mapdval2N - Mathbox for Norm Megill

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Theorem mapdval2N 41014

Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
mapdval2.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdval2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdval2.s	⊢ 𝑆 = (LSubSp‘𝑈)
mapdval2.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdval2.f	⊢ 𝐹 = (LFnl‘𝑈)
mapdval2.l	⊢ 𝐿 = (LKer‘𝑈)
mapdval2.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
mapdval2.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdval2.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdval2.t	⊢ (𝜑 → 𝑇 ∈ 𝑆)
mapdval2.c	⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}

**Assertion**
Ref	Expression
mapdval2N	⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})})

Distinct variable groups: 𝑣,𝐶 𝑓,𝑔,𝐹 𝑓,𝐾 𝑣,𝑔,𝐿 𝑣,𝑁 𝑔,𝑂,𝑣 𝑣,𝑓,𝑇 𝑣,𝑈 𝑓,𝑊 𝜑,𝑓,𝑣 Allowed substitution hints: 𝜑(𝑔) 𝐶(𝑓,𝑔) 𝑆(𝑣,𝑓,𝑔) 𝑇(𝑔) 𝑈(𝑓,𝑔) 𝐹(𝑣) 𝐻(𝑣,𝑓,𝑔) 𝐾(𝑣,𝑔) 𝐿(𝑓) 𝑀(𝑣,𝑓,𝑔) 𝑁(𝑓,𝑔) 𝑂(𝑓) 𝑊(𝑣,𝑔)

