hdmapinvlem4 - Mathbox for Norm Megill

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Theorem hdmapinvlem4 39072

Description: Part 1.1 of Proposition 1 of [Baer] p. 110. We use 𝐶, 𝐷, 𝐼, and 𝐽 for Baer's u, v, s, and t. Our unit vector 𝐸 has the required properties for his w by hdmapevec2 38987. Our ((𝑆‘𝐷)‘𝐶) means his f(u,v) (note argument reversal). (Contributed by NM, 12-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmapinvlem3.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapinvlem3.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapinvlem3.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
hdmapinvlem3.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapinvlem3.v	⊢ 𝑉 = (Base‘𝑈)
hdmapinvlem3.p	⊢ + = (+_g‘𝑈)
hdmapinvlem3.m	⊢ − = (-_g‘𝑈)
hdmapinvlem3.q	⊢ · = ( ·_𝑠 ‘𝑈)
hdmapinvlem3.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmapinvlem3.b	⊢ 𝐵 = (Base‘𝑅)
hdmapinvlem3.t	⊢ × = (._r‘𝑅)
hdmapinvlem3.z	⊢ 0 = (0_g‘𝑅)
hdmapinvlem3.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapinvlem3.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hdmapinvlem3.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapinvlem3.c	⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
hdmapinvlem3.d	⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
hdmapinvlem3.i	⊢ (𝜑 → 𝐼 ∈ 𝐵)
hdmapinvlem3.j	⊢ (𝜑 → 𝐽 ∈ 𝐵)
hdmapinvlem3.ij	⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))

