hdmap1l6lem2 - Mathbox for Norm Megill

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Theorem hdmap1l6lem2 39749

Description: Lemma for hdmap1l6 39762. 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 2738	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		hdmap1l6.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	1, 3, 5	dvhlmod 39051	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		hdmap1l6e.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
8	7	eldifad 3895	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
9		hdmap1l6.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
10		hdmap1l6.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
11	9, 4, 10	lspsncl 20154	. . . . . 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 3895	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
15	9, 4, 10	lspsncl 20154	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
16	6, 14, 15	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
17		eqid 2738	. . . . . 6 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
18	4, 17	lsmcl 20260	. . . . 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 1369	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
20		hdmap1l6cl.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3895	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
22		hdmap1l6.p	. . . . . . . . 9 ⊢ + = (+_g‘𝑈)
23	9, 22	lmodvacl 20052	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
24	6, 8, 14, 23	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
25		hdmap1l6.s	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
26	9, 25	lmodvsubcl 20083	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 + 𝑍) ∈ 𝑉) → (𝑋 − (𝑌 + 𝑍)) ∈ 𝑉)
27	6, 21, 24, 26	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝑋 − (𝑌 + 𝑍)) ∈ 𝑉)
28	9, 4, 10	lspsncl 20154	. . . . . 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 20154	. . . . . 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 20260	. . . . 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 1369	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋})) ∈ (LSubSp‘𝑈))
34	1, 2, 3, 4, 5, 19, 33	mapdin 39603	. . 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 2738	. . . . . 6 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
37	1, 2, 3, 4, 17, 35, 36, 5, 12, 16	mapdlsm 39605	. . . . 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 39050	. . . . . . . . . . . . 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 20312	. . . . . . . . . . . 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 39745	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
52	38, 51	eqeltrrd 2840	. . . . . . . . 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 39742	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)}))))
54	38, 53	mpbid 231	. . . . . . 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 20310	. . . . . . . . . . . 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 39745	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
60	56, 59	eqeltrrd 2840	. . . . . . . . 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 39742	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐿‘{(𝐹𝑅𝐸)}))))
62	56, 61	mpbid 231	. . . . . . 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 7273	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐿‘{𝐺})(LSSum‘𝐶)(𝐿‘{𝐸})))
65	37, 64	eqtrd 2778	. . . 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 39605	. . . . 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 39748	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))
70	69, 45	oveq12d 7273	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑋}))) = ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹})))
71	66, 70	eqtrd 2778	. . . 4 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋}))) = ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹})))
72	65, 71	ineq12d 4144	. . 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 2778	. 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 39656	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))})(LSSum‘𝑈)(𝑁‘{𝑋}))))
75	74	fveq2d 6760	. 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 39532	. . 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 39612	. . 3 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐿‘{𝐺, 𝐸}))
78	1, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 8, 14, 60, 63, 47	mapdncol 39611	. . 3 ⊢ (𝜑 → (𝐿‘{𝐺}) ≠ (𝐿‘{𝐸}))
79	1, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 39, 68, 7	mapdn0 39610	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 ∖ {𝑄}))
80	1, 2, 3, 9, 10, 35, 40, 42, 5, 60, 63, 39, 68, 13	mapdn0 39610	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐷 ∖ {𝑄}))
81	40, 41, 68, 36, 42, 76, 44, 77, 78, 79, 80, 67	baerlem5b 39656	. 2 ⊢ (𝜑 → (𝐿‘{(𝐺 ✚ 𝐸)}) = (((𝐿‘{𝐺})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSum‘𝐶)(𝐿‘{𝐹}))))
82	73, 75, 81	3eqtr4d 2788	1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 ✚ 𝐸)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ∩ cin 3882 {csn 4558 {cpr 4560 ⟨cotp 4566 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 +_gcplusg 16888 0_gc0g 17067 -_gcsg 18494 LSSumclsm 19154 LModclmod 20038 LSubSpclss 20108 LSpanclspn 20148 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 LCDualclcd 39527 mapdcmpd 39565 HDMap1chdma1 39732

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-hdmap1 39734

This theorem is referenced by: hdmap1l6a 39750

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