hdmap1l6d - Mathbox for Norm Megill

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Theorem hdmap1l6d 41814

Description: Lemmma for hdmap1l6 41822. (Contributed by NM, 1-May-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}))
hdmap1l6d.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
hdmap1l6d.yz	⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
hdmap1l6d.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap1l6d.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
hdmap1l6d.w	⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
hdmap1l6d.wn	⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))

**Assertion**
Ref	Expression
hdmap1l6d	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)))

**Proof of Theorem hdmap1l6d**
Step	Hyp	Ref	Expression
1		hdmap1l6.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1l6.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		hdmap1l6.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	lcdlmod 41593	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hdmap1l6.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		hdmap1l6.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
7		hdmap1l6c.o	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
8		hdmap1l6.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
9		hdmap1l6.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐶)
10		hdmap1l6.l	. . . . . 6 ⊢ 𝐿 = (LSpan‘𝐶)
11		hdmap1l6.m	. . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
12		hdmap1l6.i	. . . . . 6 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
13		hdmap1l6.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
14		hdmap1l6.mn	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
15	1, 5, 3	dvhlvec 41110	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LVec)
16		hdmap1l6d.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
17	16	eldifad 3929	. . . . . . . . 9 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
18		hdmap1l6cl.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19	18	eldifad 3929	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
20		hdmap1l6d.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3929	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
22		hdmap1l6d.wn	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
23	6, 8, 15, 17, 19, 21, 22	lspindpi 21049	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
24	23	simpld 494	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
25	24	necomd 2981	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))
26	1, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17	hdmap1cl 41805	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ∈ 𝐷)
27		hdmap1l6.a	. . . . . 6 ⊢ ✚ = (+_g‘𝐶)
28		hdmap1l6.q	. . . . . 6 ⊢ 𝑄 = (0_g‘𝐶)
29	9, 27, 28	lmod0vrid 20806	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ∈ 𝐷) → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ 𝑄) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
30	4, 26, 29	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ 𝑄) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
31	30	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ 𝑄) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
32		oteq3 4851	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → ⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩ = ⟨𝑋, 𝐹, 0 ⟩)
33	32	fveq2d 6865	. . . . 5 ⊢ ((𝑌 + 𝑍) = 0 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = (𝐼‘⟨𝑋, 𝐹, 0 ⟩))
34	1, 5, 6, 7, 2, 9, 28, 12, 3, 13, 19	hdmap1val0 41800	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
35	33, 34	sylan9eqr 2787	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = 𝑄)
36	35	oveq2d 7406	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ 𝑄))
37		oveq2 7398	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → (𝑤 + (𝑌 + 𝑍)) = (𝑤 + 0 ))
38	1, 5, 3	dvhlmod 41111	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
39		hdmap1l6.p	. . . . . . . 8 ⊢ + = (+_g‘𝑈)
40	6, 39, 7	lmod0vrid 20806	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉) → (𝑤 + 0 ) = 𝑤)
41	38, 17, 40	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑤 + 0 ) = 𝑤)
42	37, 41	sylan9eqr 2787	. . . . 5 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑤 + (𝑌 + 𝑍)) = 𝑤)
43	42	oteq3d 4854	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩ = ⟨𝑋, 𝐹, 𝑤⟩)
44	43	fveq2d 6865	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
45	31, 36, 44	3eqtr4rd 2776	. 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)))
46		hdmap1l6.s	. . 3 ⊢ − = (-_g‘𝑈)
47		hdmap1l6.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
48	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
49	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝐹 ∈ 𝐷)
50	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
51	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
52	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑤 ∈ (𝑉 ∖ { 0 }))
53		hdmap1l6d.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
54	53	eldifad 3929	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
55	6, 39	lmodvacl 20788	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
56	38, 21, 54, 55	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
57	56	anim1i 615	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
58		eldifsn 4753	. . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
59	57, 58	sylibr 234	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }))
60		hdmap1l6d.yz	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))
61		hdmap1l6d.xn	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
62	6, 8, 15, 19, 21, 54, 61	lspindpi 21049	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
63	62	simpld 494	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
64	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp1 41721	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
65	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp2 41722	. . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))
66	6, 7, 8, 15, 18, 56, 17, 64, 65	lspindp1 21050	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑤, (𝑌 + 𝑍)})))
67	66	simprd 495	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, (𝑌 + 𝑍)}))
68	67	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝑤, (𝑌 + 𝑍)}))
69	23	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
70	6, 7, 8, 15, 16, 21, 69	lspsnne1 21034	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌}))
71		eqid 2730	. . . . . . . . . 10 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
72	6, 8, 71, 38, 21, 54	lsmpr 21003	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})))
73	60	oveq2d 7406	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑌})) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})))
74		eqid 2730	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
75	6, 74, 8	lspsncl 20890	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
76	38, 21, 75	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
77	74	lsssubg 20870	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈))
78	38, 76, 77	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑈))
79	71	lsmidm 19600	. . . . . . . . . 10 ⊢ ((𝑁‘{𝑌}) ∈ (SubGrp‘𝑈) → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘{𝑌}))
80	78, 79	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘{𝑌}))
81	72, 73, 80	3eqtr2d 2771	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌}))
82	70, 81	neleqtrrd 2852	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑍}))
83	6, 39, 8, 38, 21, 54, 17, 82	lspindp4 21054	. . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌, (𝑌 + 𝑍)}))
84	6, 8, 15, 17, 21, 56, 83	lspindpi 21049	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)})))
85	84	simprd 495	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
86	85	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
87		eqidd 2731	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
88		eqidd 2731	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩))
89	1, 5, 6, 39, 46, 7, 8, 2, 9, 27, 47, 28, 10, 11, 12, 48, 49, 50, 51, 52, 59, 68, 86, 87, 88	hdmap1l6a 41810	. 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)))
90	45, 89	pm2.61dane 3013	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∖ cdif 3914 {csn 4592 {cpr 4594 ⟨cotp 4600 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 +_gcplusg 17227 0_gc0g 17409 -_gcsg 18874 SubGrpcsubg 19059 LSSumclsm 19571 LModclmod 20773 LSubSpclss 20844 LSpanclspn 20884 HLchlt 39350 LHypclh 39985 DVecHcdvh 41079 LCDualclcd 41587 mapdcmpd 41625 HDMap1chdma1 41792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 38953

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-ot 4601 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17411 df-mre 17554 df-mrc 17555 df-acs 17557 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-oppg 19285 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-nzr 20429 df-rlreg 20610 df-domn 20611 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976 df-lshyp 38977 df-lcv 39019 df-lfl 39058 df-lkr 39086 df-ldual 39124 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-tgrp 40744 df-tendo 40756 df-edring 40758 df-dveca 41004 df-disoa 41030 df-dvech 41080 df-dib 41140 df-dic 41174 df-dih 41230 df-doch 41349 df-djh 41396 df-lcdual 41588 df-mapd 41626 df-hdmap1 41794

This theorem is referenced by: hdmap1l6g 41817

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