**Assertion**
Ref	Expression
hdmapinvlem4	⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷))

**Proof of Theorem hdmapinvlem4**
Step	Hyp	Ref	Expression
1		hdmapinvlem3.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmapinvlem3.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmapinvlem3.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
4		hdmapinvlem3.m	. . . 4 ⊢ − = (-_g‘𝑈)
5		hdmapinvlem3.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑈)
6		eqid 2821	. . . 4 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
7		hdmapinvlem3.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
8		hdmapinvlem3.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	1, 2, 8	dvhlmod 38261	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		hdmapinvlem3.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐵)
11		eqid 2821	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2821	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
13		eqid 2821	. . . . . . 7 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
14		hdmapinvlem3.e	. . . . . . 7 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
15	1, 11, 12, 2, 3, 13, 14, 8	dvheveccl 38263	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
16	15	eldifad 3948	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
17		hdmapinvlem3.q	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑈)
18		hdmapinvlem3.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
19	3, 5, 17, 18	lmodvscl 19651	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐽 · 𝐸) ∈ 𝑉)
20	9, 10, 16, 19	syl3anc 1367	. . . 4 ⊢ (𝜑 → (𝐽 · 𝐸) ∈ 𝑉)
21	16	snssd 4742	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
22		hdmapinvlem3.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
23	1, 2, 3, 22	dochssv 38506	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
24	8, 21, 23	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
25		hdmapinvlem3.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
26	24, 25	sseldd 3968	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
27		hdmapinvlem3.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
28	3, 5, 17, 18	lmodvscl 19651	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐼 · 𝐸) ∈ 𝑉)
29	9, 27, 16, 28	syl3anc 1367	. . . . 5 ⊢ (𝜑 → (𝐼 · 𝐸) ∈ 𝑉)
30		hdmapinvlem3.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
31	24, 30	sseldd 3968	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
32		hdmapinvlem3.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
33	3, 32	lmodvacl 19648	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼 · 𝐸) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
34	9, 29, 31, 33	syl3anc 1367	. . . 4 ⊢ (𝜑 → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
35	1, 2, 3, 4, 5, 6, 7, 8, 20, 26, 34	hdmaplns1 39059	. . 3 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘((𝐽 · 𝐸) − 𝐷)) = (((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸))(-_g‘𝑅)((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐷)))
36		hdmapinvlem3.t	. . . . 5 ⊢ × = (._r‘𝑅)
37		hdmapinvlem3.z	. . . . 5 ⊢ 0 = (0_g‘𝑅)
38		hdmapinvlem3.g	. . . . 5 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
39		hdmapinvlem3.ij	. . . . 5 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))
40	1, 14, 22, 2, 3, 32, 4, 17, 5, 18, 36, 37, 7, 38, 8, 30, 25, 27, 10, 39	hdmapinvlem3 39071	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )
41	3, 4	lmodvsubcl 19679	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝐽 · 𝐸) ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → ((𝐽 · 𝐸) − 𝐷) ∈ 𝑉)
42	9, 20, 26, 41	syl3anc 1367	. . . . 5 ⊢ (𝜑 → ((𝐽 · 𝐸) − 𝐷) ∈ 𝑉)
43	1, 2, 3, 5, 37, 7, 8, 42, 34	hdmapip0com 39068	. . . 4 ⊢ (𝜑 → (((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 ↔ ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘((𝐽 · 𝐸) − 𝐷)) = 0 ))
44	40, 43	mpbid 234	. . 3 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘((𝐽 · 𝐸) − 𝐷)) = 0 )
45	1, 2, 3, 17, 5, 18, 36, 7, 8, 16, 34, 10	hdmaplnm1 39060	. . . . 5 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸)) = (𝐽 × ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸)))
46		eqid 2821	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
47	1, 2, 3, 32, 5, 46, 7, 8, 16, 29, 31	hdmaplna2 39061	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸) = (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅)((𝑆‘𝐶)‘𝐸)))
48	1, 14, 22, 2, 3, 5, 18, 36, 37, 7, 8, 30	hdmapinvlem2 39070	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐸) = 0 )
49	48	oveq2d 7172	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅)((𝑆‘𝐶)‘𝐸)) = (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ))
50	5	lmodring 19642	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
51	9, 50	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
52		ringgrp 19302	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
53	51, 52	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Grp)
54	1, 2, 3, 5, 18, 7, 8, 16, 29	hdmapipcl 39056	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐸) ∈ 𝐵)
