Theorem hdmap14lem6 41872

Description: Case where 𝐹 is zero. (Contributed by NM, 1-Jun-2015.)

Hypotheses

Ref	Expression
hdmap14lem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap14lem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem1.v	⊢ 𝑉 = (Base‘𝑈)
hdmap14lem1.t	⊢ · = ( ·𝑠 ‘𝑈)
hdmap14lem3.o	⊢ 0 = (0g‘𝑈)
hdmap14lem1.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmap14lem1.b	⊢ 𝐵 = (Base‘𝑅)
hdmap14lem1.z	⊢ 𝑍 = (0g‘𝑅)
hdmap14lem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem2.e	⊢ ∙ = ( ·𝑠 ‘𝐶)
hdmap14lem1.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap14lem2.p	⊢ 𝑃 = (Scalar‘𝐶)
hdmap14lem2.a	⊢ 𝐴 = (Base‘𝑃)
hdmap14lem2.q	⊢ 𝑄 = (0g‘𝑃)
hdmap14lem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap14lem3.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap14lem6.f	⊢ (𝜑 → 𝐹 = 𝑍)

Assertion

Ref	Expression
hdmap14lem6	⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))

Distinct variable groups:   𝐴,𝑔   𝐶,𝑔   ∙ ,𝑔   𝑄,𝑔   𝑆,𝑔   𝑔,𝑋   𝜑,𝑔

Allowed substitution hints:   𝐵(𝑔)   𝑃(𝑔)   𝑅(𝑔)   · (𝑔)   𝑈(𝑔)   𝐹(𝑔)   𝐻(𝑔)   𝐾(𝑔)   𝐿(𝑔)   𝑉(𝑔)   𝑊(𝑔)   0 (𝑔)   𝑍(𝑔)

