Theorem hdmap14lem6 39814

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 39533	. . . . 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 20064	. . . . 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 2738	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		hdmap14lem1.s	. . . . . . 7 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
14		hdmap14lem3.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
15	14	eldifad 3895	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
16	1, 10, 11, 2, 12, 13, 3, 15	hdmapcl 39771	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
17		hdmap14lem2.e	. . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝐶)
18		eqid 2738	. . . . . . 7 ⊢ (0g‘𝐶) = (0g‘𝐶)
19	12, 5, 17, 7, 18	lmod0vs 20071	. . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑋) ∈ (Base‘𝐶)) → (𝑄 ∙ (𝑆‘𝑋)) = (0g‘𝐶))
20	4, 16, 19	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑄 ∙ (𝑆‘𝑋)) = (0g‘𝐶))
21	20	eqcomd 2744	. . . 4 ⊢ (𝜑 → (0g‘𝐶) = (𝑄 ∙ (𝑆‘𝑋)))
22		oveq1 7262	. . . . 5 ⊢ (𝑔 = 𝑄 → (𝑔 ∙ (𝑆‘𝑋)) = (𝑄 ∙ (𝑆‘𝑋)))
23	22	rspceeqv 3567	. . . 4 ⊢ ((𝑄 ∈ 𝐴 ∧ (0g‘𝐶) = (𝑄 ∙ (𝑆‘𝑋))) → ∃𝑔 ∈ 𝐴 (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)))
24	9, 21, 23	syl2anc 583	. . 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 39786	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘𝑋) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)}))
27		eldifsni 4720	. . . . . . . . . 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 2947	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑆‘𝑋) = (0g‘𝐶))
30	29	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → ¬ (𝑆‘𝑋) = (0g‘𝐶))
31		simp3l 1199	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)))
32	31	eqcomd 2744	. . . . . . . . . 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 39532	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ LVec)
34	33	3ad2ant1 1131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → 𝐶 ∈ LVec)
35		simp2l 1197	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → 𝑔 ∈ 𝐴)
36	16	3ad2ant1 1131	. . . . . . . . . . 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 20285	. . . . . . . . . 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 231	. . . . . . . . 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 867	. . . . . . . 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 860	. . . . . . 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 1200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))
43	42	eqcomd 2744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → (ℎ ∙ (𝑆‘𝑋)) = (0g‘𝐶))
44		simp2r 1198	. . . . . . . . . . 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 20285	. . . . . . . . . 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 231	. . . . . . . . 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 867	. . . . . . . 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 860	. . . . . . 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 2781	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) ∧ ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋)))) → 𝑔 = ℎ)
51	50	3exp 1117	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → (((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))) → 𝑔 = ℎ)))
52	51	ralrimivv 3113	. . 3 ⊢ (𝜑 → ∀𝑔 ∈ 𝐴 ∀ℎ ∈ 𝐴 (((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))) → 𝑔 = ℎ))
53		oveq1 7262	. . . . 5 ⊢ (𝑔 = ℎ → (𝑔 ∙ (𝑆‘𝑋)) = (ℎ ∙ (𝑆‘𝑋)))
54	53	eqeq2d 2749	. . . 4 ⊢ (𝑔 = ℎ → ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ↔ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))))
55	54	reu4 3661	. . 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 582	. 2 ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)))
57		hdmap14lem6.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = 𝑍)
58	57	oveq1d 7270	. . . . . . 7 ⊢ (𝜑 → (𝐹 · 𝑋) = (𝑍 · 𝑋))
59	1, 10, 3	dvhlmod 39051	. . . . . . . 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 20071	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑍 · 𝑋) = 0 )
64	59, 15, 63	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑍 · 𝑋) = 0 )
65	58, 64	eqtrd 2778	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑋) = 0 )
66	65	fveq2d 6760	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝑆‘ 0 ))
67	1, 10, 25, 2, 18, 13, 3	hdmapval0 39774	. . . . 5 ⊢ (𝜑 → (𝑆‘ 0 ) = (0g‘𝐶))
68	66, 67	eqtrd 2778	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (0g‘𝐶))
69	68	eqeq1d 2740	. . 3 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋))))
70	69	reubidv 3315	. 2 ⊢ (𝜑 → (∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋))))
71	56, 70	mpbird 256	1 ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065   ∖ cdif 3880  {csn 4558  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067  LModclmod 20038  LSpanclspn 20148  LVecclvec 20279  HLchlt 37291  LHypclh 37925  DVecHcdvh 39019  LCDualclcd 39527  HDMapchdma 39733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-riotaBAD 36894

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-undef 8060  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-0g 17069  df-mre 17212  df-mrc 17213  df-acs 17215  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-cntz 18838  df-oppg 18865  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-dvr 19840  df-drng 19908  df-lmod 20040  df-lss 20109  df-lsp 20149  df-lvec 20280  df-lsatoms 36917  df-lshyp 36918  df-lcv 36960  df-lfl 36999  df-lkr 37027  df-ldual 37065  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-lplanes 37440  df-lvols 37441  df-lines 37442  df-psubsp 37444  df-pmap 37445  df-padd 37737  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100  df-tgrp 38684  df-tendo 38696  df-edring 38698  df-dveca 38944  df-disoa 38970  df-dvech 39020  df-dib 39080  df-dic 39114  df-dih 39170  df-doch 39289  df-djh 39336  df-lcdual 39528  df-mapd 39566  df-hvmap 39698  df-hdmap1 39734  df-hdmap 39735

This theorem is referenced by:  hdmap14lem7  39815