dvhvaddass - Mathbox for Norm Megill

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Theorem dvhvaddass 38393

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 6085	. . . 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 38388	. . . . . . . 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 1168	. . . . . . 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 6649	. . . . . 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 6658	. . . . . . . 8 ⊢ (1^st ‘𝐹) ∈ V
13		fvex 6658	. . . . . . . 8 ⊢ (1^st ‘𝐺) ∈ V
14	12, 13	coex 7617	. . . . . . 7 ⊢ ((1^st ‘𝐹) ∘ (1^st ‘𝐺)) ∈ V
15		ovex 7168	. . . . . . 7 ⊢ ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) ∈ V
16	14, 15	op1st 7679	. . . . . 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 2849	. . . . 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 5696	. . . 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 38388	. . . . . . . 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 1166	. . . . . . 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 6649	. . . . . 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 6658	. . . . . . . 8 ⊢ (1^st ‘𝐼) ∈ V
23	13, 22	coex 7617	. . . . . . 7 ⊢ ((1^st ‘𝐺) ∘ (1^st ‘𝐼)) ∈ V
24		ovex 7168	. . . . . . 7 ⊢ ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼)) ∈ V
25	23, 24	op1st 7679	. . . . . 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 2849	. . . . 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 5697	. . . 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 2859	. . 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 7704	. . . . . 6 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐹) ∈ 𝐸)
30		xp2nd 7704	. . . . . 6 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐺) ∈ 𝐸)
31		xp2nd 7704	. . . . . 6 ⊢ (𝐼 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐼) ∈ 𝐸)
32	29, 30, 31	3anim123i 1148	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸)) → ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸))
33		eqid 2798	. . . . . . . . . 10 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
34	2, 33, 5, 6	dvhsca 38378	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
35	2, 33	erngdv 38289	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
36	34, 35	eqeltrd 2890	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
37		drnggrp 19503	. . . . . . . 8 ⊢ (𝐷 ∈ DivRing → 𝐷 ∈ Grp)
38	36, 37	syl 17	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
39	38	adantr 484	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → 𝐷 ∈ Grp)
40		simpr1 1191	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐹) ∈ 𝐸)
41		eqid 2798	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
42	2, 4, 5, 6, 41	dvhbase 38379	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
43	42	adantr 484	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (Base‘𝐷) = 𝐸)
44	40, 43	eleqtrrd 2893	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐹) ∈ (Base‘𝐷))
45		simpr2 1192	. . . . . . 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 2893	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐺) ∈ (Base‘𝐷))
47		simpr3 1193	. . . . . . 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 2893	. . . . . 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 18104	. . . . . 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 1369	. . . . 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 595	. . . 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 6649	. . . . . 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 7680	. . . . . 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 2849	. . . . 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 7150	. . . 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 6649	. . . . . 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 7680	. . . . . 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 2849	. . . . 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 7151	. . . 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 2843	. . 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 4773	. 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 486	. . 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 38391	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64	63	3adantr3 1168	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65		simpr3 1193	. . 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 38388	. . 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 835	. 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 1191	. . 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 38391	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70	69	3adantr1 1166	. . 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 38388	. . 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 835	. 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 2843	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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⟨cop 4531 × cxp 5517 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 1^st c1st 7669 2^nd c2nd 7670 Basecbs 16475 +_gcplusg 16557 Scalarcsca 16560 Grpcgrp 18095 DivRingcdr 19495 HLchlt 36646 LHypclh 37280 LTrncltrn 37397 TEndoctendo 38048 EDRingcedring 38049 DVecHcdvh 38374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tendo 38051 df-edring 38053 df-dvech 38375

This theorem is referenced by: dvhgrp 38403

W3C validator