Theorem hdmap14lem6 41840

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 41559	. . . . 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 20770	. . . . 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 3923	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
16	1, 10, 11, 2, 12, 13, 3, 15	hdmapcl 41797	. . . . . 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 20777	. . . . . 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 7376	. . . . 5 ⊢ (𝑔 = 𝑄 → (𝑔 ∙ (𝑆‘𝑋)) = (𝑄 ∙ (𝑆‘𝑋)))
23	22	rspceeqv 3608	. . . 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 41812	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘𝑋) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)}))
27		eldifsni 4750	. . . . . . . . . 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 41558	. . . . . . . . . . . 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 20994	. . . . . . . . . 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 20994	. . . . . . . . . 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 3176	. . 3 ⊢ (𝜑 → ∀𝑔 ∈ 𝐴 ∀ℎ ∈ 𝐴 (((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))) → 𝑔 = ℎ))
53		oveq1 7376	. . . . 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 3699	. . 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 7384	. . . . . . 7 ⊢ (𝜑 → (𝐹 · 𝑋) = (𝑍 · 𝑋))
59	1, 10, 3	dvhlmod 41077	. . . . . . . 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 20777	. . . . . . . 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 6844	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝑆‘ 0 ))
67	1, 10, 25, 2, 18, 13, 3	hdmapval0 41800	. . . . 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 3369	. 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 3349   ∖ cdif 3908  {csn 4585  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  Scalarcsca 17199   ·_𝑠 cvsca 17200  0_gc0g 17378  LModclmod 20742  LSpanclspn 20853  LVecclvec 20985  HLchlt 39316  LHypclh 39951  DVecHcdvh 41045  LCDualclcd 41553  HDMapchdma 41759

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-riotaBAD 38919

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-tpos 8182  df-undef 8229  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-n0 12419  df-z 12506  df-uz 12770  df-fz 13445  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-0g 17380  df-mre 17523  df-mrc 17524  df-acs 17526  df-proset 18231  df-poset 18250  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-p1 18361  df-lat 18367  df-clat 18434  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-submnd 18687  df-grp 18844  df-minusg 18845  df-sbg 18846  df-subg 19031  df-cntz 19225  df-oppg 19254  df-lsm 19542  df-cmn 19688  df-abl 19689  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-oppr 20222  df-dvdsr 20242  df-unit 20243  df-invr 20273  df-dvr 20286  df-nzr 20398  df-rlreg 20579  df-domn 20580  df-drng 20616  df-lmod 20744  df-lss 20814  df-lsp 20854  df-lvec 20986  df-lsatoms 38942  df-lshyp 38943  df-lcv 38985  df-lfl 39024  df-lkr 39052  df-ldual 39090  df-oposet 39142  df-ol 39144  df-oml 39145  df-covers 39232  df-ats 39233  df-atl 39264  df-cvlat 39288  df-hlat 39317  df-llines 39465  df-lplanes 39466  df-lvols 39467  df-lines 39468  df-psubsp 39470  df-pmap 39471  df-padd 39763  df-lhyp 39955  df-laut 39956  df-ldil 40071  df-ltrn 40072  df-trl 40126  df-tgrp 40710  df-tendo 40722  df-edring 40724  df-dveca 40970  df-disoa 40996  df-dvech 41046  df-dib 41106  df-dic 41140  df-dih 41196  df-doch 41315  df-djh 41362  df-lcdual 41554  df-mapd 41592  df-hvmap 41724  df-hdmap1 41760  df-hdmap 41761

This theorem is referenced by:  hdmap14lem7  41841