hdmap1l6lem2 - Mathbox for Norm Megill

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Theorem hdmap1l6lem2 42268

Description: Lemma for hdmap1l6 42281. 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 2737	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		hdmap1l6.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	1, 3, 5	dvhlmod 41570	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		hdmap1l6e.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
8	7	eldifad 3902	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
9		hdmap1l6.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
10		hdmap1l6.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
11	9, 4, 10	lspsncl 20963	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
12	6, 8, 11	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
13		hdmap1l6e.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3902	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
15	9, 4, 10	lspsncl 20963	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
16	6, 14, 15	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
17		eqid 2737	. . . . . 6 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
18	4, 17	lsmcl 21070	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
20		hdmap1l6cl.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3902	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22		hdmap1l6.p	. . . . . . . . 9 ⊢ + = (+_g‘𝑈)
23	9, 22	lmodvacl 20861	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
24	6, 8, 14, 23	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
25		hdmap1l6.s	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
26	9, 25	lmodvsubcl 20893	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 + 𝑍) ∈ 𝑉) → (𝑋 − (𝑌 + 𝑍)) ∈ 𝑉)
27	6, 21, 24, 26	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝑋 − (𝑌 + 𝑍)) ∈ 𝑉)
28	9, 4, 10	lspsncl 20963	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − (𝑌 + 𝑍)) ∈ 𝑉) → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) ∈ (LSubSp‘𝑈))
29	6, 27, 28	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) ∈ (LSubSp‘𝑈))
30	9, 4, 10	lspsncl 20963	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
31	6, 21, 30	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
32	4, 17	lsmcl 21070	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋})) ∈ (LSubSp‘𝑈))
34	1, 2, 3, 4, 5, 19, 33	mapdin 42122	. . 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 2737	. . . . . 6 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
37	1, 2, 3, 4, 17, 35, 36, 5, 12, 16	mapdlsm 42124	. . . . 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 41569	. . . . . . . . . . . . 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 21125	. . . . . . . . . . . 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 42264	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
52	38, 51	eqeltrrd 2838	. . . . . . . . 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 42261	. . . . . . . 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 21123	. . . . . . . . . . . 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 42264	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
60	56, 59	eqeltrrd 2838	. . . . . . . . 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 42261	. . . . . . . 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 7378	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐿‘{𝐺})(LSSum‘𝐶)(𝐿‘{𝐸})))
65	37, 64	eqtrd 2772	. . . 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 42124	. . . . 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 42267	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))
70	69, 45	oveq12d 7378	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑋}))) = ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹})))
71	66, 70	eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋}))) = ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹})))
72	65, 71	ineq12d 4162	. . 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 2772	. 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 42175	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋}))))
75	74	fveq2d 6838	. 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 42051	. . 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 42131	. . 3 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐿‘{𝐺, 𝐸}))
78	1, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 8, 14, 60, 63, 47	mapdncol 42130	. . 3 ⊢ (𝜑 → (𝐿‘{𝐺}) ≠ (𝐿‘{𝐸}))
79	1, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 39, 68, 7	mapdn0 42129	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 ∖ {𝑄}))
80	1, 2, 3, 9, 10, 35, 40, 42, 5, 60, 63, 39, 68, 13	mapdn0 42129	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐷 ∖ {𝑄}))
81	40, 41, 68, 36, 42, 76, 44, 77, 78, 79, 80, 67	baerlem5b 42175	. 2 ⊢ (𝜑 → (𝐿‘{(𝐺 ✚ 𝐸)}) = (((𝐿‘{𝐺})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹}))))
82	73, 75, 81	3eqtr4d 2782	1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 ✚ 𝐸)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ∩ cin 3889 {csn 4568 {cpr 4570 ⟨cotp 4576 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 +_gcplusg 17211 0_gc0g 17393 -_gcsg 18902 LSSumclsm 19600 LModclmod 20846 LSubSpclss 20917 LSpanclspn 20957 HLchlt 39810 LHypclh 40444 DVecHcdvh 41538 LCDualclcd 42046 mapdcmpd 42084 HDMap1chdma1 42251

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-riotaBAD 39413

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8169 df-undef 8216 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cntz 19283 df-oppg 19312 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-nzr 20481 df-rlreg 20662 df-domn 20663 df-drng 20699 df-lmod 20848 df-lss 20918 df-lsp 20958 df-lvec 21090 df-lsatoms 39436 df-lshyp 39437 df-lcv 39479 df-lfl 39518 df-lkr 39546 df-ldual 39584 df-oposet 39636 df-ol 39638 df-oml 39639 df-covers 39726 df-ats 39727 df-atl 39758 df-cvlat 39782 df-hlat 39811 df-llines 39958 df-lplanes 39959 df-lvols 39960 df-lines 39961 df-psubsp 39963 df-pmap 39964 df-padd 40256 df-lhyp 40448 df-laut 40449 df-ldil 40564 df-ltrn 40565 df-trl 40619 df-tgrp 41203 df-tendo 41215 df-edring 41217 df-dveca 41463 df-disoa 41489 df-dvech 41539 df-dib 41599 df-dic 41633 df-dih 41689 df-doch 41808 df-djh 41855 df-lcdual 42047 df-mapd 42085 df-hdmap1 42253

This theorem is referenced by: hdmap1l6a 42269

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