hdmapinvlem3 - Mathbox for Norm Megill

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Theorem hdmapinvlem3 43169

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 2763	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
6		eqid 2763	. . . 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 42359	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		hdmapinvlem3.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝐵)
11		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2763	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
13		eqid 2763	. . . . . . 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 42361	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
16	15	eldifad 4054	. . . . 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 21183	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐽 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐽 · 𝐸) ∈ 𝑉)
21	9, 10, 16, 20	syl3anc 1399	. . . 4 ⊢ (𝜑 → (𝐽 · 𝐸) ∈ 𝑉)
22	16	snssd 4747	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
23		hdmapinvlem3.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
24	1, 2, 3, 23	dochssv 42604	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
25	8, 22, 24	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
26		hdmapinvlem3.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))
27	25, 26	sseldd 3914	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
28	1, 2, 3, 4, 5, 6, 7, 8, 21, 27	hdmapsub 43096	. . 3 ⊢ (𝜑 → (𝑆‘((𝐽 · 𝐸) − 𝐷)) = ((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷)))
29	28	fveq1d 6910	. 2 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = (((𝑆‘(𝐽 · 𝐸))(-_g‘((LCDual‘𝐾)‘𝑊))(𝑆‘𝐷))‘((𝐼 · 𝐸) + 𝐶)))
30		eqid 2763	. . . 4 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
31		eqid 2763	. . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
32	1, 2, 3, 5, 31, 7, 8, 21	hdmapcl 43079	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐽 · 𝐸)) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
33	1, 2, 3, 5, 31, 7, 8, 27	hdmapcl 43079	. . . 4 ⊢ (𝜑 → (𝑆‘𝐷) ∈ (Base‘((LCDual‘𝐾)‘𝑊)))
34		hdmapinvlem3.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
35	3, 17, 18, 19	lmodvscl 21183	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝐼 · 𝐸) ∈ 𝑉)
36	9, 34, 16, 35	syl3anc 1399	. . . . 5 ⊢ (𝜑 → (𝐼 · 𝐸) ∈ 𝑉)
37		hdmapinvlem3.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))
38	25, 37	sseldd 3914	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
39		hdmapinvlem3.p	. . . . . 6 ⊢ + = (+_g‘𝑈)
40	3, 39	lmodvacl 21180	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼 · 𝐸) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
41	9, 36, 38, 40	syl3anc 1399	. . . 4 ⊢ (𝜑 → ((𝐼 · 𝐸) + 𝐶) ∈ 𝑉)
42	1, 2, 3, 17, 30, 5, 31, 6, 8, 32, 33, 41	lcdvsubval 42867	. . 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 2763	. . . . . 6 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
44	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 21	hdmaplna1 43156	. . . . 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 43160	. . . . . . 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 43158	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘𝐸)‘(𝐼 · 𝐸)) = (𝐼 × ((𝑆‘𝐸)‘𝐸)))
49		eqid 2763	. . . . . . . . . . 11 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
50		eqid 2763	. . . . . . . . . . 11 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
51	1, 14, 49, 7, 8, 2, 17, 50	hdmapevec2 43085	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
52	51	oveq2d 7459	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐸)‘𝐸)) = (𝐼 × (1_r‘𝑅)))
53	17	lmodring 21173	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
54	9, 53	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring)
55	19, 45, 50	ringridm 20529	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → (𝐼 × (1_r‘𝑅)) = 𝐼)
56	54, 34, 55	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × (1_r‘𝑅)) = 𝐼)
57	48, 52, 56	3eqtrd 2802	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘(𝐼 · 𝐸)) = 𝐼)
58	57	oveq1d 7458	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘(𝐼 · 𝐸)) × (𝐺‘𝐽)) = (𝐼 × (𝐺‘𝐽)))
59	47, 58	eqtrd 2798	. . . . . 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 43160	. . . . . . 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 43167	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 )
63	62	oveq1d 7458	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝐸)‘𝐶) × (𝐺‘𝐽)) = ( 0 × (𝐺‘𝐽)))
64	1, 2, 17, 19, 46, 8, 10	hgmapcl 43138	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐽) ∈ 𝐵)
65	19, 45, 61	ringlz 20554	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐽) ∈ 𝐵) → ( 0 × (𝐺‘𝐽)) = 0 )
66	54, 64, 65	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ( 0 × (𝐺‘𝐽)) = 0 )
67	60, 63, 66	3eqtrd 2802	. . . . . 6 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘𝐶) = 0 )
68	59, 67	oveq12d 7461	. . . . 5 ⊢ (𝜑 → (((𝑆‘(𝐽 · 𝐸))‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘(𝐽 · 𝐸))‘𝐶)) = ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ))
69		ringgrp 20494	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
70	54, 69	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
71	17, 19, 45	lmodmcl 21178	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐼 ∈ 𝐵 ∧ (𝐺‘𝐽) ∈ 𝐵) → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
72	9, 34, 64, 71	syl3anc 1399	. . . . . 6 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) ∈ 𝐵)
73	19, 43, 61	grprid 19208	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ) = (𝐼 × (𝐺‘𝐽)))
74	70, 72, 73	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((𝐼 × (𝐺‘𝐽))(+_g‘𝑅) 0 ) = (𝐼 × (𝐺‘𝐽)))
75	44, 68, 74	3eqtrd 2802	. . . 4 ⊢ (𝜑 → ((𝑆‘(𝐽 · 𝐸))‘((𝐼 · 𝐸) + 𝐶)) = (𝐼 × (𝐺‘𝐽)))
76	1, 2, 3, 39, 17, 43, 7, 8, 36, 38, 27	hdmaplna1 43156	. . . . 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 43158	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝐷)‘(𝐼 · 𝐸)) = (𝐼 × ((𝑆‘𝐷)‘𝐸)))
78	1, 14, 23, 2, 3, 17, 19, 45, 61, 7, 8, 26	hdmapinvlem2 43168	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐸) = 0 )
79	78	oveq2d 7459	. . . . . . 7 ⊢ (𝜑 → (𝐼 × ((𝑆‘𝐷)‘𝐸)) = (𝐼 × 0 ))
80	19, 45, 61	ringrz 20555	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → (𝐼 × 0 ) = 0 )
81	54, 34, 80	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐼 × 0 ) = 0 )
82	77, 79, 81	3eqtrrd 2803	. . . . . 6 ⊢ (𝜑 → 0 = ((𝑆‘𝐷)‘(𝐼 · 𝐸)))
83		hdmapinvlem3.ij	. . . . . 6 ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))
84	82, 83	oveq12d 7461	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (((𝑆‘𝐷)‘(𝐼 · 𝐸))(+_g‘𝑅)((𝑆‘𝐷)‘𝐶)))
85	19, 43, 61	grplid 19207	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (𝐼 × (𝐺‘𝐽)))
86	70, 72, 85	syl2anc 596	. . . . 5 ⊢ (𝜑 → ( 0 (+_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = (𝐼 × (𝐺‘𝐽)))
87	76, 84, 86	3eqtr2d 2804	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐷)‘((𝐼 · 𝐸) + 𝐶)) = (𝐼 × (𝐺‘𝐽)))
88	75, 87	oveq12d 7461	. . 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 2798	. 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 19263	. . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝐼 × (𝐺‘𝐽)) ∈ 𝐵) → ((𝐼 × (𝐺‘𝐽))(-_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = 0 )
91	70, 72, 90	syl2anc 596	. 2 ⊢ (𝜑 → ((𝐼 × (𝐺‘𝐽))(-_g‘𝑅)(𝐼 × (𝐺‘𝐽))) = 0 )
92	29, 89, 91	3eqtrd 2802	1 ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 {csn 4588 ⟨cop 4838 I cid 5549 ↾ cres 5657 ‘cfv 6558 (class class class)co 7443 Basecbs 17416 +_gcplusg 17457 ._rcmulr 17458 Scalarcsca 17460 ·_𝑠 cvsca 17461 0_gc0g 17639 Grpcgrp 19173 -_gcsg 19175 1_rcur 20437 Ringcrg 20489 LModclmod 21165 HLchlt 40599 LHypclh 41233 LTrncltrn 41350 DVecHcdvh 42327 ocHcoch 42596 LCDualclcd 42835 HVMapchvm 43005 HDMapchdma 43041 HGMapchg 43132

