dvhvaddass - Mathbox for Norm Megill

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Theorem dvhvaddass 39111

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 6169	. . . 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 39106	. . . . . . . 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 1170	. . . . . . 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 6778	. . . . . 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 6787	. . . . . . . 8 ⊢ (1^st ‘𝐹) ∈ V
13		fvex 6787	. . . . . . . 8 ⊢ (1^st ‘𝐺) ∈ V
14	12, 13	coex 7777	. . . . . . 7 ⊢ ((1^st ‘𝐹) ∘ (1^st ‘𝐺)) ∈ V
15		ovex 7308	. . . . . . 7 ⊢ ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) ∈ V
16	14, 15	op1st 7839	. . . . . 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 2794	. . . . 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 5770	. . . 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 39106	. . . . . . . 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 1168	. . . . . . 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 6778	. . . . . 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 6787	. . . . . . . 8 ⊢ (1^st ‘𝐼) ∈ V
23	13, 22	coex 7777	. . . . . . 7 ⊢ ((1^st ‘𝐺) ∘ (1^st ‘𝐼)) ∈ V
24		ovex 7308	. . . . . . 7 ⊢ ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼)) ∈ V
25	23, 24	op1st 7839	. . . . . 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 2794	. . . . 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 5771	. . . 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 2804	. . 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 7864	. . . . . 6 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐹) ∈ 𝐸)
30		xp2nd 7864	. . . . . 6 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐺) ∈ 𝐸)
31		xp2nd 7864	. . . . . 6 ⊢ (𝐼 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐼) ∈ 𝐸)
32	29, 30, 31	3anim123i 1150	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸)) → ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸))
33		eqid 2738	. . . . . . . . . 10 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
34	2, 33, 5, 6	dvhsca 39096	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
35	2, 33	erngdv 39007	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
36	34, 35	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
37		drnggrp 19999	. . . . . . . 8 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
38	36, 37	syl 17	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
39	38	adantr 481	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → 𝐷 ∈ Grp)
40		simpr1 1193	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐹) ∈ 𝐸)
41		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
42	2, 4, 5, 6, 41	dvhbase 39097	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
43	42	adantr 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (Base‘𝐷) = 𝐸)
44	40, 43	eleqtrrd 2842	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐹) ∈ (Base‘𝐷))
45		simpr2 1194	. . . . . . 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 2842	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐺) ∈ (Base‘𝐷))
47		simpr3 1195	. . . . . . 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 2842	. . . . . 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 18586	. . . . . 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 1371	. . . . 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 593	. . . 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 6778	. . . . . 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 7840	. . . . . 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 2794	. . . . 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 7290	. . . 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 6778	. . . . . 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 7840	. . . . . 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 2794	. . . . 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 7291	. . . 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 2788	. . 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 4812	. 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 483	. . 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 39109	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64	63	3adantr3 1170	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65		simpr3 1195	. . 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 39106	. . 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 834	. 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 1193	. . 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 39109	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70	69	3adantr1 1168	. . 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 39106	. . 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 834	. 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 2788	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

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ⟨cop 4567 × cxp 5587 ∘ ccom 5593 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 Basecbs 16912 +_gcplusg 16962 Scalarcsca 16965 Grpcgrp 18577 DivRingcdr 19991 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 TEndoctendo 38766 EDRingcedring 38767 DVecHcdvh 39092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tendo 38769 df-edring 38771 df-dvech 39093

This theorem is referenced by: dvhgrp 39121

W3C validator