Proof of Theorem hdmap14lem6

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap14lem1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap14lem1.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		hdmap14lem1.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	lcdlmod 41591	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hdmap14lem2.p	. . . . . 6 ⊢ 𝑃 = (Scalar‘𝐶)
6		hdmap14lem2.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝑃)
7		hdmap14lem2.q	. . . . . 6 ⊢ 𝑄 = (0g‘𝑃)
8	5, 6, 7	lmod0cl 20810	. . . . 5 ⊢ (𝐶 ∈ LMod → 𝑄 ∈ 𝐴)
9	4, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
10		hdmap14lem1.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11		hdmap14lem1.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
12		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		hdmap14lem1.s	. . . . . . 7 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
14		hdmap14lem3.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
15	14	eldifad 3917	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
16	1, 10, 11, 2, 12, 13, 3, 15	hdmapcl 41829	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
17		hdmap14lem2.e	. . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝐶)
18		eqid 2729	. . . . . . 7 ⊢ (0g‘𝐶) = (0g‘𝐶)
19	12, 5, 17, 7, 18	lmod0vs 20817	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑋) ∈ (Base‘𝐶)) → (𝑄 ∙ (𝑆‘𝑋)) = (0g‘𝐶))
20	4, 16, 19	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑄 ∙ (𝑆‘𝑋)) = (0g‘𝐶))
21	20	eqcomd 2735	. . . 4 ⊢ (𝜑 → (0g‘𝐶) = (𝑄 ∙ (𝑆‘𝑋)))
22		oveq1 7360	. . . . 5 ⊢ (𝑔 = 𝑄 → (𝑔 ∙ (𝑆‘𝑋)) = (𝑄 ∙ (𝑆‘𝑋)))
23	22	rspceeqv 3602	. . . 4 ⊢ ((𝑄 ∈ 𝐴 ∧ (0g‘𝐶) = (𝑄 ∙ (𝑆‘𝑋))) → ∃𝑔 ∈ 𝐴 (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)))
24	9, 21, 23	syl2anc 584	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)))
25		hdmap14lem3.o	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑈)
26	1, 10, 11, 25, 2, 18, 12, 13, 3, 14	hdmapnzcl 41844	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘𝑋) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)}))
27		eldifsni 4744	. . . . . . . . . 10 ⊢ ((𝑆‘𝑋) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)}) → (𝑆‘𝑋) ≠ (0g‘𝐶))
28	26, 27	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘𝑋) ≠ (0g‘𝐶))
29	28	neneqd 2930	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑆‘𝑋) = (0g‘𝐶))
30	29	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → ¬ (𝑆‘𝑋) = (0g‘𝐶))
31		simp3l 1202	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)))
32	31	eqcomd 2735	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (𝑔 ∙ (𝑆‘𝑋)) = (0g‘𝐶))
33	1, 2, 3	lcdlvec 41590	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ LVec)
34	33	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → 𝐶 ∈ LVec)
35		simp2l 1200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → 𝑔 ∈ 𝐴)
36	16	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (𝑆‘𝑋) ∈ (Base‘𝐶))
37	12, 17, 5, 6, 7, 18, 34, 35, 36	lvecvs0or 21034	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → ((𝑔 ∙ (𝑆‘𝑋)) = (0g‘𝐶) ↔ (𝑔 = 𝑄 ∨ (𝑆‘𝑋) = (0g‘𝐶))))
38	32, 37	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (𝑔 = 𝑄 ∨ (𝑆‘𝑋) = (0g‘𝐶)))
39	38	orcomd 871	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → ((𝑆‘𝑋) = (0g‘𝐶) ∨ 𝑔 = 𝑄))
40	39	ord 864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (¬ (𝑆‘𝑋) = (0g‘𝐶) → 𝑔 = 𝑄))
41	30, 40	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → 𝑔 = 𝑄)
42		simp3r 1203	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))
43	42	eqcomd 2735	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (ℎ ∙ (𝑆‘𝑋)) = (0g‘𝐶))
44		simp2r 1201	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → ℎ ∈ 𝐴)
45	12, 17, 5, 6, 7, 18, 34, 44, 36	lvecvs0or 21034	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → ((ℎ ∙ (𝑆‘𝑋)) = (0g‘𝐶) ↔ (ℎ = 𝑄 ∨ (𝑆‘𝑋) = (0g‘𝐶))))
46	43, 45	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (ℎ = 𝑄 ∨ (𝑆‘𝑋) = (0g‘𝐶)))
47	46	orcomd 871	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → ((𝑆‘𝑋) = (0g‘𝐶) ∨ ℎ = 𝑄))
48	47	ord 864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (¬ (𝑆‘𝑋) = (0g‘𝐶) → ℎ = 𝑄))
49	30, 48	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → ℎ = 𝑄)
50	41, 49	eqtr4d 2767	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → 𝑔 = ℎ)
51	50	3exp 1119	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → (((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))) → 𝑔 = ℎ)))
52	51	ralrimivv 3170	. . 3 ⊢ (𝜑 → ∀𝑔 ∈ 𝐴 ∀ℎ ∈ 𝐴 (((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))) → 𝑔 = ℎ))
53		oveq1 7360	. . . . 5 ⊢ (𝑔 = ℎ → (𝑔 ∙ (𝑆‘𝑋)) = (ℎ ∙ (𝑆‘𝑋)))
54	53	eqeq2d 2740	. . . 4 ⊢ (𝑔 = ℎ → ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ↔ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))))
55	54	reu4 3693	. . 3 ⊢ (∃!𝑔 ∈ 𝐴 (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ↔ (∃𝑔 ∈ 𝐴 (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ ∀𝑔 ∈ 𝐴 ∀ℎ ∈ 𝐴 (((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))) → 𝑔 = ℎ)))
56	24, 52, 55	sylanbrc 583	. 2 ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)))
57		hdmap14lem6.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = 𝑍)
58	57	oveq1d 7368	. . . . . . 7 ⊢ (𝜑 → (𝐹 · 𝑋) = (𝑍 · 𝑋))
59	1, 10, 3	dvhlmod 41109	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
60		hdmap14lem1.r	. . . . . . . . 9 ⊢ 𝑅 = (Scalar‘𝑈)
61		hdmap14lem1.t	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑈)
62		hdmap14lem1.z	. . . . . . . . 9 ⊢ 𝑍 = (0g‘𝑅)
63	11, 60, 61, 62, 25	lmod0vs 20817	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑍 · 𝑋) = 0 )
64	59, 15, 63	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑍 · 𝑋) = 0 )
65	58, 64	eqtrd 2764	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑋) = 0 )
66	65	fveq2d 6830	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝑆‘ 0 ))
67	1, 10, 25, 2, 18, 13, 3	hdmapval0 41832	. . . . 5 ⊢ (𝜑 → (𝑆‘ 0 ) = (0g‘𝐶))
68	66, 67	eqtrd 2764	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (0g‘𝐶))
69	68	eqeq1d 2731	. . 3 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋))))
70	69	reubidv 3363	. 2 ⊢ (𝜑 → (∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋))))
71	56, 70	mpbird 257	1 ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  ∃!wreu 3343   ∖ cdif 3902  {csn 4579  ‘cfv 6486  (class class class)co 7353  Basecbs 17139  Scalarcsca 17183   ·_𝑠 cvsca 17184  0_gc0g 17362  LModclmod 20782  LSpanclspn 20893  LVecclvec 21025  HLchlt 39348  LHypclh 39983  DVecHcdvh 41077  LCDualclcd 41585  HDMapchdma 41791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-riotaBAD 38951

