Theorem hdmapval0 41008

Description: Value of map from vectors to functionals at zero. Note: we use dvh3dim 40621 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 41019 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 2731	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
4		eqid 2731	. . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
5		hdmapval0.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		eqid 2731	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		eqid 2731	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
8		hdmapval0.o	. . . . 5 ⊢ 0 = (0g‘𝑈)
9		eqid 2731	. . . . 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 40287	. . . 4 ⊢ (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ ((Base‘𝑈) ∖ { 0 }))
11	10	eldifad 3960	. . 3 ⊢ (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (Base‘𝑈))
12	1, 2, 5	dvhlmod 40285	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
13	3, 8	lmod0vcl 20646	. . . 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 40621	. 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 2731	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		eqid 2731	. . . . 5 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
19		eqid 2731	. . . . 5 ⊢ ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊)
20		hdmapval0.s	. . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
21	5	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	14	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → 0 ∈ (Base‘𝑈))
23		simp2 1136	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → 𝑥 ∈ (Base‘𝑈))
24		eqid 2731	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
25	3, 24, 4, 12, 11, 14	lspprcl 20734	. . . . . . . . . 10 ⊢ (𝜑 → ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) ∈ (LSubSp‘𝑈))
26	3, 4, 12, 11, 14	lspprid1 20753	. . . . . . . . . 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	lspsnel5a 20752	. . . . . . . . 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 20754	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
29	24, 4, 12, 25, 28	lspsnel5a 20752	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘{ 0 }) ⊆ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
30	27, 29	unssd 4186	. . . . . . . 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 3984	. . . . . . 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 1116	. . . . 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 41007	. . . 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 2731	. . . . . 6 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
37		eqid 2731	. . . . . 6 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
38	1, 2, 3, 8, 16, 17, 35, 18, 5, 10	hvmapcl2 40941	. . . . . . . 8 ⊢ (𝜑 → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ ((Base‘𝐶) ∖ {𝑄}))
39	38	eldifad 3960	. . . . . . 7 ⊢ (𝜑 → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ (Base‘𝐶))
40	39	3ad2ant1 1132	. . . . . 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 40944	. . . . . . 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 1132	. . . . . 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 40284	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LVec)
44	43	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → 𝑈 ∈ LVec)
45	11	3ad2ant1 1132	. . . . . . . . 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 1137	. . . . . . . . 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 20891	. . . . . . . 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 2995	. . . . . 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 1132	. . . . . 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 40979	. . . . 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 40974	. . . 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 2771	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑈) ∧ ¬ 𝑥 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 })) → (𝑆‘ 0 ) = 𝑄)
54	53	rexlimdv3a 3158	. 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 1540   ∈ wcel 2105   ≠ wne 2939  ∃wrex 3069   ∖ cdif 3945   ∪ cun 3946  {csn 4628  {cpr 4630  ⟨cop 4634  ⟨cotp 4636   I cid 5573   ↾ cres 5678  ‘cfv 6543  Basecbs 17149  0_gc0g 17390  LModclmod 20615  LSubSpclss 20687  LSpanclspn 20727  LVecclvec 20858  HLchlt 38524  LHypclh 39159  LTrncltrn 39276  DVecHcdvh 40253  LCDualclcd 40761  mapdcmpd 40799  HVMapchvm 40931  HDMap1chdma1 40966  HDMapchdma 40967

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-riotaBAD 38127

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-tpos 8215  df-undef 8262  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-sca 17218  df-vsca 17219  df-0g 17392  df-mre 17535  df-mrc 17536  df-acs 17538  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-submnd 18707  df-grp 18859  df-minusg 18860  df-sbg 18861  df-subg 19040  df-cntz 19223  df-oppg 19252  df-lsm 19546  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-oppr 20226  df-dvdsr 20249  df-unit 20250  df-invr 20280  df-dvr 20293  df-drng 20503  df-lmod 20617  df-lss 20688  df-lsp 20728  df-lvec 20859  df-lsatoms 38150  df-lshyp 38151  df-lcv 38193  df-lfl 38232  df-lkr 38260  df-ldual 38298  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674  df-lvols 38675  df-lines 38676  df-psubsp 38678  df-pmap 38679  df-padd 38971  df-lhyp 39163  df-laut 39164  df-ldil 39279  df-ltrn 39280  df-trl 39334  df-tgrp 39918  df-tendo 39930  df-edring 39932  df-dveca 40178  df-disoa 40204  df-dvech 40254  df-dib 40314  df-dic 40348  df-dih 40404  df-doch 40523  df-djh 40570  df-lcdual 40762  df-mapd 40800  df-hvmap 40932  df-hdmap1 40968  df-hdmap 40969

This theorem is referenced by:  hdmapval3N  41013  hdmap10  41015  hdmaprnlem17N  41038  hdmap14lem2a  41042  hdmap14lem6  41048  hdmap14lem13  41055  hgmapval0  41067