dvhvaddass - Mathbox for Norm Megill

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Theorem dvhvaddass 41099

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 6285	. . . 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 41094	. . . . . . . 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 1172	. . . . . . 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 6910	. . . . . 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 6919	. . . . . . . 8 ⊢ (1^st ‘𝐹) ∈ V
13		fvex 6919	. . . . . . . 8 ⊢ (1^st ‘𝐺) ∈ V
14	12, 13	coex 7952	. . . . . . 7 ⊢ ((1^st ‘𝐹) ∘ (1^st ‘𝐺)) ∈ V
15		ovex 7464	. . . . . . 7 ⊢ ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) ∈ V
16	14, 15	op1st 8022	. . . . . 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 2793	. . . . 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 5872	. . . 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 41094	. . . . . . . 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 1170	. . . . . . 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 6910	. . . . . 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 6919	. . . . . . . 8 ⊢ (1^st ‘𝐼) ∈ V
23	13, 22	coex 7952	. . . . . . 7 ⊢ ((1^st ‘𝐺) ∘ (1^st ‘𝐼)) ∈ V
24		ovex 7464	. . . . . . 7 ⊢ ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐼)) ∈ V
25	23, 24	op1st 8022	. . . . . 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 2793	. . . . 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 5873	. . . 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 2803	. . 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 8047	. . . . . 6 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐹) ∈ 𝐸)
30		xp2nd 8047	. . . . . 6 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐺) ∈ 𝐸)
31		xp2nd 8047	. . . . . 6 ⊢ (𝐼 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐼) ∈ 𝐸)
32	29, 30, 31	3anim123i 1152	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸)) → ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸))
33		eqid 2737	. . . . . . . . . 10 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
34	2, 33, 5, 6	dvhsca 41084	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
35	2, 33	erngdv 40995	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing)
36	34, 35	eqeltrd 2841	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ DivRing)
37		drnggrp 20739	. . . . . . . 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 1195	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐹) ∈ 𝐸)
41		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
42	2, 4, 5, 6, 41	dvhbase 41085	. . . . . . . 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 2844	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐹) ∈ (Base‘𝐷))
45		simpr2 1196	. . . . . . 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 2844	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸 ∧ (2^nd ‘𝐼) ∈ 𝐸)) → (2^nd ‘𝐺) ∈ (Base‘𝐷))
47		simpr3 1197	. . . . . . 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 2844	. . . . . 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 18960	. . . . . 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 1374	. . . . 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 6910	. . . . . 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 8023	. . . . . 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 2793	. . . . 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 7446	. . . 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 6910	. . . . . 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 8023	. . . . . 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 2793	. . . . 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 7447	. . . 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 2787	. . 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 4881	. 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 41097	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
64	63	3adantr3 1172	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
65		simpr3 1197	. . 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 41094	. . 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 837	. 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 1195	. . 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 41097	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐼) ∈ (𝑇 × 𝐸))
70	69	3adantr1 1170	. . 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 41094	. . 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 837	. 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 2787	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 1087 = wceq 1540 ∈ wcel 2108 ⟨cop 4632 × cxp 5683 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 Basecbs 17247 +_gcplusg 17297 Scalarcsca 17300 Grpcgrp 18951 DivRingcdr 20729 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 TEndoctendo 40754 EDRingcedring 40755 DVecHcdvh 41080

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tendo 40757 df-edring 40759 df-dvech 41081

This theorem is referenced by: dvhgrp 41109

W3C validator