58	grplinv 17468	. . . . . 6 |- ( ( D e. Grp /\ ( 2nd ` f ) e. ( Base ` D ) ) -> ( ( I ` ( 2nd ` f ) ) .+^ ( 2nd ` f ) ) = .0. )
70	45, 47, 69	syl2anc 693	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( ( I ` ( 2nd ` f ) ) .+^ ( 2nd ` f ) ) = .0. )
71	68, 70	opeq12d 4410	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> <. ( `' ( 1st ` f ) o. ( 1st ` f ) ) , ( ( I ` ( 2nd ` f ) ) .+^ ( 2nd ` f ) ) >. = <. ( _I |` B ) , .0. >. )
72	66, 71	eqtrd 2656	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( <. `' ( 1st ` f ) , ( I ` ( 2nd ` f ) ) >. .+ <. ( 1st ` f ) , ( 2nd ` f ) >. ) = <. ( _I |` B ) , .0. >. )
73	64, 72	eqtrd 2656	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ f e. ( T X. E ) ) -> ( <. `' ( 1st ` f ) , ( I ` ( 2nd ` f ) ) >. .+ f ) = <. ( _I |` B ) , .0. >. )
74	7, 9, 13, 14, 30, 55, 63, 73	isgrpd 17444	1 |- ( ( K e. HL /\ W e. H ) -> U e. Grp )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   DivRingcdr 18747   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   EDRingcedring 36041   DVecHcdvh 36367

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  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-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  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-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043  df-edring 36045  df-dvech 36368

This theorem is referenced by:  dvhlveclem  36397