Theorem hdmap14lem6 41891

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 41610	. . . . 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 20814	. . . . 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 2730	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
13		hdmap14lem1.s	. . . . . . 7 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
14		hdmap14lem3.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
15	14	eldifad 3912	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
16	1, 10, 11, 2, 12, 13, 3, 15	hdmapcl 41848	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
17		hdmap14lem2.e	. . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝐶)
18		eqid 2730	. . . . . . 7 ⊢ (0g‘𝐶) = (0g‘𝐶)
19	12, 5, 17, 7, 18	lmod0vs 20821	. . . . . 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 2736	. . . 4 ⊢ (𝜑 → (0g‘𝐶) = (𝑄 ∙ (𝑆‘𝑋)))
22		oveq1 7348	. . . . 5 ⊢ (𝑔 = 𝑄 → (𝑔 ∙ (𝑆‘𝑋)) = (𝑄 ∙ (𝑆‘𝑋)))
23	22	rspceeqv 3598	. . . 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 41863	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘𝑋) ∈ ((Base‘𝐶) ∖ {(0g‘𝐶)}))
27		eldifsni 4740	. . . . . . . . . 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 2931	. . . . . . . 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 2736	. . . . . . . . . 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 41609	. . . . . . . . . . . 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 21038	. . . . . . . . . 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 2736	. . . . . . . . . 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 21038	. . . . . . . . . 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 2768	. . . . 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 3171	. . 3 ⊢ (𝜑 → ∀𝑔 ∈ 𝐴 ∀ℎ ∈ 𝐴 (((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ∧ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))) → 𝑔 = ℎ))
53		oveq1 7348	. . . . 5 ⊢ (𝑔 = ℎ → (𝑔 ∙ (𝑆‘𝑋)) = (ℎ ∙ (𝑆‘𝑋)))
54	53	eqeq2d 2741	. . . 4 ⊢ (𝑔 = ℎ → ((0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋)) ↔ (0g‘𝐶) = (ℎ ∙ (𝑆‘𝑋))))
55	54	reu4 3688	. . 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 7356	. . . . . . 7 ⊢ (𝜑 → (𝐹 · 𝑋) = (𝑍 · 𝑋))
59	1, 10, 3	dvhlmod 41128	. . . . . . . 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 20821	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑍 · 𝑋) = 0 )
64	59, 15, 63	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑍 · 𝑋) = 0 )
65	58, 64	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑋) = 0 )
66	65	fveq2d 6821	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝑆‘ 0 ))
67	1, 10, 25, 2, 18, 13, 3	hdmapval0 41851	. . . . 5 ⊢ (𝜑 → (𝑆‘ 0 ) = (0g‘𝐶))
68	66, 67	eqtrd 2765	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (0g‘𝐶))
69	68	eqeq1d 2732	. . 3 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ (0g‘𝐶) = (𝑔 ∙ (𝑆‘𝑋))))
70	69	reubidv 3360	. 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 1541   ∈ wcel 2110   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  ∃!wreu 3342   ∖ cdif 3897  {csn 4574  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  Scalarcsca 17156   ·_𝑠 cvsca 17157  0_gc0g 17335  LModclmod 20786  LSpanclspn 20897  LVecclvec 21029  HLchlt 39368  LHypclh 40002  DVecHcdvh 41096  LCDualclcd 41604  HDMapchdma 41810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075  ax-riotaBAD 38971

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-ot 4583  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-om 7792  df-1st 7916  df-2nd 7917  df-tpos 8151  df-undef 8198  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-n0 12374  df-z 12461  df-uz 12725  df-fz 13400  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-mulr 17167  df-sca 17169  df-vsca 17170  df-0g 17337  df-mre 17480  df-mrc 17481  df-acs 17483  df-proset 18192  df-poset 18211  df-plt 18226  df-lub 18242  df-glb 18243  df-join 18244  df-meet 18245  df-p0 18321  df-p1 18322  df-lat 18330  df-clat 18397  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-submnd 18684  df-grp 18841  df-minusg 18842  df-sbg 18843  df-subg 19028  df-cntz 19222  df-oppg 19251  df-lsm 19541  df-cmn 19687  df-abl 19688  df-mgp 20052  df-rng 20064  df-ur 20093  df-ring 20146  df-oppr 20248  df-dvdsr 20268  df-unit 20269  df-invr 20299  df-dvr 20312  df-nzr 20421  df-rlreg 20602  df-domn 20603  df-drng 20639  df-lmod 20788  df-lss 20858  df-lsp 20898  df-lvec 21030  df-lsatoms 38994  df-lshyp 38995  df-lcv 39037  df-lfl 39076  df-lkr 39104  df-ldual 39142  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369  df-llines 39516  df-lplanes 39517  df-lvols 39518  df-lines 39519  df-psubsp 39521  df-pmap 39522  df-padd 39814  df-lhyp 40006  df-laut 40007  df-ldil 40122  df-ltrn 40123  df-trl 40177  df-tgrp 40761  df-tendo 40773  df-edring 40775  df-dveca 41021  df-disoa 41047  df-dvech 41097  df-dib 41157  df-dic 41191  df-dih 41247  df-doch 41366  df-djh 41413  df-lcdual 41605  df-mapd 41643  df-hvmap 41775  df-hdmap1 41811  df-hdmap 41812

This theorem is referenced by:  hdmap14lem7  41892