hdmapinvlem4 - Mathbox for Norm Megill

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Theorem hdmapinvlem4 41394

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 41309. 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 2728	. . . 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 40583	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		hdmapinvlem3.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐵)
11		eqid 2728	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2728	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
13		eqid 2728	. . . . . . 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 40585	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
16	15	eldifad 3959	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
17		hdmapinvlem3.q	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑈)
18		hdmapinvlem3.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
19	3, 5, 17, 18	lmodvscl 20760	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐽 · 𝐸) ∈ 𝑉)
20	9, 10, 16, 19	syl3anc 1369	. . . 4 ⊢ (𝜑 → (𝐽 · 𝐸) ∈ 𝑉)
21	16	snssd 4813	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
22		hdmapinvlem3.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
23	1, 2, 3, 22	dochssv 40828	. . . . . 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 3981	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
27		hdmapinvlem3.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
28	3, 5, 17, 18	lmodvscl 20760	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐼 · 𝐸) ∈ 𝑉)
29	9, 27, 16, 28	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (𝐼 · 𝐸) ∈ 𝑉)
30		hdmapinvlem3.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
31	24, 30	sseldd 3981	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
32		hdmapinvlem3.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
33	3, 32	lmodvacl 20757	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼 · 𝐸) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
34	9, 29, 31, 33	syl3anc 1369	. . . 4 ⊢ (𝜑 → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
35	1, 2, 3, 4, 5, 6, 7, 8, 20, 26, 34	hdmaplns1 41381	. . 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 41393	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )
41	3, 4	lmodvsubcl 20789	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝐽 · 𝐸) ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → ((𝐽 · 𝐸) − 𝐷) ∈ 𝑉)
42	9, 20, 26, 41	syl3anc 1369	. . . . 5 ⊢ (𝜑 → ((𝐽 · 𝐸) − 𝐷) ∈ 𝑉)
43	1, 2, 3, 5, 37, 7, 8, 42, 34	hdmapip0com 41390	. . . 4 ⊢ (𝜑 → (((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 ↔ ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘((𝐽 · 𝐸) − 𝐷)) = 0 ))
44	40, 43	mpbid 231	. . 3 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘((𝐽 · 𝐸) − 𝐷)) = 0 )
45	1, 2, 3, 17, 5, 18, 36, 7, 8, 16, 34, 10	hdmaplnm1 41382	. . . . 5 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸)) = (𝐽 × ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸)))
46		eqid 2728	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
47	1, 2, 3, 32, 5, 46, 7, 8, 16, 29, 31	hdmaplna2 41383	. . . . . . 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 41392	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐸) = 0 )
49	48	oveq2d 7436	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅)((𝑆‘𝐶)‘𝐸)) = (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ))
50	5	lmodring 20750	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
51	9, 50	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
52		ringgrp 20177	. . . . . . . . . 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 41378	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐸) ∈ 𝐵)
55	18, 46, 37	grprid 18924	. . . . . . . . 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 41384	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐸) = (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)))
58		eqid 2728	. . . . . . . . . . 11 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
59		eqid 2728	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
60	1, 14, 58, 7, 8, 2, 5, 59	hdmapevec2 41309	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
61	60	oveq1d 7435	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)) = ((1_r‘𝑅) × (𝐺‘𝐼)))
62	1, 2, 5, 18, 38, 8, 27	hgmapcl 41362	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝐼) ∈ 𝐵)
63	18, 36, 59	ringlidm 20204	. . . . . . . . . 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 2768	. . . . . . . 8 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐸) × (𝐺‘𝐼)) = (𝐺‘𝐼))
66	56, 57, 65	3eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘(𝐼 · 𝐸))‘𝐸)(+_g‘𝑅) 0 ) = (𝐺‘𝐼))
67	47, 49, 66	3eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸) = (𝐺‘𝐼))
68	67	oveq2d 7436	. . . . 5 ⊢ (𝜑 → (𝐽 × ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐸)) = (𝐽 × (𝐺‘𝐼)))
69	45, 68	eqtrd 2768	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘(𝐽 · 𝐸)) = (𝐽 × (𝐺‘𝐼)))
70	1, 2, 3, 32, 5, 46, 7, 8, 26, 29, 31	hdmaplna2 41383	. . . . 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 41384	. . . . . . 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 41391	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐷) = 0 )
73	72	oveq1d 7435	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐷) × (𝐺‘𝐼)) = ( 0 × (𝐺‘𝐼)))
74	18, 36, 37	ringlz 20228	. . . . . . . 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 2772	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐼 · 𝐸))‘𝐷) = 0 )
77	76	oveq1d 7435	. . . . 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 41378	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐷) ∈ 𝐵)
79	18, 46, 37	grplid 18923	. . . . . 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 2772	. . . 4 ⊢ (𝜑 → ((𝑆‘((𝐼 · 𝐸) + 𝐶))‘𝐷) = ((𝑆‘𝐶)‘𝐷))
82	69, 81	oveq12d 7438	. . 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 2777	. 2 ⊢ (𝜑 → ((𝐽 × (𝐺‘𝐼))(-_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = 0 )
84	5, 18, 36	lmodmcl 20755	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ (𝐺‘𝐼) ∈ 𝐵) → (𝐽 × (𝐺‘𝐼)) ∈ 𝐵)
85	9, 10, 62, 84	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) ∈ 𝐵)
86	18, 37, 6	grpsubeq0 18981	. . 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 1369	. 2 ⊢ (𝜑 → (((𝐽 × (𝐺‘𝐼))(-_g‘𝑅)((𝑆‘𝐶)‘𝐷)) = 0 ↔ (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷)))
88	83, 87	mpbid 231	1 ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 {csn 4629 ⟨cop 4635 I cid 5575 ↾ cres 5680 ‘cfv 6548 (class class class)co 7420 Basecbs 17179 +_gcplusg 17232 ._rcmulr 17233 Scalarcsca 17235 ·_𝑠 cvsca 17236 0_gc0g 17420 Grpcgrp 18889 -_gcsg 18891 1_rcur 20120 Ringcrg 20172 LModclmod 20742 HLchlt 38822 LHypclh 39457 LTrncltrn 39574 DVecHcdvh 40551 ocHcoch 40820 HVMapchvm 41229 HDMapchdma 41265 HGMapchg 41356

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-riotaBAD 38425

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-undef 8278 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-0g 17422 df-mre 17565 df-mrc 17566 df-acs 17568 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-cntz 19267 df-oppg 19296 df-lsm 19590 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-drng 20625 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lvec 20987 df-lsatoms 38448 df-lshyp 38449 df-lcv 38491 df-lfl 38530 df-lkr 38558 df-ldual 38596 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tgrp 40216 df-tendo 40228 df-edring 40230 df-dveca 40476 df-disoa 40502 df-dvech 40552 df-dib 40612 df-dic 40646 df-dih 40702 df-doch 40821 df-djh 40868 df-lcdual 41060 df-mapd 41098 df-hvmap 41230 df-hdmap1 41266 df-hdmap 41267 df-hgmap 41357

This theorem is referenced by: hdmapglem5 41395 hgmapvvlem1 41396

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