dvhvaddass - Mathbox for Norm Megill

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Theorem dvhvaddass 42074

Description: Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)

**Hypotheses**
Ref	Expression
dvhvaddcl.h	⊢ 𝐻 = (LHyp‘𝐾)
dvhvaddcl.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhvaddcl.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhvaddcl.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhvaddcl.d	⊢ 𝐷 = (Scalar‘𝑈)
dvhvaddcl.p	⊢ ⨣ = (+_g‘𝐷)
dvhvaddcl.a	⊢ + = (+_g‘𝑈)

**Assertion**
Ref	Expression
dvhvaddass	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼)))

**Proof of Theorem dvhvaddass**
Step	Hyp	Ref	Expression
1		coass 6256	. . . 4 ⊢ (((1^st ‘𝐹) ∘ (1^st ‘𝐺)) ∘ (1^st ‘𝐼)) = ((1^st ‘𝐹) ∘ ((1^st ‘𝐺) ∘ (1^st ‘𝐼)))
2		dvhvaddcl.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
3		dvhvaddcl.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		dvhvaddcl.e	. . . . . . . . 9 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
5		dvhvaddcl.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		dvhvaddcl.d	. . . . . . . . 9 ⊢ 𝐷 = (Scalar‘𝑈)
7		dvhvaddcl.a	. . . . . . . . 9 ⊢ + = (+_g‘𝑈)
8		dvhvaddcl.p	. . . . . . . . 9 ⊢ ⨣ = (+_g‘𝐷)
9	2, 3, 4, 5, 6, 7, 8	dvhvadd 42069	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩)
10	9	3adantr3 1190	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩)
11	10	fveq2d 6877	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1^st ‘(𝐹 + 𝐺)) = (1^st ‘⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩))
12		fvex 6886	. . . . . . . 8 ⊢ (1^st ‘𝐹) ∈ V
13		fvex 6886	. . . . . . . 8 ⊢ (1^st ‘𝐺) ∈ V
14	12, 13	coex 7925	. . . . . . 7 ⊢ ((1^st ‘𝐹) ∘ (1^st ‘𝐺)) ∈ V
15		ovex 7441	. . . . . . 7 ⊢ ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) ∈ V
16	14, 15	op1st 7992	. . . . . 6 ⊢ (1^st ‘⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩) = ((1^st ‘𝐹) ∘ (1^st ‘𝐺))
17	11, 16	eqtrdi 2811	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1^st ‘(𝐹 + 𝐺)) = ((1^st ‘𝐹) ∘ (1^st ‘𝐺)))
18	17	coeq1d 5835	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1^st ‘(𝐹 + 𝐺)) ∘ (1^st ‘𝐼)) = (((1^st ‘𝐹) ∘ (1^st ‘𝐺)) ∘ (1^st ‘𝐼)))
19	2, 3, 4, 5, 6, 7, 8	dvhvadd 42069	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐼)), ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))⟩)
20	19	3adantr1 1188	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐼)), ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))⟩)
21	20	fveq2d 6877	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1^st ‘(𝐺 + 𝐼)) = (1^st ‘⟨((1^st ‘𝐺) ∘ (1^st ‘𝐼)), ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))⟩))
22		fvex 6886	. . . . . . . 8 ⊢ (1^st ‘𝐼) ∈ V
23	13, 22	coex 7925	. . . . . . 7 ⊢ ((1^st ‘𝐺) ∘ (1^st ‘𝐼)) ∈ V
24		ovex 7441	. . . . . . 7 ⊢ ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼)) ∈ V
25	23, 24	op1st 7992	. . . . . 6 ⊢ (1^st ‘⟨((1^st ‘𝐺) ∘ (1^st ‘𝐼)), ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))⟩) = ((1^st ‘𝐺) ∘ (1^st ‘𝐼))
26	21, 25	eqtrdi 2811	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (1^st ‘(𝐺 + 𝐼)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐼)))
27	26	coeq2d 5836	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1^st ‘𝐹) ∘ (1^st ‘(𝐺 + 𝐼))) = ((1^st ‘𝐹) ∘ ((1^st ‘𝐺) ∘ (1^st ‘𝐼))))
28	1, 18, 27	3eqtr4a 2821	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((1^st ‘(𝐹 + 𝐺)) ∘ (1^st ‘𝐼)) = ((1^st ‘𝐹) ∘ (1^st ‘(𝐺 + 𝐼))))
29		xp2nd 8017	. . . . . 6 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐹) ∈ 𝐸)
30		xp2nd 8017	. . . . . 6 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐺) ∈ 𝐸)
31		xp2nd 8017	. . . . . 6 ⊢ (𝐼 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐼) ∈ 𝐸)
32	29, 30, 31	3anim123i 1169	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸)) → ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸))
33		eqid 2760	. . . . . . . . . 10 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
34	2, 33, 5, 6	dvhsca 42059	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
35	2, 33	erngdv 41970	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
36	34, 35	eqeltrd 2860	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
37		drnggrp 20951	. . . . . . . 8 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
38	36, 37	syl 18	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
39	38	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → 𝐷 ∈ Grp)
40		simpr1 1213	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐹) ∈ 𝐸)
41		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
