hdmap1l6lem2 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1l6lem2		Structured version Visualization version GIF version

Theorem hdmap1l6lem2 41765

Description: Lemma for hdmap1l6 41778. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.)

**Hypotheses**
Ref	Expression
hdmap1l6.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1l6.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1l6.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1l6.p	⊢ + = (+_g‘𝑈)
hdmap1l6.s	⊢ − = (-_g‘𝑈)
hdmap1l6c.o	⊢ 0 = (0_g‘𝑈)
hdmap1l6.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1l6.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1l6.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1l6.a	⊢ ✚ = (+_g‘𝐶)
hdmap1l6.r	⊢ 𝑅 = (-_g‘𝐶)
hdmap1l6.q	⊢ 𝑄 = (0_g‘𝐶)
hdmap1l6.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap1l6.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1l6.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1l6.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap1l6.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
hdmap1l6cl.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap1l6.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
hdmap1l6e.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap1l6e.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
hdmap1l6e.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
hdmap1l6.yz	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
hdmap1l6.fg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
hdmap1l6.fe	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)

**Assertion**
Ref	Expression
hdmap1l6lem2	⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 ✚ 𝐸)}))

**Proof of Theorem hdmap1l6lem2**
Step	Hyp	Ref	Expression
1		hdmap1l6.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
3		hdmap1l6.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		eqid 2740	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		hdmap1l6.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	1, 3, 5	dvhlmod 41067	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		hdmap1l6e.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
8	7	eldifad 3988	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
9		hdmap1l6.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
10		hdmap1l6.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
11	9, 4, 10	lspsncl 20998	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
12	6, 8, 11	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
13		hdmap1l6e.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3988	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
15	9, 4, 10	lspsncl 20998	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
16	6, 14, 15	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
17		eqid 2740	. . . . . 6 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
18	4, 17	lsmcl 21105	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
19	6, 12, 16, 18	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
20		hdmap1l6cl.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3988	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22		hdmap1l6.p	. . . . . . . . 9 ⊢ + = (+_g‘𝑈)
23	9, 22	lmodvacl 20895	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
24	6, 8, 14, 23	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
25		hdmap1l6.s	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
26	9, 25	lmodvsubcl 20927	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 + 𝑍) ∈ 𝑉) → (𝑋 − (𝑌 + 𝑍)) ∈ 𝑉)
27	6, 21, 24, 26	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝑋 − (𝑌 + 𝑍)) ∈ 𝑉)
28	9, 4, 10	lspsncl 20998	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − (𝑌 + 𝑍)) ∈ 𝑉) → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) ∈ (LSubSp‘𝑈))
29	6, 27, 28	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) ∈ (LSubSp‘𝑈))
30	9, 4, 10	lspsncl 20998	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
31	6, 21, 30	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
32	4, 17	lsmcl 21105	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋})) ∈ (LSubSp‘𝑈))
33	6, 29, 31, 32	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋})) ∈ (LSubSp‘𝑈))
34	1, 2, 3, 4, 5, 19, 33	mapdin 41619	. . 3 ⊢ (𝜑 → (𝑀‘(((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋})))) = ((𝑀‘((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) ∩ (𝑀‘((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋})))))
35		hdmap1l6.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
36		eqid 2740	. . . . . 6 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
37	1, 2, 3, 4, 17, 35, 36, 5, 12, 16	mapdlsm 41621	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) = ((𝑀‘(𝑁‘{𝑌}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))))
38		hdmap1l6.fg	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
39		hdmap1l6c.o	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
40		hdmap1l6.d	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐶)
41		hdmap1l6.r	. . . . . . . . 9 ⊢ 𝑅 = (-_g‘𝐶)
42		hdmap1l6.l	. . . . . . . . 9 ⊢ 𝐿 = (LSpan‘𝐶)
43		hdmap1l6.i	. . . . . . . . 9 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
44		hdmap1l6.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
45		hdmap1l6.mn	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
46	1, 3, 5	dvhlvec 41066	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ LVec)
47		hdmap1l6.yz	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
