hdmap1l6d - Mathbox for Norm Megill

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Theorem hdmap1l6d 40032

Description: Lemmma for hdmap1l6 40040. (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 39811	. . . . 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 39328	. . . . . . . . 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 20466	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})))
24	23	simpld 495	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
25	24	necomd 2997	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))
26	1, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17	hdmap1cl 40023	. . . . 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 20226	. . . . 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 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ 𝑄) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
32		oteq3 4826	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → ⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩ = ⟨𝑋, 𝐹, 0 ⟩)
33	32	fveq2d 6815	. . . . 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 40018	. . . . 5 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
35	33, 34	sylan9eqr 2799	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = 𝑄)
36	35	oveq2d 7331	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ 𝑄))
37		oveq2 7323	. . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → (𝑤 + (𝑌 + 𝑍)) = (𝑤 + 0 ))
38	1, 5, 3	dvhlmod 39329	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
39		hdmap1l6.p	. . . . . . . 8 ⊢ + = (+_g‘𝑈)
40	6, 39, 7	lmod0vrid 20226	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉) → (𝑤 + 0 ) = 𝑤)
41	38, 17, 40	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑤 + 0 ) = 𝑤)
42	37, 41	sylan9eqr 2799	. . . . 5 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑤 + (𝑌 + 𝑍)) = 𝑤)
43	42	oteq3d 4829	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩ = ⟨𝑋, 𝐹, 𝑤⟩)
44	43	fveq2d 6815	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
45	31, 36, 44	3eqtr4rd 2788	. 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 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
49	13	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝐹 ∈ 𝐷)
50	18	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
51	14	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))
52	16	adantr 481	. . 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 20209	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉)
56	38, 21, 54, 55	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉)
57	56	anim1i 615	. . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
58		eldifsn 4732	. . . 4 ⊢ ((𝑌 + 𝑍) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑌 + 𝑍) ∈ 𝑉 ∧ (𝑌 + 𝑍) ≠ 0 ))
59	57, 58	sylibr 233	. . 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 20466	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
63	62	simpld 495	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
64	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp1 39939	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
65	6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22	mapdindp2 39940	. . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))
66	6, 7, 8, 15, 18, 56, 17, 64, 65	lspindp1 20467	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑤, (𝑌 + 𝑍)})))
67	66	simprd 496	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, (𝑌 + 𝑍)}))
68	67	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝑤, (𝑌 + 𝑍)}))
69	23	simprd 496	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))
70	6, 7, 8, 15, 16, 21, 69	lspsnne1 20451	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌}))
71		eqid 2737	. . . . . . . . . 10 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
72	6, 8, 71, 38, 21, 54	lsmpr 20423	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})))
73	60	oveq2d 7331	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑌})) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})))
74		eqid 2737	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
75	6, 74, 8	lspsncl 20311	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
76	38, 21, 75	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
77	74	lsssubg 20291	. . . . . . . . . . 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 19337	. . . . . . . . . 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 2783	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌}))
82	70, 81	neleqtrrd 2860	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑍}))
83	6, 39, 8, 38, 21, 54, 17, 82	lspindp4 20471	. . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑌, (𝑌 + 𝑍)}))
84	6, 8, 15, 17, 21, 56, 83	lspindpi 20466	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)})))
85	84	simprd 496	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
86	85	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑤}) ≠ (𝑁‘{(𝑌 + 𝑍)}))
87		eqidd 2738	. . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
88		eqidd 2738	. . 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 40028	. 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 3030	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 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2941 ∖ cdif 3894 {csn 4571 {cpr 4573 ⟨cotp 4579 ‘cfv 6465 (class class class)co 7315 Basecbs 16982 +_gcplusg 17032 0_gc0g 17220 -_gcsg 18648 SubGrpcsubg 18818 LSSumclsm 19308 LModclmod 20195 LSubSpclss 20265 LSpanclspn 20305 HLchlt 37568 LHypclh 38203 DVecHcdvh 39297 LCDualclcd 39805 mapdcmpd 39843 HDMap1chdma1 40010

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5224 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7628 ax-cnex 11000 ax-resscn 11001 ax-1cn 11002 ax-icn 11003 ax-addcl 11004 ax-addrcl 11005 ax-mulcl 11006 ax-mulrcl 11007 ax-mulcom 11008 ax-addass 11009 ax-mulass 11010 ax-distr 11011 ax-i2m1 11012 ax-1ne0 11013 ax-1rid 11014 ax-rnegex 11015 ax-rrecex 11016 ax-cnre 11017 ax-pre-lttri 11018 ax-pre-lttrn 11019 ax-pre-ltadd 11020 ax-pre-mulgt0 11021 ax-riotaBAD 37171

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-ot 4580 df-uni 4851 df-int 4893 df-iun 4939 df-iin 4940 df-br 5088 df-opab 5150 df-mpt 5171 df-tr 5205 df-id 5507 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5562 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-riota 7272 df-ov 7318 df-oprab 7319 df-mpo 7320 df-of 7573 df-om 7758 df-1st 7876 df-2nd 7877 df-tpos 8089 df-undef 8136 df-frecs 8144 df-wrecs 8175 df-recs 8249 df-rdg 8288 df-1o 8344 df-er 8546 df-map 8665 df-en 8782 df-dom 8783 df-sdom 8784 df-fin 8785 df-pnf 11084 df-mnf 11085 df-xr 11086 df-ltxr 11087 df-le 11088 df-sub 11280 df-neg 11281 df-nn 12047 df-2 12109 df-3 12110 df-4 12111 df-5 12112 df-6 12113 df-n0 12307 df-z 12393 df-uz 12656 df-fz 13313 df-struct 16918 df-sets 16935 df-slot 16953 df-ndx 16965 df-base 16983 df-ress 17012 df-plusg 17045 df-mulr 17046 df-sca 17048 df-vsca 17049 df-0g 17222 df-mre 17365 df-mrc 17366 df-acs 17368 df-proset 18083 df-poset 18101 df-plt 18118 df-lub 18134 df-glb 18135 df-join 18136 df-meet 18137 df-p0 18213 df-p1 18214 df-lat 18220 df-clat 18287 df-mgm 18396 df-sgrp 18445 df-mnd 18456 df-submnd 18501 df-grp 18649 df-minusg 18650 df-sbg 18651 df-subg 18821 df-cntz 18992 df-oppg 19019 df-lsm 19310 df-cmn 19456 df-abl 19457 df-mgp 19789 df-ur 19806 df-ring 19853 df-oppr 19930 df-dvdsr 19951 df-unit 19952 df-invr 19982 df-dvr 19993 df-drng 20065 df-lmod 20197 df-lss 20266 df-lsp 20306 df-lvec 20437 df-lsatoms 37194 df-lshyp 37195 df-lcv 37237 df-lfl 37276 df-lkr 37304 df-ldual 37342 df-oposet 37394 df-ol 37396 df-oml 37397 df-covers 37484 df-ats 37485 df-atl 37516 df-cvlat 37540 df-hlat 37569 df-llines 37717 df-lplanes 37718 df-lvols 37719 df-lines 37720 df-psubsp 37722 df-pmap 37723 df-padd 38015 df-lhyp 38207 df-laut 38208 df-ldil 38323 df-ltrn 38324 df-trl 38378 df-tgrp 38962 df-tendo 38974 df-edring 38976 df-dveca 39222 df-disoa 39248 df-dvech 39298 df-dib 39358 df-dic 39392 df-dih 39448 df-doch 39567 df-djh 39614 df-lcdual 39806 df-mapd 39844 df-hdmap1 40012

This theorem is referenced by: hdmap1l6g 40035

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