42	2, 4, 5, 6, 41	dvhbase 42060	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
43	42	adantr 486	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (Base‘𝐷) = 𝐸)
44	40, 43	eleqtrrd 2863	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐹) ∈ (Base‘𝐷))
45		simpr2 1214	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐺) ∈ 𝐸)
46	45, 43	eleqtrrd 2863	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐺) ∈ (Base‘𝐷))
47		simpr3 1215	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐼) ∈ 𝐸)
48	47, 43	eleqtrrd 2863	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐼) ∈ (Base‘𝐷))
49	41, 8	grpass 19114	. . . . . 6 ⊢ ((𝐷 ∈ Grp ∧ ((2^nd ‘𝐹) ∈ (Base‘𝐷) ∧ (2^nd ‘𝐺) ∈ (Base‘𝐷) ∧ (2^nd ‘𝐼) ∈ (Base‘𝐷))) → (((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) ⨣ (2^nd ‘𝐼)) = ((2^nd ‘𝐹) ⨣ ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))))
50	39, 44, 46, 48, 49	syl13anc 1399	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) ⨣ (2^nd ‘𝐼)) = ((2^nd ‘𝐹) ⨣ ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))))
51	32, 50	sylan2 605	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) ⨣ (2^nd ‘𝐼)) = ((2^nd ‘𝐹) ⨣ ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))))
52	10	fveq2d 6877	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2^nd ‘(𝐹 + 𝐺)) = (2^nd ‘⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩))
53	14, 15	op2nd 7993	. . . . . 6 ⊢ (2^nd ‘⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))
54	52, 53	eqtrdi 2811	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2^nd ‘(𝐹 + 𝐺)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)))
55	54	oveq1d 7423	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2^nd ‘(𝐹 + 𝐺)) ⨣ (2^nd ‘𝐼)) = (((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) ⨣ (2^nd ‘𝐼)))
56	20	fveq2d 6877	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2^nd ‘(𝐺 + 𝐼)) = (2^nd ‘⟨((1^st ‘𝐺) ∘ (1^st ‘𝐼)), ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))⟩))
57	23, 24	op2nd 7993	. . . . . 6 ⊢ (2^nd ‘⟨((1^st ‘𝐺) ∘ (1^st ‘𝐼)), ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))⟩) = ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))
58	56, 57	eqtrdi 2811	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (2^nd ‘(𝐺 + 𝐼)) = ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼)))
59	58	oveq2d 7424	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2^nd ‘𝐹) ⨣ (2^nd ‘(𝐺 + 𝐼))) = ((2^nd ‘𝐹) ⨣ ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼))))
60	51, 55, 59	3eqtr4d 2805	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((2^nd ‘(𝐹 + 𝐺)) ⨣ (2^nd ‘𝐼)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘(𝐺 + 𝐼))))
61	28, 60	opeq12d 4840	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ⟨((1^st ‘(𝐹 + 𝐺)) ∘ (1^st ‘𝐼)), ((2^nd ‘(𝐹 + 𝐺)) ⨣ (2^nd ‘𝐼))⟩ = ⟨((1^st ‘𝐹) ∘ (1^st ‘(𝐺 + 𝐼))), ((2^nd ‘𝐹) ⨣ (2^nd ‘(𝐺 + 𝐼)))⟩)
62		simpl 488	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
63	2, 3, 4, 5, 6, 8, 7	dvhvaddcl 42072	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64	63	3adantr3 1190	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65		simpr3 1215	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐼 ∈ (𝑇 × 𝐸))
66	2, 3, 4, 5, 6, 7, 8	dvhvadd 42069	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹 + 𝐺) ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = ⟨((1^st ‘(𝐹 + 𝐺)) ∘ (1^st ‘𝐼)), ((2^nd ‘(𝐹 + 𝐺)) ⨣ (2^nd ‘𝐼))⟩)
67	62, 64, 65, 66	syl12anc 850	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = ⟨((1^st ‘(𝐹 + 𝐺)) ∘ (1^st ‘𝐼)), ((2^nd ‘(𝐹 + 𝐺)) ⨣ (2^nd ‘𝐼))⟩)
68		simpr1 1213	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → 𝐹 ∈ (𝑇 × 𝐸))
69	2, 3, 4, 5, 6, 8, 7	dvhvaddcl 42072	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70	69	3adantr1 1188	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
71	2, 3, 4, 5, 6, 7, 8	dvhvadd 42069	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))) → (𝐹 + (𝐺 + 𝐼)) = ⟨((1^st ‘𝐹) ∘ (1^st ‘(𝐺 + 𝐼))), ((2^nd ‘𝐹) ⨣ (2^nd ‘(𝐺 + 𝐼)))⟩)
72	62, 68, 70, 71	syl12anc 850	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + (𝐺 + 𝐼)) = ⟨((1^st ‘𝐹) ∘ (1^st ‘(𝐺 + 𝐼))), ((2^nd ‘𝐹) ⨣ (2^nd ‘(𝐺 + 𝐼)))⟩)
73	61, 67, 72	3eqtr4d 2805	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ⟨cop 4589 × cxp 5645 ∘ ccom 5651 ‘cfv 6527 (class class class)co 7408 1^st c1st 7982 2^nd c2nd 7983 Basecbs 17348 +_gcplusg 17389 Scalarcsca 17392 Grpcgrp 19105 DivRingcdr 20941 HLchlt 40327 LHypclh 40961 LTrncltrn 41078 TEndoctendo 41729 EDRingcedring 41730 DVecHcdvh 42055