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-riotaBAD 40202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-ot 4862 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-tpos 8271 df-undef 8318 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-n0 12634 df-z 12721 df-uz 12993 df-fz 13667 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-sca 17473 df-vsca 17474 df-0g 17641 df-mre 17785 df-mrc 17786 df-acs 17788 df-proset 18497 df-poset 18516 df-plt 18531 df-lub 18547 df-glb 18548 df-join 18549 df-meet 18550 df-p0 18626 df-p1 18627 df-lat 18635 df-clat 18702 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-subg 19362 df-cntz 19561 df-oppg 19590 df-lsm 19880 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-invr 20648 df-dvr 20661 df-nzr 20793 df-rlreg 20976 df-domn 20977 df-drng 21012 df-lmod 21167 df-lss 21237 df-lsp 21277 df-lvec 21408 df-lsatoms 40225 df-lshyp 40226 df-lcv 40268 df-lfl 40307 df-lkr 40335 df-ldual 40373 df-oposet 40425 df-ol 40427 df-oml 40428 df-covers 40515 df-ats 40516 df-atl 40547 df-cvlat 40571 df-hlat 40600 df-llines 40747 df-lplanes 40748 df-lvols 40749 df-lines 40750 df-psubsp 40752 df-pmap 40753 df-padd 41045 df-lhyp 41237 df-laut 41238 df-ldil 41353 df-ltrn 41354 df-trl 41408 df-tgrp 41992 df-tendo 42004 df-edring 42006 df-dveca 42252 df-disoa 42278 df-dvech 42328 df-dib 42388 df-dic 42422 df-dih 42478 df-doch 42597 df-djh 42644 df-lcdual 42836 df-mapd 42874 df-hvmap 43006 df-hdmap1 43042 df-hdmap 43043 df-hgmap 43133

This theorem is used by: hdmapinvlem4 43170

W3C validator