hdmapinvlem3 - Mathbox for Norm Megill

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Theorem hdmapinvlem3 41285

Description: Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (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
hdmapinvlem3	⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )

**Proof of Theorem hdmapinvlem3**
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		eqid 2724	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
6		eqid 2724	. . . 4 ⊢ (-_g‘((LCDual‘𝐾)‘𝑊)) = (-_g‘((LCDual‘𝐾)‘𝑊))
7		hdmapinvlem3.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
8		hdmapinvlem3.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	1, 2, 8	dvhlmod 40475	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		hdmapinvlem3.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐵)
11		eqid 2724	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2724	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
13		eqid 2724	. . . . . . 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 40477	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
16	15	eldifad 3953	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
17		hdmapinvlem3.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
18		hdmapinvlem3.q	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑈)
19		hdmapinvlem3.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
20	3, 17, 18, 19	lmodvscl 20716	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐽 · 𝐸) ∈ 𝑉)
21	9, 10, 16, 20	syl3anc 1368	. . . 4 ⊢ (𝜑 → (𝐽 · 𝐸) ∈ 𝑉)
22	16	snssd 4805	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
23		hdmapinvlem3.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
24	1, 2, 3, 23	dochssv 40720	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
25	8, 22, 24	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
26		hdmapinvlem3.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
27	25, 26	sseldd 3976	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
28	1, 2, 3, 4, 5, 6, 7, 8, 21, 27	hdmapsub 41212	. . 3 ⊢ (𝜑 → (𝑆‘((𝐽 · 𝐸) − 𝐷)) = ((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷)))
29	28	fveq1d 6884	. 2 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = (((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷))‘((𝐼 · 𝐸) + 𝐶)))
30		eqid 2724	. . . 4 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
31		eqid 2724	. . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
32	1, 2, 3, 5, 31, 7, 8, 21	hdmapcl 41195	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐽 · 𝐸)) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
33	1, 2, 3, 5, 31, 7, 8, 27	hdmapcl 41195	. . . 4 ⊢ (𝜑 → (𝑆‘𝐷) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
34		hdmapinvlem3.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
35	3, 17, 18, 19	lmodvscl 20716	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐼 · 𝐸) ∈ 𝑉)
36	9, 34, 16, 35	syl3anc 1368	. . . . 5 ⊢ (𝜑 → (𝐼 · 𝐸) ∈ 𝑉)
37		hdmapinvlem3.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
38	25, 37	sseldd 3976	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
39		hdmapinvlem3.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
40	3, 39	lmodvacl 20713	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼 · 𝐸) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
41	9, 36, 38, 40	syl3anc 1368	. . . 4 ⊢ (𝜑 → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
42	1, 2, 3, 17, 30, 5, 31, 6, 8, 32, 33, 41	lcdvsubval 40983	. . 3 ⊢ (𝜑 → (((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷))‘((𝐼 · 𝐸) + 𝐶)) = (((𝑆‘(𝐽 · 𝐸))‘((𝐼 · 𝐸) + 𝐶))(-_g‘𝑅)((𝑆‘𝐷)‘((𝐼 · 𝐸) + 𝐶))))
43		eqid 2724	. . . . . 6 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
44	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 21	hdmaplna1 41272	. . . . 5 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘((𝐼 · 𝐸) + 𝐶)) = (((𝑆‘(𝐽 · 𝐸))‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘(𝐽 · 𝐸))‘𝐶)))
45		hdmapinvlem3.t	. . . . . . . 8 ⊢ × = (._r‘𝑅)
46		hdmapinvlem3.g	. . . . . . . 8 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
47	1, 2, 3, 18, 17, 19, 45, 7, 46, 8, 36, 16, 10	hdmapglnm2 41276	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘(𝐼 · 𝐸)) = (((𝑆‘𝐸)‘(𝐼 · 𝐸)) × (𝐺‘𝐽)))
48	1, 2, 3, 18, 17, 19, 45, 7, 8, 16, 16, 34	hdmaplnm1 41274	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘𝐸)‘(𝐼 · 𝐸)) = (𝐼 × ((𝑆‘𝐸)‘𝐸)))
49		eqid 2724	. . . . . . . . . . 11 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
50		eqid 2724	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
51	1, 14, 49, 7, 8, 2, 17, 50	hdmapevec2 41201	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
52	51	oveq2d 7418	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐸)‘𝐸)) = (𝐼 × (1_r‘𝑅)))
53	17	lmodring 20706	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
54	9, 53	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
55	19, 45, 50	ringridm 20161	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → (𝐼 × (1_r‘𝑅)) = 𝐼)
56	54, 34, 55	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × (1_r‘𝑅)) = 𝐼)
57	48, 52, 56	3eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘(𝐼 · 𝐸)) = 𝐼)
58	57	oveq1d 7417	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘(𝐼 · 𝐸)) × (𝐺‘𝐽)) = (𝐼 × (𝐺‘𝐽)))
59	47, 58	eqtrd 2764	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘(𝐼 · 𝐸)) = (𝐼 × (𝐺‘𝐽)))
60	1, 2, 3, 18, 17, 19, 45, 7, 46, 8, 38, 16, 10	hdmapglnm2 41276	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘𝐶) = (((𝑆‘𝐸)‘𝐶) × (𝐺‘𝐽)))
61		hdmapinvlem3.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
62	1, 14, 23, 2, 3, 17, 19, 45, 61, 7, 8, 37	hdmapinvlem1 41283	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 )
63	62	oveq1d 7417	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐶) × (𝐺‘𝐽)) = ( 0 × (𝐺‘𝐽)))
64	1, 2, 17, 19, 46, 8, 10	hgmapcl 41254	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐽) ∈ 𝐵)
65	19, 45, 61	ringlz 20184	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐽) ∈ 𝐵) → ( 0 × (𝐺‘𝐽)) = 0 )
66	54, 64, 65	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ( 0 × (𝐺‘𝐽)) = 0 )
67	60, 63, 66	3eqtrd 2768	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘𝐶) = 0 )
68	59, 67	oveq12d 7420	. . . . 5 ⊢ (𝜑 → (((𝑆‘(𝐽 · 𝐸))‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘(𝐽 · 𝐸))‘𝐶)) = ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ))
69		ringgrp 20135	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
70	54, 69	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
71	17, 19, 45	lmodmcl 20711	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ (𝐺‘𝐽) ∈ 𝐵) → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
72	9, 34, 64, 71	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
73	19, 43, 61	grprid 18890	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ) = (𝐼 × (𝐺‘𝐽)))
74	70, 72, 73	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ) = (𝐼 × (𝐺‘𝐽)))
75	44, 68, 74	3eqtrd 2768	. . . 4 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘((𝐼 · 𝐸) + 𝐶)) = (𝐼 × (𝐺‘𝐽)))
76	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 27	hdmaplna1 41272	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐷)‘((𝐼 · 𝐸) + 𝐶)) = (((𝑆‘𝐷)‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘𝐷)‘𝐶)))
77	1, 2, 3, 18, 17, 19, 45, 7, 8, 16, 27, 34	hdmaplnm1 41274	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘(𝐼 · 𝐸)) = (𝐼 × ((𝑆‘𝐷)‘𝐸)))
78	1, 14, 23, 2, 3, 17, 19, 45, 61, 7, 8, 26	hdmapinvlem2 41284	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐸) = 0 )
79	78	oveq2d 7418	. . . . . . 7 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐷)‘𝐸)) = (𝐼 × 0 ))
80	19, 45, 61	ringrz 20185	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → (𝐼 × 0 ) = 0 )
81	54, 34, 80	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐼 × 0 ) = 0 )
82	77, 79, 81	3eqtrrd 2769	. . . . . 6 ⊢ (𝜑 → 0 = ((𝑆‘𝐷)‘(𝐼 · 𝐸)))
83		hdmapinvlem3.ij	. . . . . 6 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))
84	82, 83	oveq12d 7420	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (((𝑆‘𝐷)‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘𝐷)‘𝐶)))
85	19, 43, 61	grplid 18889	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (𝐼 × (𝐺‘𝐽)))
86	70, 72, 85	syl2anc 583	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (𝐼 × (𝐺‘𝐽)))
87	76, 84, 86	3eqtr2d 2770	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐷)‘((𝐼 · 𝐸) + 𝐶)) = (𝐼 × (𝐺‘𝐽)))
88	75, 87	oveq12d 7420	. . 3 ⊢ (𝜑 → (((𝑆‘(𝐽 · 𝐸))‘((𝐼 · 𝐸) + 𝐶))(-_g‘𝑅)((𝑆‘𝐷)‘((𝐼 · 𝐸) + 𝐶))) = ((𝐼 × (𝐺‘𝐽))(-_g‘𝑅)(𝐼 × (𝐺‘𝐽))))
89	42, 88	eqtrd 2764	. 2 ⊢ (𝜑 → (((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷))‘((𝐼 · 𝐸) + 𝐶)) = ((𝐼 × (𝐺‘𝐽))(-_g‘𝑅)(𝐼 × (𝐺‘𝐽))))
90	19, 61, 30	grpsubid 18944	. . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ((𝐼 × (𝐺‘𝐽))(-_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = 0 )
91	70, 72, 90	syl2anc 583	. 2 ⊢ (𝜑 → ((𝐼 × (𝐺‘𝐽))(-_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = 0 )
92	29, 89, 91	3eqtrd 2768	1 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 {csn 4621 ⟨cop 4627 I cid 5564 ↾ cres 5669 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 +_gcplusg 17198 ._rcmulr 17199 Scalarcsca 17201 ·_𝑠 cvsca 17202 0_gc0g 17386 Grpcgrp 18855 -_gcsg 18857 1_rcur 20078 Ringcrg 20130 LModclmod 20698 HLchlt 38714 LHypclh 39349 LTrncltrn 39466 DVecHcdvh 40443 ocHcoch 40712 LCDualclcd 40951 HVMapchvm 41121 HDMapchdma 41157 HGMapchg 41248

