hdmapinvlem4 - Mathbox for Norm Megill

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Theorem hdmapinvlem4 41878

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 41793. 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 2740	. . . 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 41067	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		hdmapinvlem3.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐵)
11		eqid 2740	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2740	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
13		eqid 2740	. . . . . . 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 41069	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
16	15	eldifad 3988	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
17		hdmapinvlem3.q	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑈)
18		hdmapinvlem3.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
19	3, 5, 17, 18	lmodvscl 20898	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐽 · 𝐸) ∈ 𝑉)
20	9, 10, 16, 19	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐽 · 𝐸) ∈ 𝑉)
21	16	snssd 4834	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
22		hdmapinvlem3.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
23	1, 2, 3, 22	dochssv 41312	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
24	8, 21, 23	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
25		hdmapinvlem3.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
26	24, 25	sseldd 4009	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
27		hdmapinvlem3.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
28	3, 5, 17, 18	lmodvscl 20898	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐼 · 𝐸) ∈ 𝑉)
29	9, 27, 16, 28	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (𝐼 · 𝐸) ∈ 𝑉)
30		hdmapinvlem3.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
31	24, 30	sseldd 4009	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
32		hdmapinvlem3.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
33	3, 32	lmodvacl 20895	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼 · 𝐸) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
34	9, 29, 31, 33	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
35	1, 2, 3, 4, 5, 6, 7, 8, 20, 26, 34	hdmaplns1 41865	. . 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 41877	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )
41	3, 4	lmodvsubcl 20927	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝐽 · 𝐸) ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → ((𝐽 · 𝐸) − 𝐷) ∈ 𝑉)
42	9, 20, 26, 41	syl3anc 1371	. . . . 5 ⊢ (𝜑 → ((𝐽 · 𝐸) − 𝐷) ∈ 𝑉)
43	1, 2, 3, 5, 37, 7, 8, 42, 34	hdmapip0com 41874	. . . 4 ⊢ (𝜑 → (((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 ↔ ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘((𝐽 · 𝐸) − 𝐷)) = 0 ))
44	40, 43	mpbid 232	. . 3 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘((𝐽 · 𝐸) − 𝐷)) = 0 )
45	1, 2, 3, 17, 5, 18, 36, 7, 8, 16, 34, 10	hdmaplnm1 41866	. . . . 5 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸)) = (𝐽 × ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸)))
46		eqid 2740	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
47	1, 2, 3, 32, 5, 46, 7, 8, 16, 29, 31	hdmaplna2 41867	. . . . . . 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 41876	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐸) = 0 )
49	48	oveq2d 7464	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅)((𝑆‘𝐶)‘𝐸)) = (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ))
50	5	lmodring 20888	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
51	9, 50	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
52		ringgrp 20265	. . . . . . . . . 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 41862	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐸) ∈ 𝐵)
55	18, 46, 37	grprid 19008	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘(𝐼 · 𝐸))‘𝐸) ∈ 𝐵) → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ) = ((𝑆‘(𝐼 · 𝐸))‘𝐸))
56	53, 54, 55	syl2anc 583	. . . . . . . 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 41868	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐸) = (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)))
58		eqid 2740	. . . . . . . . . . 11 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
59		eqid 2740	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
