Theorem hdmapval0 41836

Description: Value of map from vectors to functionals at zero. Note: we use dvh3dim 41449 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 41847 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 2736	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
4		eqid 2736	. . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
5		hdmapval0.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		eqid 2736	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		eqid 2736	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
8		hdmapval0.o	. . . . 5 ⊢ 0 = (0g‘𝑈)
9		eqid 2736	. . . . 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 41115	. . . 4 ⊢ (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ ((Base‘𝑈) ∖ { 0 }))
11	10	eldifad 3962	. . 3 ⊢ (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (Base‘𝑈))
12	1, 2, 5	dvhlmod 41113	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
13	3, 8	lmod0vcl 20890	. . . 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 41449	. 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 2736	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		eqid 2736	. . . . 5 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
19		eqid 2736	. . . . 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 2736	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
25	3, 24, 4, 12, 11, 14	lspprcl 20977	. . . . . . . . . 10 ⊢ (𝜑 → ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }) ∈ (LSubSp‘𝑈))
26	3, 4, 12, 11, 14	lspprid1 20996	. . . . . . . . . 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 20995	. . . . . . . . 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 20997	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
29	24, 4, 12, 25, 28	ellspsn5 20995	. . . . . . . . 9 ⊢ (𝜑 → ((LSpan‘𝑈)‘{ 0 }) ⊆ ((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, 0 }))
30	27, 29	unssd 4191	. . . . . . . 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 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 41835	. . . 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 2736	. . . . . 6 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
37		eqid 2736	. . . . . 6 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
38	1, 2, 3, 8, 16, 17, 35, 18, 5, 10	hvmapcl2 41769	. . . . . . . 8 ⊢ (𝜑 → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ ((Base‘𝐶) ∖ {𝑄}))
39	38	eldifad 3962	. . . . . . 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 41772	. . . . . . 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 41112	. . . . . . . . . 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 21135	. . . . . . . 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 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 41807	. . . . 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 41802	. . . 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 2776	. . 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 1539   ∈ wcel 2107   ≠ wne 2939  ∃wrex 3069   ∖ cdif 3947   ∪ cun 3948  {csn 4625  {cpr 4627  ⟨cop 4631  ⟨cotp 4633   I cid 5576   ↾ cres 5686  ‘cfv 6560  Basecbs 17248  0_gc0g 17485  LModclmod 20859  LSubSpclss 20930  LSpanclspn 20970  LVecclvec 21102  HLchlt 39352  LHypclh 39987  LTrncltrn 40104  DVecHcdvh 41081  LCDualclcd 41589  mapdcmpd 41627  HVMapchvm 41759  HDMap1chdma1 41794  HDMapchdma 41795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-riotaBAD 38955

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-undef 8299  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-0g 17487  df-mre 17630  df-mrc 17631  df-acs 17633  df-proset 18341  df-poset 18360  df-plt 18376  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-p0 18471  df-p1 18472  df-lat 18478  df-clat 18545  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-subg 19142  df-cntz 19336  df-oppg 19365  df-lsm 19655  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-dvr 20402  df-nzr 20514  df-rlreg 20695  df-domn 20696  df-drng 20732  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lvec 21103  df-lsatoms 38978  df-lshyp 38979  df-lcv 39021  df-lfl 39060  df-lkr 39088  df-ldual 39126  df-oposet 39178  df-ol 39180  df-oml 39181  df-covers 39268  df-ats 39269  df-atl 39300  df-cvlat 39324  df-hlat 39353  df-llines 39501  df-lplanes 39502  df-lvols 39503  df-lines 39504  df-psubsp 39506  df-pmap 39507  df-padd 39799  df-lhyp 39991  df-laut 39992  df-ldil 40107  df-ltrn 40108  df-trl 40162  df-tgrp 40746  df-tendo 40758  df-edring 40760  df-dveca 41006  df-disoa 41032  df-dvech 41082  df-dib 41142  df-dic 41176  df-dih 41232  df-doch 41351  df-djh 41398  df-lcdual 41590  df-mapd 41628  df-hvmap 41760  df-hdmap1 41796  df-hdmap 41797

This theorem is referenced by:  hdmapval3N  41841  hdmap10  41843  hdmaprnlem17N  41866  hdmap14lem2a  41870  hdmap14lem6  41876  hdmap14lem13  41883  hgmapval0  41895