55	18, 46, 37	grprid 18134	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘(𝐼 · 𝐸))‘𝐸) ∈ 𝐵) → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ) = ((𝑆‘(𝐼 · 𝐸))‘𝐸))
56	53, 54, 55	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ) = ((𝑆‘(𝐼 · 𝐸))‘𝐸))
57	1, 2, 3, 17, 5, 18, 36, 7, 38, 8, 16, 16, 27	hdmapglnm2 39062	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐸) = (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)))
58		eqid 2821	. . . . . . . . . . 11 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
59		eqid 2821	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
60	1, 14, 58, 7, 8, 2, 5, 59	hdmapevec2 38987	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
61	60	oveq1d 7171	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)) = ((1_r‘𝑅) × (𝐺‘𝐼)))
62	1, 2, 5, 18, 38, 8, 27	hgmapcl 39040	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝐼) ∈ 𝐵)
63	18, 36, 59	ringlidm 19321	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐼) ∈ 𝐵) → ((1_r‘𝑅) × (𝐺‘𝐼)) = (𝐺‘𝐼))
64	51, 62, 63	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → ((1_r‘𝑅) × (𝐺‘𝐼)) = (𝐺‘𝐼))
65	61, 64	eqtrd 2856	. . . . . . . 8 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)) = (𝐺‘𝐼))
66	56, 57, 65	3eqtrd 2860	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ) = (𝐺‘𝐼))
67	47, 49, 66	3eqtrd 2860	. . . . . 6 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸) = (𝐺‘𝐼))
68	67	oveq2d 7172	. . . . 5 ⊢ (𝜑 → (𝐽 × ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸)) = (𝐽 × (𝐺‘𝐼)))
69	45, 68	eqtrd 2856	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸)) = (𝐽 × (𝐺‘𝐼)))
70	1, 2, 3, 32, 5, 46, 7, 8, 26, 29, 31	hdmaplna2 39061	. . . . 5 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐷) = (((𝑆‘(𝐼 · 𝐸))‘𝐷)(+_g‘𝑅)((𝑆‘𝐶)‘𝐷)))
71	1, 2, 3, 17, 5, 18, 36, 7, 38, 8, 26, 16, 27	hdmapglnm2 39062	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐷) = (((𝑆‘𝐸)‘𝐷) × (𝐺‘𝐼)))
72	1, 14, 22, 2, 3, 5, 18, 36, 37, 7, 8, 25	hdmapinvlem1 39069	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐷) = 0 )
73	72	oveq1d 7171	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐷) × (𝐺‘𝐼)) = ( 0 × (𝐺‘𝐼)))
74	18, 36, 37	ringlz 19337	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐼) ∈ 𝐵) → ( 0 × (𝐺‘𝐼)) = 0 )
75	51, 62, 74	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → ( 0 × (𝐺‘𝐼)) = 0 )
76	71, 73, 75	3eqtrd 2860	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐷) = 0 )
77	76	oveq1d 7171	. . . . 5 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐷)(+_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = ( 0 (+_g‘𝑅)((𝑆‘𝐶)‘𝐷)))
78	1, 2, 3, 5, 18, 7, 8, 26, 31	hdmapipcl 39056	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐷) ∈ 𝐵)
79	18, 46, 37	grplid 18133	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘𝐶)‘𝐷) ∈ 𝐵) → ( 0 (+_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = ((𝑆‘𝐶)‘𝐷))
80	53, 78, 79	syl2anc 586	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = ((𝑆‘𝐶)‘𝐷))
81	70, 77, 80	3eqtrd 2860	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐷) = ((𝑆‘𝐶)‘𝐷))
82	69, 81	oveq12d 7174	. . 3 ⊢ (𝜑 → (((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸))(-_g‘𝑅)((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐷)) = ((𝐽 × (𝐺‘𝐼))(-_g‘𝑅)((𝑆‘𝐶)‘𝐷)))
83	35, 44, 82	3eqtr3rd 2865	. 2 ⊢ (𝜑 → ((𝐽 × (𝐺‘𝐼))(-_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = 0 )
84	5, 18, 36	lmodmcl 19646	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ (𝐺‘𝐼) ∈ 𝐵) → (𝐽 × (𝐺‘𝐼)) ∈ 𝐵)
85	9, 10, 62, 84	syl3anc 1367	. . 3 ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) ∈ 𝐵)
86	18, 37, 6	grpsubeq0 18185	. . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝐽 × (𝐺‘𝐼)) ∈ 𝐵 ∧ ((𝑆‘𝐶)‘𝐷) ∈ 𝐵) → (((𝐽 × (𝐺‘𝐼))(-_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = 0 ↔ (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷)))
87	53, 85, 78, 86	syl3anc 1367	. 2 ⊢ (𝜑 → (((𝐽 × (𝐺‘𝐼))(-_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = 0 ↔ (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷)))
88	83, 87	mpbid 234	1 ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 {csn 4567 ⟨cop 4573 I cid 5459 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +_gcplusg 16565 ._rcmulr 16566 Scalarcsca 16568 ·_𝑠 cvsca 16569 0_gc0g 16713 Grpcgrp 18103 -_gcsg 18105 1_rcur 19251 Ringcrg 19297 LModclmod 19634 HLchlt 36501 LHypclh 37135 LTrncltrn 37252 DVecHcdvh 38229 ocHcoch 38498 HVMapchvm 38907 HDMapchdma 38943 HGMapchg 39034

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-mre 16857 df-mrc 16858 df-acs 16860 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-oppg 18474 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lsatoms 36127 df-lshyp 36128 df-lcv 36170 df-lfl 36209 df-lkr 36237 df-ldual 36275 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tgrp 37894 df-tendo 37906 df-edring 37908 df-dveca 38154 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 df-doch 38499 df-djh 38546 df-lcdual 38738 df-mapd 38776 df-hvmap 38908 df-hdmap1 38944 df-hdmap 38945 df-hgmap 39035

This theorem is referenced by: hdmapglem5 39073 hgmapvvlem1 39074

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