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  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 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-undef 8213  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-n0 12404  df-z 12491  df-uz 12755  df-fz 13430  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17140  df-ress 17161  df-plusg 17193  df-mulr 17194  df-sca 17196  df-vsca 17197  df-0g 17364  df-mre 17507  df-mrc 17508  df-acs 17510  df-proset 18219  df-poset 18238  df-plt 18253  df-lub 18269  df-glb 18270  df-join 18271  df-meet 18272  df-p0 18348  df-p1 18349  df-lat 18357  df-clat 18424  df-mgm 18533  df-sgrp 18612  df-mnd 18628  df-submnd 18677  df-grp 18834  df-minusg 18835  df-sbg 18836  df-subg 19021  df-cntz 19215  df-oppg 19244  df-lsm 19534  df-cmn 19680  df-abl 19681  df-mgp 20045  df-rng 20057  df-ur 20086  df-ring 20139  df-oppr 20241  df-dvdsr 20261  df-unit 20262  df-invr 20292  df-dvr 20305  df-nzr 20417  df-rlreg 20598  df-domn 20599  df-drng 20635  df-lmod 20784  df-lss 20854  df-lsp 20894  df-lvec 21026  df-lsatoms 38974  df-lshyp 38975  df-lcv 39017  df-lfl 39056  df-lkr 39084  df-ldual 39122  df-oposet 39174  df-ol 39176  df-oml 39177  df-covers 39264  df-ats 39265  df-atl 39296  df-cvlat 39320  df-hlat 39349  df-llines 39497  df-lplanes 39498  df-lvols 39499  df-lines 39500  df-psubsp 39502  df-pmap 39503  df-padd 39795  df-lhyp 39987  df-laut 39988  df-ldil 40103  df-ltrn 40104  df-trl 40158  df-tgrp 40742  df-tendo 40754  df-edring 40756  df-dveca 41002  df-disoa 41028  df-dvech 41078  df-dib 41138  df-dic 41172  df-dih 41228  df-doch 41347  df-djh 41394  df-lcdual 41586  df-mapd 41624  df-hvmap 41756  df-hdmap1 41792  df-hdmap 41793

This theorem is referenced by:  hdmap14lem7  41873