Theorem hdmapval0 41952

Description: Value of map from vectors to functionals at zero. Note: we use dvh3dim 41565 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 41963 could be derived from the other to shorten proof. (Contributed by NM, 17-May-2015.)

Hypotheses

Ref	Expression
hdmapval0.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapval0.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapval0.o	⊢ 0 = (0g‘𝑈)
hdmapval0.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmapval0.q	⊢ 𝑄 = (0g‘𝐶)
hdmapval0.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapval0.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

Assertion

Ref	Expression
hdmapval0	⊢ (𝜑 → (𝑆‘ 0 ) = 𝑄)

Proof of Theorem hdmapval0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmapval0.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmapval0.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		eqid 2733	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
4		eqid 2733	. . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
5		hdmapval0.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		eqid 2733	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		eqid 2733	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
8		hdmapval0.o	. . . . 5 ⊢ 0 = (0g‘𝑈)
9		eqid 2733	. . . . 5 ⊢ ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
10	1, 6, 7, 2, 3, 8, 9, 5	dvheveccl 41231	. . . 4 ⊢ (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ ((Base‘𝑈) ∖ { 0 }))
11	10	eldifad 3910	. . 3 ⊢ (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (Base‘𝑈))
12	1, 2, 5	dvhlmod 41229	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
13	3, 8	lmod0vcl 20826	. . . 4 ⊢ (𝑈 ∈ LMod → 0 ∈ (Base‘𝑈))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝑈))
15	1, 2, 3, 4, 5, 11, 14	dvh3dim 41565	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝑈) ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
16		hdmapval0.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
17		eqid 2733	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		eqid 2733	. . . . 5 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
19		eqid 2733	. . . . 5 ⊢ ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊)
20		hdmapval0.s	. . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
21	5	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	14	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → 0 ∈ (Base‘𝑈))
23		simp2 1137	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → 𝑥 ∈ (Base‘𝑈))
24		eqid 2733	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
25	3, 24, 4, 12, 11, 14	lspprcl 20913	. . . . . . . . . 10 ⊢ (𝜑 → ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) ∈ (LSubSp‘𝑈))
26	3, 4, 12, 11, 14	lspprid1 20932	. . . . . . . . . 10 ⊢ (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
27	24, 4, 12, 25, 26	ellspsn5 20931	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ⊆ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
28	3, 4, 12, 11, 14	lspprid2 20933	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
29	24, 4, 12, 25, 28	ellspsn5 20931	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘{ 0 }) ⊆ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
30	27, 29	unssd 4141	. . . . . . . 8 ⊢ (𝜑 → (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{ 0 })) ⊆ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
31	30	ssneld 3932	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) → ¬ 𝑥 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{ 0 }))))
32	31	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈)) → (¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) → ¬ 𝑥 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{ 0 }))))
33	32	3impia 1117	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ¬ 𝑥 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{ 0 })))
34	1, 9, 2, 3, 4, 16, 17, 18, 19, 20, 21, 22, 23, 33	hdmapval2 41951	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (𝑆‘ 0 ) = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑥, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑥⟩), 0 ⟩))
35		hdmapval0.q	. . . . 5 ⊢ 𝑄 = (0g‘𝐶)
36		eqid 2733	. . . . . 6 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
37		eqid 2733	. . . . . 6 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
38	1, 2, 3, 8, 16, 17, 35, 18, 5, 10	hvmapcl2 41885	. . . . . . . 8 ⊢ (𝜑 → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ ((Base‘𝐶) ∖ {𝑄}))
39	38	eldifad 3910	. . . . . . 7 ⊢ (𝜑 → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ (Base‘𝐶))
40	39	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ (Base‘𝐶))
41	1, 2, 3, 8, 4, 16, 36, 37, 18, 5, 10	mapdhvmap 41888	. . . . . . 7 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩})) = ((LSpan‘𝐶)‘{(((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)}))
42	41	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩})) = ((LSpan‘𝐶)‘{(((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)}))
43	1, 2, 5	dvhlvec 41228	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LVec)
44	43	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → 𝑈 ∈ LVec)
45	11	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (Base‘𝑈))
46		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
47	3, 4, 44, 23, 45, 22, 46	lspindpi 21071	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((LSpan‘𝑈)‘{𝑥}) ≠ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∧ ((LSpan‘𝑈)‘{𝑥}) ≠ ((LSpan‘𝑈)‘{ 0 })))
48	47	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ((LSpan‘𝑈)‘{𝑥}) ≠ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}))
49	48	necomd 2984	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ≠ ((LSpan‘𝑈)‘{𝑥}))
50	10	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ ((Base‘𝑈) ∖ { 0 }))
51	1, 2, 3, 8, 4, 16, 17, 36, 37, 19, 21, 40, 42, 49, 50, 23	hdmap1cl 41923	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑥⟩) ∈ (Base‘𝐶))
52	1, 2, 3, 8, 16, 17, 35, 19, 21, 51, 23	hdmap1val0 41918	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (((HDMap1‘𝐾)‘𝑊)‘⟨𝑥, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑥⟩), 0 ⟩) = 𝑄)
53	34, 52	eqtrd 2768	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (𝑆‘ 0 ) = 𝑄)
54	53	rexlimdv3a 3138	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (Base‘𝑈) ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) → (𝑆‘ 0 ) = 𝑄))
55	15, 54	mpd 15	1 ⊢ (𝜑 → (𝑆‘ 0 ) = 𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∃wrex 3057   ∖ cdif 3895   ∪ cun 3896  {csn 4575  {cpr 4577  ⟨cop 4581  ⟨cotp 4583   I cid 5513   ↾ cres 5621  ‘cfv 6486  Basecbs 17122  0_gc0g 17345  LModclmod 20795  LSubSpclss 20866  LSpanclspn 20906  LVecclvec 21038  HLchlt 39469  LHypclh 40103  LTrncltrn 40220  DVecHcdvh 41197  LCDualclcd 41705  mapdcmpd 41743  HVMapchvm 41875  HDMap1chdma1 41910  HDMapchdma 41911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-riotaBAD 39072

