hdmap1l6lem1 - Mathbox for Norm Megill

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Theorem hdmap1l6lem1 42562

Description: Lemma for hdmap1l6 42576. 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 2763	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		hdmap1l6.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	1, 3, 5	dvhlmod 41865	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		hdmap1l6cl.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
8	7	eldifad 3918	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
9		hdmap1l6e.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
10	9	eldifad 3918	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11		hdmap1l6.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
12		hdmap1l6.s	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
13	11, 12	lmodvsubcl 21009	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
14	6, 8, 10, 13	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
15		hdmap1l6.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
16	11, 4, 15	lspsncl 21079	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
17	6, 14, 16	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
18		hdmap1l6e.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
19	18	eldifad 3918	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
20	11, 4, 15	lspsncl 21079	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
21	6, 19, 20	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
22		eqid 2763	. . . . . 6 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
23	4, 22	lsmcl 21185	. . . . 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 1398	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
25	11, 12	lmodvsubcl 21009	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑋 − 𝑍) ∈ 𝑉)
26	6, 8, 19, 25	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑍) ∈ 𝑉)
27	11, 4, 15	lspsncl 21079	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑍) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈))
28	6, 26, 27	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈))
29	11, 4, 15	lspsncl 21079	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
30	6, 10, 29	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
31	4, 22	lsmcl 21185	. . . . 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 1398	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
33	1, 2, 3, 4, 5, 24, 32	mapdin 42417	. . 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 2763	. . . . . 6 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
36	1, 2, 3, 4, 22, 34, 35, 5, 17, 21	mapdlsm 42419	. . . . 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 42419	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌}))) = ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌}))))
38	36, 37	ineq12d 4175	. . . 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 41864	. . . . . . . . . . . . 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 21240	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
51	50	simpld 499	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
52	1, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 51, 7, 10	hdmap1cl 42559	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
53	39, 52	eqeltrrd 2864	. . . . . . . . 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 42556	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)}))))
55	39, 54	mpbid 235	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)})))
56	55	simprd 500	. . . . . 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 21238	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
59	58	simpld 499	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
60	1, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 59, 7, 19	hdmap1cl 42559	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
61	57, 60	eqeltrrd 2864	. . . . . . . . 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 42556	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐿‘{(𝐹𝑅𝐸)}))))
63	57, 62	mpbid 235	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐿‘{(𝐹𝑅𝐸)})))
64	63	simpld 499	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐿‘{𝐸}))
65	56, 64	oveq12d 7430	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐿‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐿‘{𝐸})))
66	63	simprd 500	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐿‘{(𝐹𝑅𝐸)}))
67	55	simpld 499	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}))
68	66, 67	oveq12d 7430	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌}))) = ((𝐿‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐿‘{𝐺})))
69	65, 68	ineq12d 4175	. . . 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 2798	. . 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 2798	. 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 42469	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌}))))
74	73	fveq2d 6887	. 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 42346	. . 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 42426	. . 3 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐿‘{𝐺, 𝐸}))
78	1, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 10, 19, 61, 64, 48	mapdncol 42425	. . 3 ⊢ (𝜑 → (𝐿‘{𝐺}) ≠ (𝐿‘{𝐸}))
79	1, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 40, 75, 9	mapdn0 42424	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 ∖ {𝑄}))
80	1, 2, 3, 11, 15, 34, 41, 43, 5, 61, 64, 40, 75, 18	mapdn0 42424	. . 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 42469	. 2 ⊢ (𝜑 → (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}) = (((𝐿‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐿‘{𝐸})) ∩ ((𝐿‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐿‘{𝐺}))))
83	71, 74, 82	3eqtr4d 2808	1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3903 ∩ cin 3905 {csn 4590 {cpr 4592 ⟨cotp 4598 ‘cfv 6538 (class class class)co 7412 Basecbs 17270 +_gcplusg 17311 0_gc0g 17493 -_gcsg 19003 LSSumclsm 19705 LModclmod 20962 LSubSpclss 21033 LSpanclspn 21073 HLchlt 40105 LHypclh 40739 DVecHcdvh 41833 LCDualclcd 42341 mapdcmpd 42379 HDMap1chdma1 42546

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-riotaBAD 39708

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-ot 4599 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8223 df-undef 8270 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-n0 12506 df-z 12593 df-uz 12864 df-fz 13537 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17495 df-mre 17639 df-mrc 17640 df-acs 17642 df-proset 18351 df-poset 18370 df-plt 18385 df-lub 18401 df-glb 18402 df-join 18403 df-meet 18404 df-p0 18480 df-p1 18481 df-lat 18489 df-clat 18556 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-grp 19004 df-minusg 19005 df-sbg 19006 df-subg 19190 df-cntz 19388 df-oppg 19417 df-lsm 19707 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-oppr 20420 df-dvdsr 20440 df-unit 20441 df-invr 20471 df-dvr 20484 df-nzr 20597 df-rlreg 20780 df-domn 20781 df-drng 20816 df-lmod 20964 df-lss 21034 df-lsp 21074 df-lvec 21205 df-lsatoms 39731 df-lshyp 39732 df-lcv 39774 df-lfl 39813 df-lkr 39841 df-ldual 39879 df-oposet 39931 df-ol 39933 df-oml 39934 df-covers 40021 df-ats 40022 df-atl 40053 df-cvlat 40077 df-hlat 40106 df-llines 40253 df-lplanes 40254 df-lvols 40255 df-lines 40256 df-psubsp 40258 df-pmap 40259 df-padd 40551 df-lhyp 40743 df-laut 40744 df-ldil 40859 df-ltrn 40860 df-trl 40914 df-tgrp 41498 df-tendo 41510 df-edring 41512 df-dveca 41758 df-disoa 41784 df-dvech 41834 df-dib 41894 df-dic 41928 df-dih 41984 df-doch 42103 df-djh 42150 df-lcdual 42342 df-mapd 42380 df-hdmap1 42548

This theorem is referenced by: hdmap1l6lem2 42563 hdmap1l6a 42564

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