hdmapinvlem3 - Mathbox for Norm Megill

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Theorem hdmapinvlem3 39861

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 2738	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
6		eqid 2738	. . . 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 39051	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		hdmapinvlem3.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐵)
11		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2738	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
13		eqid 2738	. . . . . . 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 39053	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
16	15	eldifad 3895	. . . . 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 20055	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐽 · 𝐸) ∈ 𝑉)
21	9, 10, 16, 20	syl3anc 1369	. . . 4 ⊢ (𝜑 → (𝐽 · 𝐸) ∈ 𝑉)
22	16	snssd 4739	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
23		hdmapinvlem3.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
24	1, 2, 3, 23	dochssv 39296	. . . . . 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 3918	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
28	1, 2, 3, 4, 5, 6, 7, 8, 21, 27	hdmapsub 39788	. . 3 ⊢ (𝜑 → (𝑆‘((𝐽 · 𝐸) − 𝐷)) = ((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷)))
29	28	fveq1d 6758	. 2 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = (((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷))‘((𝐼 · 𝐸) + 𝐶)))
30		eqid 2738	. . . 4 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
31		eqid 2738	. . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
32	1, 2, 3, 5, 31, 7, 8, 21	hdmapcl 39771	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐽 · 𝐸)) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
33	1, 2, 3, 5, 31, 7, 8, 27	hdmapcl 39771	. . . 4 ⊢ (𝜑 → (𝑆‘𝐷) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
34		hdmapinvlem3.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
35	3, 17, 18, 19	lmodvscl 20055	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐼 · 𝐸) ∈ 𝑉)
36	9, 34, 16, 35	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (𝐼 · 𝐸) ∈ 𝑉)
37		hdmapinvlem3.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
38	25, 37	sseldd 3918	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
39		hdmapinvlem3.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
40	3, 39	lmodvacl 20052	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼 · 𝐸) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
41	9, 36, 38, 40	syl3anc 1369	. . . 4 ⊢ (𝜑 → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
42	1, 2, 3, 17, 30, 5, 31, 6, 8, 32, 33, 41	lcdvsubval 39559	. . 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 2738	. . . . . 6 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
44	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 21	hdmaplna1 39848	. . . . 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 39852	. . . . . . 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 39850	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘𝐸)‘(𝐼 · 𝐸)) = (𝐼 × ((𝑆‘𝐸)‘𝐸)))
49		eqid 2738	. . . . . . . . . . 11 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
50		eqid 2738	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
51	1, 14, 49, 7, 8, 2, 17, 50	hdmapevec2 39777	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
52	51	oveq2d 7271	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐸)‘𝐸)) = (𝐼 × (1_r‘𝑅)))
53	17	lmodring 20046	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
54	9, 53	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
55	19, 45, 50	ringridm 19726	. . . . . . . . . 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 2782	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘(𝐼 · 𝐸)) = 𝐼)
58	57	oveq1d 7270	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘(𝐼 · 𝐸)) × (𝐺‘𝐽)) = (𝐼 × (𝐺‘𝐽)))
59	47, 58	eqtrd 2778	. . . . . 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 39852	. . . . . . 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 39859	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 )
63	62	oveq1d 7270	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐶) × (𝐺‘𝐽)) = ( 0 × (𝐺‘𝐽)))
64	1, 2, 17, 19, 46, 8, 10	hgmapcl 39830	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐽) ∈ 𝐵)
65	19, 45, 61	ringlz 19741	. . . . . . . 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 2782	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘𝐶) = 0 )
68	59, 67	oveq12d 7273	. . . . 5 ⊢ (𝜑 → (((𝑆‘(𝐽 · 𝐸))‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘(𝐽 · 𝐸))‘𝐶)) = ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ))
69		ringgrp 19703	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
70	54, 69	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
71	17, 19, 45	lmodmcl 20050	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ (𝐺‘𝐽) ∈ 𝐵) → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
72	9, 34, 64, 71	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
73	19, 43, 61	grprid 18525	. . . . . 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 2782	. . . 4 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘((𝐼 · 𝐸) + 𝐶)) = (𝐼 × (𝐺‘𝐽)))
76	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 27	hdmaplna1 39848	. . . . 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 39850	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘(𝐼 · 𝐸)) = (𝐼 × ((𝑆‘𝐷)‘𝐸)))
78	1, 14, 23, 2, 3, 17, 19, 45, 61, 7, 8, 26	hdmapinvlem2 39860	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐸) = 0 )
79	78	oveq2d 7271	. . . . . . 7 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐷)‘𝐸)) = (𝐼 × 0 ))
80	19, 45, 61	ringrz 19742	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → (𝐼 × 0 ) = 0 )
81	54, 34, 80	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐼 × 0 ) = 0 )
82	77, 79, 81	3eqtrrd 2783	. . . . . 6 ⊢ (𝜑 → 0 = ((𝑆‘𝐷)‘(𝐼 · 𝐸)))
83		hdmapinvlem3.ij	. . . . . 6 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))
84	82, 83	oveq12d 7273	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (((𝑆‘𝐷)‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘𝐷)‘𝐶)))
85	19, 43, 61	grplid 18524	. . . . . 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 2784	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐷)‘((𝐼 · 𝐸) + 𝐶)) = (𝐼 × (𝐺‘𝐽)))
88	75, 87	oveq12d 7273	. . 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 2778	. 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 18574	. . 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 2782	1 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ⊆ wss 3883 {csn 4558 ⟨cop 4564 I cid 5479 ↾ cres 5582 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 +_gcplusg 16888 ._rcmulr 16889 Scalarcsca 16891 ·_𝑠 cvsca 16892 0_gc0g 17067 Grpcgrp 18492 -_gcsg 18494 1_rcur 19652 Ringcrg 19698 LModclmod 20038 HLchlt 37291 LHypclh 37925 LTrncltrn 38042 DVecHcdvh 39019 ocHcoch 39288 LCDualclcd 39527 HVMapchvm 39697 HDMapchdma 39733 HGMapchg 39824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-mre 17212 df-mrc 17213 df-acs 17215 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-oppg 18865 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-lshyp 36918 df-lcv 36960 df-lfl 36999 df-lkr 37027 df-ldual 37065 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tgrp 38684 df-tendo 38696 df-edring 38698 df-dveca 38944 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289 df-djh 39336 df-lcdual 39528 df-mapd 39566 df-hvmap 39698 df-hdmap1 39734 df-hdmap 39735 df-hgmap 39825

This theorem is referenced by: hdmapinvlem4 39862

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