hdmapinvlem4 - Mathbox for Norm Megill

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Theorem hdmapinvlem4 42367

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 42282. 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 2736	. . . 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 41556	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		hdmapinvlem3.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐵)
11		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2736	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
13		eqid 2736	. . . . . . 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 41558	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
16	15	eldifad 3901	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
17		hdmapinvlem3.q	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑈)
18		hdmapinvlem3.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
19	3, 5, 17, 18	lmodvscl 20873	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐽 · 𝐸) ∈ 𝑉)
20	9, 10, 16, 19	syl3anc 1374	. . . 4 ⊢ (𝜑 → (𝐽 · 𝐸) ∈ 𝑉)
21	16	snssd 4730	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
22		hdmapinvlem3.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
23	1, 2, 3, 22	dochssv 41801	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
24	8, 21, 23	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
25		hdmapinvlem3.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
26	24, 25	sseldd 3922	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
27		hdmapinvlem3.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
28	3, 5, 17, 18	lmodvscl 20873	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐼 · 𝐸) ∈ 𝑉)
29	9, 27, 16, 28	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝐼 · 𝐸) ∈ 𝑉)
30		hdmapinvlem3.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
31	24, 30	sseldd 3922	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
32		hdmapinvlem3.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
33	3, 32	lmodvacl 20870	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼 · 𝐸) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
34	9, 29, 31, 33	syl3anc 1374	. . . 4 ⊢ (𝜑 → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
35	1, 2, 3, 4, 5, 6, 7, 8, 20, 26, 34	hdmaplns1 42354	. . 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 42366	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )
41	3, 4	lmodvsubcl 20902	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝐽 · 𝐸) ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → ((𝐽 · 𝐸) − 𝐷) ∈ 𝑉)
42	9, 20, 26, 41	syl3anc 1374	. . . . 5 ⊢ (𝜑 → ((𝐽 · 𝐸) − 𝐷) ∈ 𝑉)
43	1, 2, 3, 5, 37, 7, 8, 42, 34	hdmapip0com 42363	. . . 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 42355	. . . . 5 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸)) = (𝐽 × ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸)))
46		eqid 2736	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
47	1, 2, 3, 32, 5, 46, 7, 8, 16, 29, 31	hdmaplna2 42356	. . . . . . 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 42365	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐸) = 0 )
49	48	oveq2d 7383	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅)((𝑆‘𝐶)‘𝐸)) = (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ))
50	5	lmodring 20863	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
51	9, 50	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
52		ringgrp 20219	. . . . . . . . . 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 42351	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐸) ∈ 𝐵)
55	18, 46, 37	grprid 18944	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘(𝐼 · 𝐸))‘𝐸) ∈ 𝐵) → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ) = ((𝑆‘(𝐼 · 𝐸))‘𝐸))
56	53, 54, 55	syl2anc 585	. . . . . . . 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 42357	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐸) = (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)))
58		eqid 2736	. . . . . . . . . . 11 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
59		eqid 2736	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
60	1, 14, 58, 7, 8, 2, 5, 59	hdmapevec2 42282	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
61	60	oveq1d 7382	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)) = ((1_r‘𝑅) × (𝐺‘𝐼)))
62	1, 2, 5, 18, 38, 8, 27	hgmapcl 42335	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝐼) ∈ 𝐵)
63	18, 36, 59	ringlidm 20250	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐼) ∈ 𝐵) → ((1_r‘𝑅) × (𝐺‘𝐼)) = (𝐺‘𝐼))
64	51, 62, 63	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ((1_r‘𝑅) × (𝐺‘𝐼)) = (𝐺‘𝐼))
65	61, 64	eqtrd 2771	. . . . . . . 8 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)) = (𝐺‘𝐼))
66	56, 57, 65	3eqtrd 2775	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ) = (𝐺‘𝐼))
67	47, 49, 66	3eqtrd 2775	. . . . . 6 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸) = (𝐺‘𝐼))
68	67	oveq2d 7383	. . . . 5 ⊢ (𝜑 → (𝐽 × ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸)) = (𝐽 × (𝐺‘𝐼)))
69	45, 68	eqtrd 2771	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸)) = (𝐽 × (𝐺‘𝐼)))
70	1, 2, 3, 32, 5, 46, 7, 8, 26, 29, 31	hdmaplna2 42356	. . . . 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 42357	. . . . . . 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 42364	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐷) = 0 )
73	72	oveq1d 7382	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐷) × (𝐺‘𝐼)) = ( 0 × (𝐺‘𝐼)))
74	18, 36, 37	ringlz 20274	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐼) ∈ 𝐵) → ( 0 × (𝐺‘𝐼)) = 0 )
75	51, 62, 74	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ( 0 × (𝐺‘𝐼)) = 0 )
76	71, 73, 75	3eqtrd 2775	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐷) = 0 )
77	76	oveq1d 7382	. . . . 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 42351	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐷) ∈ 𝐵)
79	18, 46, 37	grplid 18943	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘𝐶)‘𝐷) ∈ 𝐵) → ( 0 (+_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = ((𝑆‘𝐶)‘𝐷))
80	53, 78, 79	syl2anc 585	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = ((𝑆‘𝐶)‘𝐷))
81	70, 77, 80	3eqtrd 2775	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐷) = ((𝑆‘𝐶)‘𝐷))
82	69, 81	oveq12d 7385	. . 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 2780	. 2 ⊢ (𝜑 → ((𝐽 × (𝐺‘𝐼))(-_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = 0 )
84	5, 18, 36	lmodmcl 20868	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ (𝐺‘𝐼) ∈ 𝐵) → (𝐽 × (𝐺‘𝐼)) ∈ 𝐵)
85	9, 10, 62, 84	syl3anc 1374	. . 3 ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) ∈ 𝐵)
86	18, 37, 6	grpsubeq0 19002	. . 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 1374	. 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 1542 ∈ wcel 2114 ⊆ wss 3889 {csn 4567 ⟨cop 4573 I cid 5525 ↾ cres 5633 ‘cfv 6498 (class class class)co 7367 Basecbs 17179 +_gcplusg 17220 ._rcmulr 17221 Scalarcsca 17223 ·_𝑠 cvsca 17224 0_gc0g 17402 Grpcgrp 18909 -_gcsg 18911 1_rcur 20162 Ringcrg 20214 LModclmod 20855 HLchlt 39796 LHypclh 40430 LTrncltrn 40547 DVecHcdvh 41524 ocHcoch 41793 HVMapchvm 42202 HDMapchdma 42238 HGMapchg 42329