**Proof of Theorem mapdval2N**
Step	Hyp	Ref	Expression
1		mapdval2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdval2.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdval2.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑈)
4		mapdval2.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
5		mapdval2.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
6		mapdval2.o	. . 3 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
7		mapdval2.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
8		mapdval2.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		mapdval2.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
10		mapdval2.c	. . 3 ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	mapdvalc 41013	. 2 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇})
12	1, 2, 8	dvhlmod 40494	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
13	12	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) → 𝑈 ∈ LMod)
14		simplr 766	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) → (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈))
15		eqid 2726	. . . . . . . . 9 ⊢ (Base‘𝑈) = (Base‘𝑈)
16		mapdval2.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
17		eqid 2726	. . . . . . . . 9 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
18	15, 16, 17	islsati 38377	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) → ∃𝑣 ∈ (Base‘𝑈)(𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))
19	13, 14, 18	syl2anc 583	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) → ∃𝑣 ∈ (Base‘𝑈)(𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))
20		simprr 770	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ∧ (𝑣 ∈ (Base‘𝑈) ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))) → (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))
21		simplr 766	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ∧ (𝑣 ∈ (Base‘𝑈) ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))) → (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)
22	20, 21	eqsstrrd 4016	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ∧ (𝑣 ∈ (Base‘𝑈) ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))) → (𝑁‘{𝑣}) ⊆ 𝑇)
23	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑈 ∈ LMod)
24	23	ad3antrrr 727	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ∧ (𝑣 ∈ (Base‘𝑈) ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))) → 𝑈 ∈ LMod)
25	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑇 ∈ 𝑆)
26	25	ad3antrrr 727	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ∧ (𝑣 ∈ (Base‘𝑈) ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))) → 𝑇 ∈ 𝑆)
27		simprl 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ∧ (𝑣 ∈ (Base‘𝑈) ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))) → 𝑣 ∈ (Base‘𝑈))
28	15, 3, 16, 24, 26, 27	lspsnel5 20842	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ∧ (𝑣 ∈ (Base‘𝑈) ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))) → (𝑣 ∈ 𝑇 ↔ (𝑁‘{𝑣}) ⊆ 𝑇))
29	22, 28	mpbird 257	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ∧ (𝑣 ∈ (Base‘𝑈) ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))) → 𝑣 ∈ 𝑇)
30	19, 29, 20	reximssdv 3166	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) → ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))
31	30	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈)) → ((𝑂‘(𝐿‘𝑓)) ⊆ 𝑇 → ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})))
32		eqid 2726	. . . . . . . . . . 11 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
33	32, 3	lss0cl 20794	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑇 ∈ 𝑆) → (0_g‘𝑈) ∈ 𝑇)
34	12, 9, 33	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (0_g‘𝑈) ∈ 𝑇)
35	34	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)}) → (0_g‘𝑈) ∈ 𝑇)
36		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)}) → (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)})
37	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)}) → 𝑈 ∈ LMod)
38	32, 16	lspsn0 20855	. . . . . . . . . 10 ⊢ (𝑈 ∈ LMod → (𝑁‘{(0_g‘𝑈)}) = {(0_g‘𝑈)})
39	37, 38	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)}) → (𝑁‘{(0_g‘𝑈)}) = {(0_g‘𝑈)})
40	36, 39	eqtr4d 2769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)}) → (𝑂‘(𝐿‘𝑓)) = (𝑁‘{(0_g‘𝑈)}))
41		sneq 4633	. . . . . . . . . 10 ⊢ (𝑣 = (0_g‘𝑈) → {𝑣} = {(0_g‘𝑈)})
42	41	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑣 = (0_g‘𝑈) → (𝑁‘{𝑣}) = (𝑁‘{(0_g‘𝑈)}))
43	42	rspceeqv 3628	. . . . . . . 8 ⊢ (((0_g‘𝑈) ∈ 𝑇 ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{(0_g‘𝑈)})) → ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))
44	35, 40, 43	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)}) → ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))
45	44	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)}) → ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))
46	45	a1d 25	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)}) → ((𝑂‘(𝐿‘𝑓)) ⊆ 𝑇 → ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})))
47	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
48	10	lcfl1lem 40875	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐶 ↔ (𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)))
49	48	simplbi 497	. . . . . . 7 ⊢ (𝑓 ∈ 𝐶 → 𝑓 ∈ 𝐹)
50	49	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ 𝐹)
51	1, 6, 2, 32, 17, 4, 5, 47, 50	dochsat0 40841	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → ((𝑂‘(𝐿‘𝑓)) ∈ (LSAtoms‘𝑈) ∨ (𝑂‘(𝐿‘𝑓)) = {(0_g‘𝑈)}))
52	31, 46, 51	mpjaodan 955	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → ((𝑂‘(𝐿‘𝑓)) ⊆ 𝑇 → ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})))
53		simp3 1135	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ 𝑣 ∈ 𝑇 ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})) → (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}))
54	23	3ad2ant1 1130	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ 𝑣 ∈ 𝑇 ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})) → 𝑈 ∈ LMod)
55	25	3ad2ant1 1130	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ 𝑣 ∈ 𝑇 ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})) → 𝑇 ∈ 𝑆)
56		simp2 1134	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ 𝑣 ∈ 𝑇 ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})) → 𝑣 ∈ 𝑇)
57	3, 16, 54, 55, 56	lspsnel5a 20843	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ 𝑣 ∈ 𝑇 ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})) → (𝑁‘{𝑣}) ⊆ 𝑇)
58	53, 57	eqsstrd 4015	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐶) ∧ 𝑣 ∈ 𝑇 ∧ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})) → (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)
59	58	rexlimdv3a 3153	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣}) → (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))
60	52, 59	impbid 211	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → ((𝑂‘(𝐿‘𝑓)) ⊆ 𝑇 ↔ ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})))
61	60	rabbidva 3433	. 2 ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇} = {𝑓 ∈ 𝐶 ∣ ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})})
62	11, 61	eqtrd 2766	1 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 {crab 3426 ⊆ wss 3943 {csn 4623 ‘cfv 6537 Basecbs 17153 0_gc0g 17394 LModclmod 20706 LSubSpclss 20778 LSpanclspn 20818 LSAtomsclsa 38357 LFnlclfn 38440 LKerclk 38468 HLchlt 38733 LHypclh 39368 DVecHcdvh 40462 ocHcoch 40731 mapdcmpd 41008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38336

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-0g 17396 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cntz 19233 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lvec 20951 df-lsatoms 38359 df-lshyp 38360 df-lfl 38441 df-lkr 38469 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tgrp 40127 df-tendo 40139 df-edring 40141 df-dveca 40387 df-disoa 40413 df-dvech 40463 df-dib 40523 df-dic 40557 df-dih 40613 df-doch 40732 df-djh 40779 df-mapd 41009

This theorem is referenced by: mapdval3N 41015 mapdval4N 41016

W3C validator