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38317

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-ot 4630 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-0g 17388 df-mre 17531 df-mrc 17532 df-acs 17534 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-cntz 19225 df-oppg 19254 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-invr 20282 df-dvr 20295 df-drng 20581 df-lmod 20700 df-lss 20771 df-lsp 20811 df-lvec 20943 df-lsatoms 38340 df-lshyp 38341 df-lcv 38383 df-lfl 38422 df-lkr 38450 df-ldual 38488 df-oposet 38540 df-ol 38542 df-oml 38543 df-covers 38630 df-ats 38631 df-atl 38662 df-cvlat 38686 df-hlat 38715 df-llines 38863 df-lplanes 38864 df-lvols 38865 df-lines 38866 df-psubsp 38868 df-pmap 38869 df-padd 39161 df-lhyp 39353 df-laut 39354 df-ldil 39469 df-ltrn 39470 df-trl 39524 df-tgrp 40108 df-tendo 40120 df-edring 40122 df-dveca 40368 df-disoa 40394 df-dvech 40444 df-dib 40504 df-dic 40538 df-dih 40594 df-doch 40713 df-djh 40760 df-lcdual 40952 df-mapd 40990 df-hvmap 41122 df-hdmap1 41158 df-hdmap 41159 df-hgmap 41249

This theorem is referenced by: hdmapinvlem4 41286

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