hdmap1l6d - Mathbox for Norm Megill

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Theorem hdmap1l6d 41918

Description: Lemmma for hdmap1l6 41926. (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 41697	. . . . 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 41214	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LVec)
16		hdmap1l6d.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))
17	16	eldifad 3909	. . . . . . . . 9 ⊢ (𝜑 → 𝑤 ∈ 𝑉)
18		hdmap1l6cl.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19	18	eldifad 3909	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
20		hdmap1l6d.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
21	20	eldifad 3909	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
22		hdmap1l6d.wn	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
23	6, 8, 15, 17, 19, 21, 22	lspindpi 21075	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
24	23	simpld 494	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
25	24	necomd 2983	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))
26	1, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17	hdmap1cl 41909	. . . . 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 20832	. . . . 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 4835	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → ⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩ = ⟨𝑋, 𝐹, 0 ⟩)
33	32	fveq2d 6832	. . . . 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 41904	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
35	33, 34	sylan9eqr 2788	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = 𝑄)
36	35	oveq2d 7368	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ 𝑄))
37		oveq2 7360	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → (𝑤 + (𝑌 + 𝑍)) = (𝑤 + 0 ))
38	1, 5, 3	dvhlmod 41215	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
39		hdmap1l6.p	. . . . . . . 8 ⊢ + = (+_g‘𝑈)
40	6, 39, 7	lmod0vrid 20832	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉) → (𝑤 + 0 ) = 𝑤)
41	38, 17, 40	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑤 + 0 ) = 𝑤)
42	37, 41	sylan9eqr 2788	. . . . 5 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑤 + (𝑌 + 𝑍)) = 𝑤)
43	42	oteq3d 4838	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩ = ⟨𝑋, 𝐹, 𝑤⟩)
44	43	fveq2d 6832	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
45	31, 36, 44	3eqtr4rd 2777	. 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 3909	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
55	6, 39	lmodvacl 20814	. . . . . 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 4737	. . . 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 21075	. . . . . . . 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 41825	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
65	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp2 41826	. . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))
66	6, 7, 8, 15, 18, 56, 17, 64, 65	lspindp1 21076	. . . . 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 21060	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌}))
71		eqid 2731	. . . . . . . . . 10 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
72	6, 8, 71, 38, 21, 54	lsmpr 21029	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})))
73	60	oveq2d 7368	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑌})) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})))
74		eqid 2731	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
75	6, 74, 8	lspsncl 20916	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
76	38, 21, 75	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
77	74	lsssubg 20896	. . . . . . . . . . 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 19581	. . . . . . . . . 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 2772	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌}))
82	70, 81	neleqtrrd 2854	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑍}))
83	6, 39, 8, 38, 21, 54, 17, 82	lspindp4 21080	. . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌, (𝑌 + 𝑍)}))
84	6, 8, 15, 17, 21, 56, 83	lspindpi 21075	. . . . 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 2732	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
88		eqidd 2732	. . 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 41914	. 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 3015	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 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3894 {csn 4575 {cpr 4577 ⟨cotp 4583 ‘cfv 6487 (class class class)co 7352 Basecbs 17126 +_gcplusg 17167 0_gc0g 17349 -_gcsg 18854 SubGrpcsubg 19039 LSSumclsm 19552 LModclmod 20799 LSubSpclss 20870 LSpanclspn 20910 HLchlt 39455 LHypclh 40089 DVecHcdvh 41183 LCDualclcd 41691 mapdcmpd 41729 HDMap1chdma1 41896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-riotaBAD 39058

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-ot 4584 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-tpos 8162 df-undef 8209 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-n0 12388 df-z 12475 df-uz 12739 df-fz 13414 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-sca 17183 df-vsca 17184 df-0g 17351 df-mre 17494 df-mrc 17495 df-acs 17497 df-proset 18206 df-poset 18225 df-plt 18240 df-lub 18256 df-glb 18257 df-join 18258 df-meet 18259 df-p0 18335 df-p1 18336 df-lat 18344 df-clat 18411 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-submnd 18698 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19042 df-cntz 19235 df-oppg 19264 df-lsm 19554 df-cmn 19700 df-abl 19701 df-mgp 20065 df-rng 20077 df-ur 20106 df-ring 20159 df-oppr 20261 df-dvdsr 20281 df-unit 20282 df-invr 20312 df-dvr 20325 df-nzr 20434 df-rlreg 20615 df-domn 20616 df-drng 20652 df-lmod 20801 df-lss 20871 df-lsp 20911 df-lvec 21043 df-lsatoms 39081 df-lshyp 39082 df-lcv 39124 df-lfl 39163 df-lkr 39191 df-ldual 39229 df-oposet 39281 df-ol 39283 df-oml 39284 df-covers 39371 df-ats 39372 df-atl 39403 df-cvlat 39427 df-hlat 39456 df-llines 39603 df-lplanes 39604 df-lvols 39605 df-lines 39606 df-psubsp 39608 df-pmap 39609 df-padd 39901 df-lhyp 40093 df-laut 40094 df-ldil 40209 df-ltrn 40210 df-trl 40264 df-tgrp 40848 df-tendo 40860 df-edring 40862 df-dveca 41108 df-disoa 41134 df-dvech 41184 df-dib 41244 df-dic 41278 df-dih 41334 df-doch 41453 df-djh 41500 df-lcdual 41692 df-mapd 41730 df-hdmap1 41898

This theorem is referenced by: hdmap1l6g 41921

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