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248 ax-riotaBAD 39930

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-tpos 8221 df-undef 8268 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-er 8695 df-map 8827 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-nn 12305 df-2 12374 df-3 12375 df-4 12376 df-5 12377 df-6 12378 df-n0 12576 df-z 12663 df-uz 12935 df-fz 13609 df-struct 17286 df-sets 17303 df-slot 17321 df-ndx 17333 df-base 17349 df-ress 17370 df-plusg 17402 df-mulr 17403 df-sca 17405 df-vsca 17406 df-0g 17573 df-proset 18429 df-poset 18448 df-plt 18463 df-lub 18479 df-glb 18480 df-join 18481 df-meet 18482 df-p0 18558 df-p1 18559 df-lat 18567 df-clat 18634 df-mgm 18777 df-sgrp 18869 df-mnd 18885 df-grp 19108 df-minusg 19109 df-cmn 19957 df-abl 19958 df-mgp 20322 df-rng 20336 df-ur 20369 df-ring 20422 df-oppr 20528 df-dvdsr 20548 df-unit 20549 df-invr 20579 df-dvr 20592 df-drng 20943 df-oposet 40153 df-ol 40155 df-oml 40156 df-covers 40243 df-ats 40244 df-atl 40275 df-cvlat 40299 df-hlat 40328 df-llines 40475 df-lplanes 40476 df-lvols 40477 df-lines 40478 df-psubsp 40480 df-pmap 40481 df-padd 40773 df-lhyp 40965 df-laut 40966 df-ldil 41081 df-ltrn 41082 df-trl 41136 df-tendo 41732 df-edring 41734 df-dvech 42056

This theorem is used by: dvhgrp 42084

W3C validator