60	1, 14, 58, 7, 8, 2, 5, 59	hdmapevec2 41793	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
61	60	oveq1d 7463	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)) = ((1_r‘𝑅) × (𝐺‘𝐼)))
62	1, 2, 5, 18, 38, 8, 27	hgmapcl 41846	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝐼) ∈ 𝐵)
63	18, 36, 59	ringlidm 20292	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐼) ∈ 𝐵) → ((1_r‘𝑅) × (𝐺‘𝐼)) = (𝐺‘𝐼))
64	51, 62, 63	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((1_r‘𝑅) × (𝐺‘𝐼)) = (𝐺‘𝐼))
65	61, 64	eqtrd 2780	. . . . . . . 8 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)) = (𝐺‘𝐼))
66	56, 57, 65	3eqtrd 2784	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ) = (𝐺‘𝐼))
67	47, 49, 66	3eqtrd 2784	. . . . . 6 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸) = (𝐺‘𝐼))
68	67	oveq2d 7464	. . . . 5 ⊢ (𝜑 → (𝐽 × ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸)) = (𝐽 × (𝐺‘𝐼)))
69	45, 68	eqtrd 2780	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸)) = (𝐽 × (𝐺‘𝐼)))
70	1, 2, 3, 32, 5, 46, 7, 8, 26, 29, 31	hdmaplna2 41867	. . . . 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 41868	. . . . . . 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 41875	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐷) = 0 )
73	72	oveq1d 7463	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐷) × (𝐺‘𝐼)) = ( 0 × (𝐺‘𝐼)))
74	18, 36, 37	ringlz 20316	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐼) ∈ 𝐵) → ( 0 × (𝐺‘𝐼)) = 0 )
75	51, 62, 74	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ( 0 × (𝐺‘𝐼)) = 0 )
76	71, 73, 75	3eqtrd 2784	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐷) = 0 )
77	76	oveq1d 7463	. . . . 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 41862	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐷) ∈ 𝐵)
79	18, 46, 37	grplid 19007	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘𝐶)‘𝐷) ∈ 𝐵) → ( 0 (+_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = ((𝑆‘𝐶)‘𝐷))
80	53, 78, 79	syl2anc 583	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = ((𝑆‘𝐶)‘𝐷))
81	70, 77, 80	3eqtrd 2784	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐷) = ((𝑆‘𝐶)‘𝐷))
82	69, 81	oveq12d 7466	. . 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 2789	. 2 ⊢ (𝜑 → ((𝐽 × (𝐺‘𝐼))(-_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = 0 )
84	5, 18, 36	lmodmcl 20893	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ (𝐺‘𝐼) ∈ 𝐵) → (𝐽 × (𝐺‘𝐼)) ∈ 𝐵)
85	9, 10, 62, 84	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) ∈ 𝐵)
86	18, 37, 6	grpsubeq0 19066	. . 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 1371	. 2 ⊢ (𝜑 → (((𝐽 × (𝐺‘𝐼))(-_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = 0 ↔ (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷)))
88	83, 87	mpbid 232	1 ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 {csn 4648 ⟨cop 4654 I cid 5592 ↾ cres 5702 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 +_gcplusg 17311 ._rcmulr 17312 Scalarcsca 17314 ·_𝑠 cvsca 17315 0_gc0g 17499 Grpcgrp 18973 -_gcsg 18975 1_rcur 20208 Ringcrg 20260 LModclmod 20880 HLchlt 39306 LHypclh 39941 LTrncltrn 40058 DVecHcdvh 41035 ocHcoch 41304 HVMapchvm 41713 HDMapchdma 41749 HGMapchg 41840

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-riotaBAD 38909

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-undef 8314 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17501 df-mre 17644 df-mrc 17645 df-acs 17647 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-cntz 19357 df-oppg 19386 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-nzr 20539 df-rlreg 20716 df-domn 20717 df-drng 20753 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lvec 21125 df-lsatoms 38932 df-lshyp 38933 df-lcv 38975 df-lfl 39014 df-lkr 39042 df-ldual 39080 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-llines 39455 df-lplanes 39456 df-lvols 39457 df-lines 39458 df-psubsp 39460 df-pmap 39461 df-padd 39753 df-lhyp 39945 df-laut 39946 df-ldil 40061 df-ltrn 40062 df-trl 40116 df-tgrp 40700 df-tendo 40712 df-edring 40714 df-dveca 40960 df-disoa 40986 df-dvech 41036 df-dib 41096 df-dic 41130 df-dih 41186 df-doch 41305 df-djh 41352 df-lcdual 41544 df-mapd 41582 df-hvmap 41714 df-hdmap1 41750 df-hdmap 41751 df-hgmap 41841

This theorem is referenced by: hdmapglem5 41879 hgmapvvlem1 41880

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