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39399

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-undef 8223 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17548 df-mrc 17549 df-acs 17551 df-proset 18260 df-poset 18279 df-plt 18294 df-lub 18310 df-glb 18311 df-join 18312 df-meet 18313 df-p0 18389 df-p1 18390 df-lat 18398 df-clat 18465 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-cntz 19292 df-oppg 19321 df-lsm 19611 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-nzr 20490 df-rlreg 20671 df-domn 20672 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-lvec 21098 df-lsatoms 39422 df-lshyp 39423 df-lcv 39465 df-lfl 39504 df-lkr 39532 df-ldual 39570 df-oposet 39622 df-ol 39624 df-oml 39625 df-covers 39712 df-ats 39713 df-atl 39744 df-cvlat 39768 df-hlat 39797 df-llines 39944 df-lplanes 39945 df-lvols 39946 df-lines 39947 df-psubsp 39949 df-pmap 39950 df-padd 40242 df-lhyp 40434 df-laut 40435 df-ldil 40550 df-ltrn 40551 df-trl 40605 df-tgrp 41189 df-tendo 41201 df-edring 41203 df-dveca 41449 df-disoa 41475 df-dvech 41525 df-dib 41585 df-dic 41619 df-dih 41675 df-doch 41794 df-djh 41841 df-lcdual 42033 df-mapd 42071 df-hvmap 42203 df-hdmap1 42239 df-hdmap 42240 df-hgmap 42330

This theorem is referenced by: hdmapglem5 42368 hgmapvvlem1 42369

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