Theorem hgmapadd 37186

Description: Part 15 of [Baer] p. 50 line 13. (Contributed by NM, 6-Jun-2015.)

Hypotheses

Ref	Expression
hgmapadd.h	|- H = ( LHyp ` K )
hgmapadd.u	|- U = ( ( DVecH ` K ) ` W )
hgmapadd.r	|- R = (Scalar ` U )
hgmapadd.b	|- B = ( Base ` R )
hgmapadd.p	|- .+ = ( +g ` R )
hgmapadd.g	|- G = ( (HGMap ` K ) ` W )
hgmapadd.k	|- ( ph -> ( K e. HL /\ W e. H ) )
hgmapadd.x	|- ( ph -> X e. B )
hgmapadd.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
hgmapadd	|- ( ph -> ( G ` ( X .+ Y ) ) = ( ( G ` X ) .+ ( G ` Y ) ) )

Proof of Theorem hgmapadd

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hgmapadd.h	. . . 4 |- H = ( LHyp ` K )
2		hgmapadd.u	. . . 4 |- U = ( ( DVecH ` K ) ` W )
3		eqid 2622	. . . 4 |- ( Base ` U ) = ( Base ` U )
4		eqid 2622	. . . 4 |- ( 0g ` U ) = ( 0g ` U )
5		hgmapadd.k	. . . 4 |- ( ph -> ( K e. HL /\ W e. H ) )
6	1, 2, 3, 4, 5	dvh1dim 36731	. . 3 |- ( ph -> E. t e. ( Base ` U ) t =/= ( 0g ` U ) )
7		eqid 2622	. . . . . . . . 9 |- ( (LCDual ` K ) ` W ) = ( (LCDual ` K ) ` W )
8	1, 7, 5	lcdlmod 36881	. . . . . . . 8 |- ( ph -> ( (LCDual ` K ) ` W ) e. LMod )
9	8	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( (LCDual ` K ) ` W ) e. LMod )
10		hgmapadd.r	. . . . . . . 8 |- R = (Scalar ` U )
11		hgmapadd.b	. . . . . . . 8 |- B = ( Base ` R )
12		eqid 2622	. . . . . . . 8 |- (Scalar ` ( (LCDual ` K ) ` W ) ) = (Scalar ` ( (LCDual ` K ) ` W ) )
13		eqid 2622	. . . . . . . 8 |- ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) = ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) )
14		hgmapadd.g	. . . . . . . 8 |- G = ( (HGMap ` K ) ` W )
15	5	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( K e. HL /\ W e. H ) )
16		hgmapadd.x	. . . . . . . . 9 |- ( ph -> X e. B )
17	16	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> X e. B )
18	1, 2, 10, 11, 7, 12, 13, 14, 15, 17	hgmapdcl 37182	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( G ` X ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) )
19		hgmapadd.y	. . . . . . . . 9 |- ( ph -> Y e. B )
20	1, 2, 10, 11, 7, 12, 13, 14, 5, 19	hgmapdcl 37182	. . . . . . . 8 |- ( ph -> ( G ` Y ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) )
21	20	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( G ` Y ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) )
22		eqid 2622	. . . . . . . 8 |- ( Base ` ( (LCDual ` K ) ` W ) ) = ( Base ` ( (LCDual ` K ) ` W ) )
23		eqid 2622	. . . . . . . 8 |- ( (HDMap ` K ) ` W ) = ( (HDMap ` K ) ` W )
24		simp2 1062	. . . . . . . 8 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> t e. ( Base ` U ) )
25	1, 2, 3, 7, 22, 23, 15, 24	hdmapcl 37122	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( (HDMap ` K ) ` W ) ` t ) e. ( Base ` ( (LCDual ` K ) ` W ) ) )
26		eqid 2622	. . . . . . . 8 |- ( +g ` ( (LCDual ` K ) ` W ) ) = ( +g ` ( (LCDual ` K ) ` W ) )
27		eqid 2622	. . . . . . . 8 |- ( .s ` ( (LCDual ` K ) ` W ) ) = ( .s ` ( (LCDual ` K ) ` W ) )
28		eqid 2622	. . . . . . . 8 |- ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) = ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) )
29	22, 26, 12, 27, 13, 28	lmodvsdir 18887	. . . . . . 7 |- ( ( ( (LCDual ` K ) ` W ) e. LMod /\ ( ( G ` X ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) /\ ( G ` Y ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) /\ ( ( (HDMap ` K ) ` W ) ` t ) e. ( Base ` ( (LCDual ` K ) ` W ) ) ) ) -> ( ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) = ( ( ( G ` X ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) ( +g ` ( (LCDual ` K ) ` W ) ) ( ( G ` Y ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) ) )
30	9, 18, 21, 25, 29	syl13anc 1328	. . . . . 6 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) = ( ( ( G ` X ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) ( +g ` ( (LCDual ` K ) ` W ) ) ( ( G ` Y ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) ) )
