hdmapinvlem3 - Mathbox for Norm Megill

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Theorem hdmapinvlem3 42556

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 2765	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
6		eqid 2765	. . . 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 41746	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		hdmapinvlem3.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐵)
11		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2765	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
13		eqid 2765	. . . . . . 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 41748	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
16	15	eldifad 3919	. . . . 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 20968	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐽 · 𝐸) ∈ 𝑉)
21	9, 10, 16, 20	syl3anc 1394	. . . 4 ⊢ (𝜑 → (𝐽 · 𝐸) ∈ 𝑉)
22	16	snssd 4748	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
23		hdmapinvlem3.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
24	1, 2, 3, 23	dochssv 41991	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
25	8, 22, 24	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
26		hdmapinvlem3.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
27	25, 26	sseldd 3940	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
28	1, 2, 3, 4, 5, 6, 7, 8, 21, 27	hdmapsub 42483	. . 3 ⊢ (𝜑 → (𝑆‘((𝐽 · 𝐸) − 𝐷)) = ((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷)))
29	28	fveq1d 6873	. 2 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = (((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷))‘((𝐼 · 𝐸) + 𝐶)))
30		eqid 2765	. . . 4 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
31		eqid 2765	. . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
32	1, 2, 3, 5, 31, 7, 8, 21	hdmapcl 42466	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐽 · 𝐸)) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
33	1, 2, 3, 5, 31, 7, 8, 27	hdmapcl 42466	. . . 4 ⊢ (𝜑 → (𝑆‘𝐷) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
34		hdmapinvlem3.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
35	3, 17, 18, 19	lmodvscl 20968	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐼 · 𝐸) ∈ 𝑉)
36	9, 34, 16, 35	syl3anc 1394	. . . . 5 ⊢ (𝜑 → (𝐼 · 𝐸) ∈ 𝑉)
37		hdmapinvlem3.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
38	25, 37	sseldd 3940	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
39		hdmapinvlem3.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
40	3, 39	lmodvacl 20965	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼 · 𝐸) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
41	9, 36, 38, 40	syl3anc 1394	. . . 4 ⊢ (𝜑 → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
42	1, 2, 3, 17, 30, 5, 31, 6, 8, 32, 33, 41	lcdvsubval 42254	. . 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 2765	. . . . . 6 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
44	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 21	hdmaplna1 42543	. . . . 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 42547	. . . . . . 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 42545	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘𝐸)‘(𝐼 · 𝐸)) = (𝐼 × ((𝑆‘𝐸)‘𝐸)))
49		eqid 2765	. . . . . . . . . . 11 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
50		eqid 2765	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
51	1, 14, 49, 7, 8, 2, 17, 50	hdmapevec2 42472	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
52	51	oveq2d 7416	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐸)‘𝐸)) = (𝐼 × (1_r‘𝑅)))
53	17	lmodring 20958	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
54	9, 53	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
55	19, 45, 50	ringridm 20344	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → (𝐼 × (1_r‘𝑅)) = 𝐼)
56	54, 34, 55	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × (1_r‘𝑅)) = 𝐼)
57	48, 52, 56	3eqtrd 2804	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘(𝐼 · 𝐸)) = 𝐼)
58	57	oveq1d 7415	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘(𝐼 · 𝐸)) × (𝐺‘𝐽)) = (𝐼 × (𝐺‘𝐽)))
59	47, 58	eqtrd 2800	. . . . . 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 42547	. . . . . . 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 42554	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 )
63	62	oveq1d 7415	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐶) × (𝐺‘𝐽)) = ( 0 × (𝐺‘𝐽)))
64	1, 2, 17, 19, 46, 8, 10	hgmapcl 42525	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐽) ∈ 𝐵)
65	19, 45, 61	ringlz 20367	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐽) ∈ 𝐵) → ( 0 × (𝐺‘𝐽)) = 0 )
66	54, 64, 65	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → ( 0 × (𝐺‘𝐽)) = 0 )
67	60, 63, 66	3eqtrd 2804	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘𝐶) = 0 )
68	59, 67	oveq12d 7418	. . . . 5 ⊢ (𝜑 → (((𝑆‘(𝐽 · 𝐸))‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘(𝐽 · 𝐸))‘𝐶)) = ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ))
69		ringgrp 20311	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
70	54, 69	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
71	17, 19, 45	lmodmcl 20963	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ (𝐺‘𝐽) ∈ 𝐵) → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
72	9, 34, 64, 71	syl3anc 1394	. . . . . 6 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
73	19, 43, 61	grprid 19025	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ) = (𝐼 × (𝐺‘𝐽)))
74	70, 72, 73	syl2anc 595	. . . . 5 ⊢ (𝜑 → ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ) = (𝐼 × (𝐺‘𝐽)))
75	44, 68, 74	3eqtrd 2804	. . . 4 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘((𝐼 · 𝐸) + 𝐶)) = (𝐼 × (𝐺‘𝐽)))
76	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 27	hdmaplna1 42543	. . . . 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 42545	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘(𝐼 · 𝐸)) = (𝐼 × ((𝑆‘𝐷)‘𝐸)))
78	1, 14, 23, 2, 3, 17, 19, 45, 61, 7, 8, 26	hdmapinvlem2 42555	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐸) = 0 )
79	78	oveq2d 7416	. . . . . . 7 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐷)‘𝐸)) = (𝐼 × 0 ))
80	19, 45, 61	ringrz 20368	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → (𝐼 × 0 ) = 0 )
81	54, 34, 80	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝐼 × 0 ) = 0 )
82	77, 79, 81	3eqtrrd 2805	. . . . . 6 ⊢ (𝜑 → 0 = ((𝑆‘𝐷)‘(𝐼 · 𝐸)))
83		hdmapinvlem3.ij	. . . . . 6 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))
84	82, 83	oveq12d 7418	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (((𝑆‘𝐷)‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘𝐷)‘𝐶)))
85	19, 43, 61	grplid 19024	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (𝐼 × (𝐺‘𝐽)))
86	70, 72, 85	syl2anc 595	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (𝐼 × (𝐺‘𝐽)))
87	76, 84, 86	3eqtr2d 2806	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐷)‘((𝐼 · 𝐸) + 𝐶)) = (𝐼 × (𝐺‘𝐽)))
88	75, 87	oveq12d 7418	. . 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 2800	. 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 19081	. . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ((𝐼 × (𝐺‘𝐽))(-_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = 0 )
91	70, 72, 90	syl2anc 595	. 2 ⊢ (𝜑 → ((𝐼 × (𝐺‘𝐽))(-_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = 0 )
92	29, 89, 91	3eqtrd 2804	1 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ⊆ wss 3907 {csn 4585 ⟨cop 4591 I cid 5546 ↾ cres 5654 ‘cfv 6525 (class class class)co 7400 Basecbs 17259 +_gcplusg 17300 ._rcmulr 17301 Scalarcsca 17303 ·_𝑠 cvsca 17304 0_gc0g 17482 Grpcgrp 18990 -_gcsg 18992 1_rcur 20254 Ringcrg 20306 LModclmod 20950 HLchlt 39986 LHypclh 40620 LTrncltrn 40737 DVecHcdvh 41714 ocHcoch 41983 LCDualclcd 42222 HVMapchvm 42392 HDMapchdma 42428 HGMapchg 42519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-riotaBAD 39589