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-ot 4584  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-tpos 8162  df-undef 8209  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-n0 12389  df-z 12476  df-uz 12739  df-fz 13410  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-sca 17179  df-vsca 17180  df-0g 17347  df-mre 17490  df-mrc 17491  df-acs 17493  df-proset 18202  df-poset 18221  df-plt 18236  df-lub 18252  df-glb 18253  df-join 18254  df-meet 18255  df-p0 18331  df-p1 18332  df-lat 18340  df-clat 18407  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-submnd 18694  df-grp 18851  df-minusg 18852  df-sbg 18853  df-subg 19038  df-cntz 19231  df-oppg 19260  df-lsm 19550  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-oppr 20257  df-dvdsr 20277  df-unit 20278  df-invr 20308  df-dvr 20321  df-nzr 20430  df-rlreg 20611  df-domn 20612  df-drng 20648  df-lmod 20797  df-lss 20867  df-lsp 20907  df-lvec 21039  df-lsatoms 39095  df-lshyp 39096  df-lcv 39138  df-lfl 39177  df-lkr 39205  df-ldual 39243  df-oposet 39295  df-ol 39297  df-oml 39298  df-covers 39385  df-ats 39386  df-atl 39417  df-cvlat 39441  df-hlat 39470  df-llines 39617  df-lplanes 39618  df-lvols 39619  df-lines 39620  df-psubsp 39622  df-pmap 39623  df-padd 39915  df-lhyp 40107  df-laut 40108  df-ldil 40223  df-ltrn 40224  df-trl 40278  df-tgrp 40862  df-tendo 40874  df-edring 40876  df-dveca 41122  df-disoa 41148  df-dvech 41198  df-dib 41258  df-dic 41292  df-dih 41348  df-doch 41467  df-djh 41514  df-lcdual 41706  df-mapd 41744  df-hvmap 41876  df-hdmap1 41912  df-hdmap 41913

This theorem is referenced by:  hdmapval3N  41957  hdmap10  41959  hdmaprnlem17N  41982  hdmap14lem2a  41986  hdmap14lem6  41992  hdmap14lem13  41999  hgmapval0  42011