31	1, 2, 5	dvhlmod 36399	. . . . . . . . . 10 |- ( ph -> U e. LMod )
32	31	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> U e. LMod )
33	19	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> Y e. B )
34		eqid 2622	. . . . . . . . . 10 |- ( +g ` U ) = ( +g ` U )
35		eqid 2622	. . . . . . . . . 10 |- ( .s ` U ) = ( .s ` U )
36		hgmapadd.p	. . . . . . . . . 10 |- .+ = ( +g ` R )
37	3, 34, 10, 35, 11, 36	lmodvsdir 18887	. . . . . . . . 9 |- ( ( U e. LMod /\ ( X e. B /\ Y e. B /\ t e. ( Base ` U ) ) ) -> ( ( X .+ Y ) ( .s ` U ) t ) = ( ( X ( .s ` U ) t ) ( +g ` U ) ( Y ( .s ` U ) t ) ) )
38	32, 17, 33, 24, 37	syl13anc 1328	. . . . . . . 8 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( X .+ Y ) ( .s ` U ) t ) = ( ( X ( .s ` U ) t ) ( +g ` U ) ( Y ( .s ` U ) t ) ) )
39	38	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( (HDMap ` K ) ` W ) ` ( ( X .+ Y ) ( .s ` U ) t ) ) = ( ( (HDMap ` K ) ` W ) ` ( ( X ( .s ` U ) t ) ( +g ` U ) ( Y ( .s ` U ) t ) ) ) )
40	3, 10, 35, 11	lmodvscl 18880	. . . . . . . . 9 |- ( ( U e. LMod /\ X e. B /\ t e. ( Base ` U ) ) -> ( X ( .s ` U ) t ) e. ( Base ` U ) )
41	32, 17, 24, 40	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( X ( .s ` U ) t ) e. ( Base ` U ) )
42	3, 10, 35, 11	lmodvscl 18880	. . . . . . . . 9 |- ( ( U e. LMod /\ Y e. B /\ t e. ( Base ` U ) ) -> ( Y ( .s ` U ) t ) e. ( Base ` U ) )
43	32, 33, 24, 42	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( Y ( .s ` U ) t ) e. ( Base ` U ) )
44	1, 2, 3, 34, 7, 26, 23, 15, 41, 43	hdmapadd 37135	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( (HDMap ` K ) ` W ) ` ( ( X ( .s ` U ) t ) ( +g ` U ) ( Y ( .s ` U ) t ) ) ) = ( ( ( (HDMap ` K ) ` W ) ` ( X ( .s ` U ) t ) ) ( +g ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` ( Y ( .s ` U ) t ) ) ) )
45	1, 2, 3, 35, 10, 11, 7, 27, 23, 14, 15, 24, 17	hgmapvs 37183	. . . . . . . 8 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( (HDMap ` K ) ` W ) ` ( X ( .s ` U ) t ) ) = ( ( G ` X ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) )
46	1, 2, 3, 35, 10, 11, 7, 27, 23, 14, 15, 24, 33	hgmapvs 37183	. . . . . . . 8 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( (HDMap ` K ) ` W ) ` ( Y ( .s ` U ) t ) ) = ( ( G ` Y ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) )
47	45, 46	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( ( (HDMap ` K ) ` W ) ` ( X ( .s ` U ) t ) ) ( +g ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` ( Y ( .s ` U ) t ) ) ) = ( ( ( G ` X ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) ( +g ` ( (LCDual ` K ) ` W ) ) ( ( G ` Y ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) ) )
48	39, 44, 47	3eqtrrd 2661	. . . . . 6 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( ( G ` X ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) ( +g ` ( (LCDual ` K ) ` W ) ) ( ( G ` Y ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) ) = ( ( (HDMap ` K ) ` W ) ` ( ( X .+ Y ) ( .s ` U ) t ) ) )
49	10, 11, 36	lmodacl 18874	. . . . . . . . 9 |- ( ( U e. LMod /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
50	31, 16, 19, 49	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( X .+ Y ) e. B )
51	50	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( X .+ Y ) e. B )
52	1, 2, 3, 35, 10, 11, 7, 27, 23, 14, 15, 24, 51	hgmapvs 37183	. . . . . 6 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( (HDMap ` K ) ` W ) ` ( ( X .+ Y ) ( .s ` U ) t ) ) = ( ( G ` ( X .+ Y ) ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) )
53	30, 48, 52	3eqtrrd 2661	. . . . 5 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( G ` ( X .+ Y ) ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) = ( ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) )
54		eqid 2622	. . . . . 6 |- ( 0g ` ( (LCDual ` K ) ` W ) ) = ( 0g ` ( (LCDual ` K ) ` W ) )
55	1, 7, 5	lcdlvec 36880	. . . . . . 7 |- ( ph -> ( (LCDual ` K ) ` W ) e. LVec )
