hdmap1l6lem1 - Mathbox for Norm Megill

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Theorem hdmap1l6lem1 38935

Description: Lemma for hdmap1l6 38949. Part (6) in [Baer] p. 47, lines 16-18. (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
hdmap1l6lem1	⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))

**Proof of Theorem hdmap1l6lem1**
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 2819	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		hdmap1l6.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	1, 3, 5	dvhlmod 38238	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		hdmap1l6cl.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
8	7	eldifad 3946	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
9		hdmap1l6e.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
10	9	eldifad 3946	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11		hdmap1l6.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
12		hdmap1l6.s	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
13	11, 12	lmodvsubcl 19671	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
14	6, 8, 10, 13	syl3anc 1366	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
15		hdmap1l6.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
16	11, 4, 15	lspsncl 19741	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
17	6, 14, 16	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
18		hdmap1l6e.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
19	18	eldifad 3946	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
20	11, 4, 15	lspsncl 19741	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
21	6, 19, 20	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
22		eqid 2819	. . . . . 6 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
23	4, 22	lsmcl 19847	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
24	6, 17, 21, 23	syl3anc 1366	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
25	11, 12	lmodvsubcl 19671	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑋 − 𝑍) ∈ 𝑉)
26	6, 8, 19, 25	syl3anc 1366	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑍) ∈ 𝑉)
27	11, 4, 15	lspsncl 19741	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑍) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈))
28	6, 26, 27	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈))
29	11, 4, 15	lspsncl 19741	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
30	6, 10, 29	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
31	4, 22	lsmcl 19847	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
32	6, 28, 30, 31	syl3anc 1366	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
33	1, 2, 3, 4, 5, 24, 32	mapdin 38790	. . 3 ⊢ (𝜑 → (𝑀‘(((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))) = ((𝑀‘((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍}))) ∩ (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))))
34		hdmap1l6.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
35		eqid 2819	. . . . . 6 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
36	1, 2, 3, 4, 22, 34, 35, 5, 17, 21	mapdlsm 38792	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍}))) = ((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))))
37	1, 2, 3, 4, 22, 34, 35, 5, 28, 30	mapdlsm 38792	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌}))) = ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌}))))
38	36, 37	ineq12d 4188	. . . 4 ⊢ (𝜑 → ((𝑀‘((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍}))) ∩ (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))) = (((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) ∩ ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌})))))
39		hdmap1l6.fg	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
40		hdmap1l6c.o	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
41		hdmap1l6.d	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐶)
42		hdmap1l6.r	. . . . . . . . 9 ⊢ 𝑅 = (-_g‘𝐶)
43		hdmap1l6.l	. . . . . . . . 9 ⊢ 𝐿 = (LSpan‘𝐶)
44		hdmap1l6.i	. . . . . . . . 9 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
45		hdmap1l6.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
46		hdmap1l6.mn	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
47	1, 3, 5	dvhlvec 38237	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ LVec)
48		hdmap1l6.yz	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
49		hdmap1l6e.xn	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
50	11, 40, 15, 47, 10, 18, 8, 48, 49	lspindp2 19899	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
51	50	simpld 497	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
52	1, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 51, 7, 10	hdmap1cl 38932	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
53	39, 52	eqeltrrd 2912	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
