hdmapinvlem3 - Mathbox for Norm Megill

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Theorem hdmapinvlem3 39934

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 39124	. . . . 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 39126	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
16	15	eldifad 3899	. . . . 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 20140	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐽 · 𝐸) ∈ 𝑉)
21	9, 10, 16, 20	syl3anc 1370	. . . 4 ⊢ (𝜑 → (𝐽 · 𝐸) ∈ 𝑉)
22	16	snssd 4742	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
23		hdmapinvlem3.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
24	1, 2, 3, 23	dochssv 39369	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
25	8, 22, 24	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
26		hdmapinvlem3.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
27	25, 26	sseldd 3922	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
28	1, 2, 3, 4, 5, 6, 7, 8, 21, 27	hdmapsub 39861	. . 3 ⊢ (𝜑 → (𝑆‘((𝐽 · 𝐸) − 𝐷)) = ((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷)))
29	28	fveq1d 6776	. 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 39844	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐽 · 𝐸)) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
33	1, 2, 3, 5, 31, 7, 8, 27	hdmapcl 39844	. . . 4 ⊢ (𝜑 → (𝑆‘𝐷) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
34		hdmapinvlem3.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
35	3, 17, 18, 19	lmodvscl 20140	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐼 · 𝐸) ∈ 𝑉)
36	9, 34, 16, 35	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝐼 · 𝐸) ∈ 𝑉)
37		hdmapinvlem3.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
38	25, 37	sseldd 3922	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
39		hdmapinvlem3.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
40	3, 39	lmodvacl 20137	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼 · 𝐸) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
41	9, 36, 38, 40	syl3anc 1370	. . . 4 ⊢ (𝜑 → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
42	1, 2, 3, 17, 30, 5, 31, 6, 8, 32, 33, 41	lcdvsubval 39632	. . 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 39921	. . . . 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 39925	. . . . . . 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 39923	. . . . . . . . 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 39850	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
52	51	oveq2d 7291	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐸)‘𝐸)) = (𝐼 × (1_r‘𝑅)))
53	17	lmodring 20131	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
54	9, 53	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
55	19, 45, 50	ringridm 19811	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → (𝐼 × (1_r‘𝑅)) = 𝐼)
56	54, 34, 55	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × (1_r‘𝑅)) = 𝐼)
57	48, 52, 56	3eqtrd 2782	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘(𝐼 · 𝐸)) = 𝐼)
58	57	oveq1d 7290	. . . . . . 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 39925	. . . . . . 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 39932	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 )
63	62	oveq1d 7290	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐶) × (𝐺‘𝐽)) = ( 0 × (𝐺‘𝐽)))
64	1, 2, 17, 19, 46, 8, 10	hgmapcl 39903	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐽) ∈ 𝐵)
65	19, 45, 61	ringlz 19826	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐽) ∈ 𝐵) → ( 0 × (𝐺‘𝐽)) = 0 )
66	54, 64, 65	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ( 0 × (𝐺‘𝐽)) = 0 )
67	60, 63, 66	3eqtrd 2782	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘𝐶) = 0 )
68	59, 67	oveq12d 7293	. . . . 5 ⊢ (𝜑 → (((𝑆‘(𝐽 · 𝐸))‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘(𝐽 · 𝐸))‘𝐶)) = ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ))
69		ringgrp 19788	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
70	54, 69	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
71	17, 19, 45	lmodmcl 20135	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ (𝐺‘𝐽) ∈ 𝐵) → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
72	9, 34, 64, 71	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
73	19, 43, 61	grprid 18610	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ) = (𝐼 × (𝐺‘𝐽)))
74	70, 72, 73	syl2anc 584	. . . . 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 39921	. . . . 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 39923	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘(𝐼 · 𝐸)) = (𝐼 × ((𝑆‘𝐷)‘𝐸)))
78	1, 14, 23, 2, 3, 17, 19, 45, 61, 7, 8, 26	hdmapinvlem2 39933	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐸) = 0 )
79	78	oveq2d 7291	. . . . . . 7 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐷)‘𝐸)) = (𝐼 × 0 ))
80	19, 45, 61	ringrz 19827	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → (𝐼 × 0 ) = 0 )
81	54, 34, 80	syl2anc 584	. . . . . . 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 7293	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (((𝑆‘𝐷)‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘𝐷)‘𝐶)))
85	19, 43, 61	grplid 18609	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (𝐼 × (𝐺‘𝐽)))
86	70, 72, 85	syl2anc 584	. . . . 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 7293	. . 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 18659	. . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ((𝐼 × (𝐺‘𝐽))(-_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = 0 )
91	70, 72, 90	syl2anc 584	. 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 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 {csn 4561 ⟨cop 4567 I cid 5488 ↾ cres 5591 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +_gcplusg 16962 ._rcmulr 16963 Scalarcsca 16965 ·_𝑠 cvsca 16966 0_gc0g 17150 Grpcgrp 18577 -_gcsg 18579 1_rcur 19737 Ringcrg 19783 LModclmod 20123 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 DVecHcdvh 39092 ocHcoch 39361 LCDualclcd 39600 HVMapchvm 39770 HDMapchdma 39806 HGMapchg 39897

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lshyp 36991 df-lcv 37033 df-lfl 37072 df-lkr 37100 df-ldual 37138 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409 df-lcdual 39601 df-mapd 39639 df-hvmap 39771 df-hdmap1 39807 df-hdmap 39808 df-hgmap 39898

This theorem is referenced by: hdmapinvlem4 39935

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