56	55	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( (LCDual ` K ) ` W ) e. LVec )
57	1, 2, 10, 11, 7, 12, 13, 14, 5, 50	hgmapdcl 37182	. . . . . . 7 |- ( ph -> ( G ` ( X .+ Y ) ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) )
58	57	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( G ` ( X .+ Y ) ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) )
59	1, 2, 10, 11, 7, 12, 13, 14, 5, 16	hgmapdcl 37182	. . . . . . . 8 |- ( ph -> ( G ` X ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) )
60	12, 13, 28	lmodacl 18874	. . . . . . . 8 |- ( ( ( (LCDual ` K ) ` W ) e. LMod /\ ( G ` X ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) /\ ( G ` Y ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ) -> ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) )
61	8, 59, 20, 60	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) )
62	61	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) e. ( Base ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) )
63		simp3 1063	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> t =/= ( 0g ` U ) )
64	1, 2, 3, 4, 7, 54, 23, 15, 24	hdmapeq0 37136	. . . . . . . 8 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( ( (HDMap ` K ) ` W ) ` t ) = ( 0g ` ( (LCDual ` K ) ` W ) ) <-> t = ( 0g ` U ) ) )
65	64	necon3bid 2838	. . . . . . 7 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( ( (HDMap ` K ) ` W ) ` t ) =/= ( 0g ` ( (LCDual ` K ) ` W ) ) <-> t =/= ( 0g ` U ) ) )
66	63, 65	mpbird 247	. . . . . 6 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( (HDMap ` K ) ` W ) ` t ) =/= ( 0g ` ( (LCDual ` K ) ` W ) ) )
67	22, 27, 12, 13, 54, 56, 58, 62, 25, 66	lvecvscan2 19112	. . . . 5 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( ( ( G ` ( X .+ Y ) ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) = ( ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) ( .s ` ( (LCDual ` K ) ` W ) ) ( ( (HDMap ` K ) ` W ) ` t ) ) <-> ( G ` ( X .+ Y ) ) = ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) ) )
68	53, 67	mpbid 222	. . . 4 |- ( ( ph /\ t e. ( Base ` U ) /\ t =/= ( 0g ` U ) ) -> ( G ` ( X .+ Y ) ) = ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) )
69	68	rexlimdv3a 3033	. . 3 |- ( ph -> ( E. t e. ( Base ` U ) t =/= ( 0g ` U ) -> ( G ` ( X .+ Y ) ) = ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) ) )
70	6, 69	mpd 15	. 2 |- ( ph -> ( G ` ( X .+ Y ) ) = ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) )
71	1, 2, 10, 36, 7, 12, 28, 5	lcdsadd 36890	. . 3 |- ( ph -> ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) = .+ )
72	71	oveqd 6667	. 2 |- ( ph -> ( ( G ` X ) ( +g ` (Scalar ` ( (LCDual ` K ) ` W ) ) ) ( G ` Y ) ) = ( ( G ` X ) .+ ( G ` Y ) ) )
73	70, 72	eqtrd 2656	1 |- ( ph -> ( G ` ( X .+ Y ) ) = ( ( G ` X ) .+ ( G ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LModclmod 18863   LVecclvec 19102   HLchlt 34637   LHypclh 35270   DVecHcdvh 36367  LCDualclcd 36875  HDMapchdma 37082  HGMapchg 37175

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-ot 4186  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-of 6897  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-mre 16246  df-mrc 16247  df-acs 16249  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-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-oppg 17776  df-lsm 18051  df-cmn 18195  df-abl 18196  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-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  df-lsatoms 34263  df-lshyp 34264  df-lcv 34306  df-lfl 34345  df-lkr 34373  df-ldual 34411  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-tgrp 36031  df-tendo 36043  df-edring 36045  df-dveca 36291  df-disoa 36318  df-dvech 36368  df-dib 36428  df-dic 36462  df-dih 36518  df-doch 36637  df-djh 36684  df-lcdual 36876  df-mapd 36914  df-hvmap 37046  df-hdmap1 37083  df-hdmap 37084  df-hgmap 37176

This theorem is referenced by:  hdmapglem7  37221  hlhilsrnglem  37245