54	1, 3, 11, 12, 40, 15, 34, 41, 42, 43, 2, 44, 5, 7, 45, 9, 53, 51, 46	hdmap1eq 38929	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)}))))
55	39, 54	mpbid 234	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)})))
56	55	simprd 498	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)}))
57		hdmap1l6.fe	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
58	11, 40, 15, 47, 9, 19, 8, 48, 49	lspindp1 19897	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
59	58	simpld 497	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
60	1, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 59, 7, 19	hdmap1cl 38932	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
61	57, 60	eqeltrrd 2912	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐷)
62	1, 3, 11, 12, 40, 15, 34, 41, 42, 43, 2, 44, 5, 7, 45, 18, 61, 59, 46	hdmap1eq 38929	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐿‘{(𝐹𝑅𝐸)}))))
63	57, 62	mpbid 234	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐿‘{(𝐹𝑅𝐸)})))
64	63	simpld 497	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}))
65	56, 64	oveq12d 7166	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐿‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐿‘{𝐸})))
66	63	simprd 498	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐿‘{(𝐹𝑅𝐸)}))
67	55	simpld 497	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}))
68	66, 67	oveq12d 7166	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌}))) = ((𝐿‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐿‘{𝐺})))
69	65, 68	ineq12d 4188	. . . 4 ⊢ (𝜑 → (((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) ∩ ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌})))) = (((𝐿‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐿‘{𝐺}))))
70	38, 69	eqtrd 2854	. . 3 ⊢ (𝜑 → ((𝑀‘((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍}))) ∩ (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))) = (((𝐿‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐿‘{𝐺}))))
71	33, 70	eqtrd 2854	. 2 ⊢ (𝜑 → (𝑀‘(((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))) = (((𝐿‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐿‘{𝐺}))))
72		hdmap1l6.p	. . . 4 ⊢ + = (+_g‘𝑈)
73	11, 12, 40, 22, 15, 47, 8, 49, 48, 9, 18, 72	baerlem5a 38842	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌}))))
74	73	fveq2d 6667	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝑀‘(((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))))
75		hdmap1l6.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
76	1, 34, 5	lcdlvec 38719	. . 3 ⊢ (𝜑 → 𝐶 ∈ LVec)
77	1, 2, 3, 11, 15, 34, 41, 43, 5, 45, 46, 8, 10, 53, 67, 19, 61, 64, 49	mapdindp 38799	. . 3 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐿‘{𝐺, 𝐸}))
78	1, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 10, 19, 61, 64, 48	mapdncol 38798	. . 3 ⊢ (𝜑 → (𝐿‘{𝐺}) ≠ (𝐿‘{𝐸}))
79	1, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 40, 75, 9	mapdn0 38797	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 ∖ {𝑄}))
80	1, 2, 3, 11, 15, 34, 41, 43, 5, 61, 64, 40, 75, 18	mapdn0 38797	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐷 ∖ {𝑄}))
81		hdmap1l6.a	. . 3 ⊢ ✚ = (+_g‘𝐶)
82	41, 42, 75, 35, 43, 76, 45, 77, 78, 79, 80, 81	baerlem5a 38842	. 2 ⊢ (𝜑 → (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}) = (((𝐿‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐿‘{𝐺}))))
83	71, 74, 82	3eqtr4d 2864	1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∖ cdif 3931 ∩ cin 3933 {csn 4559 {cpr 4561 ⟨cotp 4567 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 +_gcplusg 16557 0_gc0g 16705 -_gcsg 18097 LSSumclsm 18751 LModclmod 19626 LSubSpclss 19695 LSpanclspn 19735 HLchlt 36478 LHypclh 37112 DVecHcdvh 38206 LCDualclcd 38714 mapdcmpd 38752 HDMap1chdma1 38919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-riotaBAD 36081

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-undef 7931 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-dvr 19425 df-drng 19496 df-lmod 19628 df-lss 19696 df-lsp 19736 df-lvec 19867 df-lsatoms 36104 df-lshyp 36105 df-lcv 36147 df-lfl 36186 df-lkr 36214 df-ldual 36252 df-oposet 36304 df-ol 36306 df-oml 36307 df-covers 36394 df-ats 36395 df-atl 36426 df-cvlat 36450 df-hlat 36479 df-llines 36626 df-lplanes 36627 df-lvols 36628 df-lines 36629 df-psubsp 36631 df-pmap 36632 df-padd 36924 df-lhyp 37116 df-laut 37117 df-ldil 37232 df-ltrn 37233 df-trl 37287 df-tgrp 37871 df-tendo 37883 df-edring 37885 df-dveca 38131 df-disoa 38157 df-dvech 38207 df-dib 38267 df-dic 38301 df-dih 38357 df-doch 38476 df-djh 38523 df-lcdual 38715 df-mapd 38753 df-hdmap1 38921

This theorem is referenced by: hdmap1l6lem2 38936 hdmap1l6a 38937

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