48		hdmap1l6e.xn	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
49	9, 39, 10, 46, 8, 13, 21, 47, 48	lspindp2 21160	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
50	49	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
51	1, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 50, 20, 8	hdmap1cl 41761	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
52	38, 51	eqeltrrd 2845	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
53	1, 3, 9, 25, 39, 10, 35, 40, 41, 42, 2, 43, 5, 20, 44, 7, 52, 50, 45	hdmap1eq 41758	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)}))))
54	38, 53	mpbid 232	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)})))
55	54	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}))
56		hdmap1l6.fe	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
57	9, 39, 10, 46, 7, 14, 21, 47, 48	lspindp1 21158	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
58	57	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
59	1, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 58, 20, 14	hdmap1cl 41761	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
60	56, 59	eqeltrrd 2845	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐷)
61	1, 3, 9, 25, 39, 10, 35, 40, 41, 42, 2, 43, 5, 20, 44, 13, 60, 58, 45	hdmap1eq 41758	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐿‘{(𝐹𝑅𝐸)}))))
62	56, 61	mpbid 232	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐿‘{(𝐹𝑅𝐸)})))
63	62	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}))
64	55, 63	oveq12d 7466	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐿‘{𝐺})(LSSum‘𝐶)(𝐿‘{𝐸})))
65	37, 64	eqtrd 2780	. . . 4 ⊢ (𝜑 → (𝑀‘((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) = ((𝐿‘{𝐺})(LSSum‘𝐶)(𝐿‘{𝐸})))
66	1, 2, 3, 4, 17, 35, 36, 5, 29, 31	mapdlsm 41621	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋}))) = ((𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑋}))))
67		hdmap1l6.a	. . . . . . 7 ⊢ ✚ = (+_g‘𝐶)
68		hdmap1l6.q	. . . . . . 7 ⊢ 𝑄 = (0_g‘𝐶)
69	1, 3, 9, 22, 25, 39, 10, 35, 40, 67, 41, 68, 42, 2, 43, 5, 44, 20, 45, 7, 13, 48, 47, 38, 56	hdmap1l6lem1 41764	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))
70	69, 45	oveq12d 7466	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑋}))) = ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹})))
71	66, 70	eqtrd 2780	. . . 4 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋}))) = ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹})))
72	65, 71	ineq12d 4242	. . 3 ⊢ (𝜑 → ((𝑀‘((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) ∩ (𝑀‘((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋})))) = (((𝐿‘{𝐺})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹}))))
73	34, 72	eqtrd 2780	. 2 ⊢ (𝜑 → (𝑀‘(((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋})))) = (((𝐿‘{𝐺})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹}))))
74	9, 25, 39, 17, 10, 46, 21, 48, 47, 7, 13, 22	baerlem5b 41672	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋}))))
75	74	fveq2d 6924	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝑀‘(((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋})))))
76	1, 35, 5	lcdlvec 41548	. . 3 ⊢ (𝜑 → 𝐶 ∈ LVec)
77	1, 2, 3, 9, 10, 35, 40, 42, 5, 44, 45, 21, 8, 52, 55, 14, 60, 63, 48	mapdindp 41628	. . 3 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐿‘{𝐺, 𝐸}))
78	1, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 8, 14, 60, 63, 47	mapdncol 41627	. . 3 ⊢ (𝜑 → (𝐿‘{𝐺}) ≠ (𝐿‘{𝐸}))
79	1, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 39, 68, 7	mapdn0 41626	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 ∖ {𝑄}))
80	1, 2, 3, 9, 10, 35, 40, 42, 5, 60, 63, 39, 68, 13	mapdn0 41626	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐷 ∖ {𝑄}))
81	40, 41, 68, 36, 42, 76, 44, 77, 78, 79, 80, 67	baerlem5b 41672	. 2 ⊢ (𝜑 → (𝐿‘{(𝐺 ✚ 𝐸)}) = (((𝐿‘{𝐺})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹}))))
82	73, 75, 81	3eqtr4d 2790	1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 ✚ 𝐸)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∖ cdif 3973 ∩ cin 3975 {csn 4648 {cpr 4650 ⟨cotp 4656 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 +_gcplusg 17311 0_gc0g 17499 -_gcsg 18975 LSSumclsm 19676 LModclmod 20880 LSubSpclss 20952 LSpanclspn 20992 HLchlt 39306 LHypclh 39941 DVecHcdvh 41035 LCDualclcd 41543 mapdcmpd 41581 HDMap1chdma1 41748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-riotaBAD 38909

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-undef 8314 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17501 df-mre 17644 df-mrc 17645 df-acs 17647 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-cntz 19357 df-oppg 19386 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-nzr 20539 df-rlreg 20716 df-domn 20717 df-drng 20753 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lvec 21125 df-lsatoms 38932 df-lshyp 38933 df-lcv 38975 df-lfl 39014 df-lkr 39042 df-ldual 39080 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-llines 39455 df-lplanes 39456 df-lvols 39457 df-lines 39458 df-psubsp 39460 df-pmap 39461 df-padd 39753 df-lhyp 39945 df-laut 39946 df-ldil 40061 df-ltrn 40062 df-trl 40116 df-tgrp 40700 df-tendo 40712 df-edring 40714 df-dveca 40960 df-disoa 40986 df-dvech 41036 df-dib 41096 df-dic 41130 df-dih 41186 df-doch 41305 df-djh 41352 df-lcdual 41544 df-mapd 41582 df-hdmap1 41750

This theorem is referenced by: hdmap1l6a 41766

W3C validator