dvhvaddass - Mathbox for Norm Megill

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Theorem dvhvaddass 39038

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 6158	. . . 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 39033	. . . . . . . 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 1169	. . . . . . 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 6760	. . . . . 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 6769	. . . . . . . 8 ⊢ (1^st ‘𝐹) ∈ V
13		fvex 6769	. . . . . . . 8 ⊢ (1^st ‘𝐺) ∈ V
14	12, 13	coex 7751	. . . . . . 7 ⊢ ((1^st ‘𝐹) ∘ (1^st ‘𝐺)) ∈ V
15		ovex 7288	. . . . . . 7 ⊢ ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) ∈ V
16	14, 15	op1st 7812	. . . . . 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 2795	. . . . 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 5759	. . . 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 39033	. . . . . . . 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 1167	. . . . . . 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 6760	. . . . . 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 6769	. . . . . . . 8 ⊢ (1^st ‘𝐼) ∈ V
23	13, 22	coex 7751	. . . . . . 7 ⊢ ((1^st ‘𝐺) ∘ (1^st ‘𝐼)) ∈ V
24		ovex 7288	. . . . . . 7 ⊢ ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼)) ∈ V
25	23, 24	op1st 7812	. . . . . 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 2795	. . . . 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 5760	. . . 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 2805	. . 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 7837	. . . . . 6 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐹) ∈ 𝐸)
30		xp2nd 7837	. . . . . 6 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐺) ∈ 𝐸)
31		xp2nd 7837	. . . . . 6 ⊢ (𝐼 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐼) ∈ 𝐸)
32	29, 30, 31	3anim123i 1149	. . . . 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 39023	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
35	2, 33	erngdv 38934	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
36	34, 35	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
37		drnggrp 19914	. . . . . . . 8 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
38	36, 37	syl 17	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
39	38	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → 𝐷 ∈ Grp)
40		simpr1 1192	. . . . . . 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 39024	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
43	42	adantr 480	. . . . . . 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 1193	. . . . . . 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 1194	. . . . . . 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 18501	. . . . . 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 1370	. . . . 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 592	. . . 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 6760	. . . . . 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 7813	. . . . . 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 2795	. . . . 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 7270	. . . 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 6760	. . . . . 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 7813	. . . . . 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 2795	. . . . 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 7271	. . . 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 4809	. 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 482	. . 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 39036	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64	63	3adantr3 1169	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65		simpr3 1194	. . 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 39033	. . 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 833	. 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 1192	. . 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 39036	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70	69	3adantr1 1167	. . 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 39033	. . 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 833	. 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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ⟨cop 4564 × cxp 5578 ∘ ccom 5584 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 Basecbs 16840 +_gcplusg 16888 Scalarcsca 16891 Grpcgrp 18492 DivRingcdr 19906 HLchlt 37291 LHypclh 37925 LTrncltrn 38042 TEndoctendo 38693 EDRingcedring 38694 DVecHcdvh 39019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tendo 38696 df-edring 38698 df-dvech 39020

This theorem is referenced by: dvhgrp 39048

W3C validator