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-tpos 8210 df-undef 8257 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-sca 17316 df-vsca 17317 df-0g 17484 df-mre 17628 df-mrc 17629 df-acs 17631 df-proset 18340 df-poset 18359 df-plt 18374 df-lub 18390 df-glb 18391 df-join 18392 df-meet 18393 df-p0 18469 df-p1 18470 df-lat 18478 df-clat 18545 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-grp 18993 df-minusg 18994 df-sbg 18995 df-subg 19180 df-cntz 19378 df-oppg 19407 df-lsm 19697 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-oppr 20410 df-dvdsr 20430 df-unit 20431 df-invr 20461 df-dvr 20474 df-nzr 20587 df-rlreg 20770 df-domn 20771 df-drng 20806 df-lmod 20952 df-lss 21022 df-lsp 21062 df-lvec 21193 df-lsatoms 39612 df-lshyp 39613 df-lcv 39655 df-lfl 39694 df-lkr 39722 df-ldual 39760 df-oposet 39812 df-ol 39814 df-oml 39815 df-covers 39902 df-ats 39903 df-atl 39934 df-cvlat 39958 df-hlat 39987 df-llines 40134 df-lplanes 40135 df-lvols 40136 df-lines 40137 df-psubsp 40139 df-pmap 40140 df-padd 40432 df-lhyp 40624 df-laut 40625 df-ldil 40740 df-ltrn 40741 df-trl 40795 df-tgrp 41379 df-tendo 41391 df-edring 41393 df-dveca 41639 df-disoa 41665 df-dvech 41715 df-dib 41775 df-dic 41809 df-dih 41865 df-doch 41984 df-djh 42031 df-lcdual 42223 df-mapd 42261 df-hvmap 42393 df-hdmap1 42429 df-hdmap 42430 df-hgmap 42520

This theorem is referenced by: hdmapinvlem4 42557

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