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Theorem mavmulass 22042

Description: Associativity of the multiplication of two NxN matrices with an N-dimensional vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 25-Feb-2019.) (Proof shortened by AV, 22-Jul-2019.)

**Hypotheses**
Ref	Expression
1mavmul.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
1mavmul.b	⊢ 𝐵 = (Base‘𝑅)
1mavmul.t	⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
1mavmul.r	⊢ (𝜑 → 𝑅 ∈ Ring)
1mavmul.n	⊢ (𝜑 → 𝑁 ∈ Fin)
1mavmul.y	⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑_m 𝑁))
mavmulass.m	⊢ × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
mavmulass.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴))
mavmulass.z	⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))

**Assertion**
Ref	Expression
mavmulass	⊢ (𝜑 → ((𝑋 × 𝑍) · 𝑌) = (𝑋 · (𝑍 · 𝑌)))

Proof of Theorem mavmulass Dummy variables 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1mavmul.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		1mavmul.t	. . . 4 ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
3		1mavmul.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
4		eqid 2732	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
5		1mavmul.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
6		1mavmul.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Fin)
7		mavmulass.m	. . . . . 6 ⊢ × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
8		mavmulass.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴))
9	1, 3	matbas2 21914	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐵 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
10	6, 5, 9	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐵 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
11	8, 10	eleqtrrd 2836	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
12		mavmulass.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))
13	12, 10	eleqtrrd 2836	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
14	3, 5, 7, 6, 6, 6, 11, 13	mamucl 21892	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑍) ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
15	14, 10	eleqtrd 2835	. . . 4 ⊢ (𝜑 → (𝑋 × 𝑍) ∈ (Base‘𝐴))
16		1mavmul.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑_m 𝑁))
17	1, 2, 3, 4, 5, 6, 15, 16	mavmulcl 22040	. . 3 ⊢ (𝜑 → ((𝑋 × 𝑍) · 𝑌) ∈ (𝐵 ↑_m 𝑁))
18		elmapi 8839	. . 3 ⊢ (((𝑋 × 𝑍) · 𝑌) ∈ (𝐵 ↑_m 𝑁) → ((𝑋 × 𝑍) · 𝑌):𝑁⟶𝐵)
19		ffn 6714	. . 3 ⊢ (((𝑋 × 𝑍) · 𝑌):𝑁⟶𝐵 → ((𝑋 × 𝑍) · 𝑌) Fn 𝑁)
20	17, 18, 19	3syl 18	. 2 ⊢ (𝜑 → ((𝑋 × 𝑍) · 𝑌) Fn 𝑁)
21	1, 2, 3, 4, 5, 6, 12, 16	mavmulcl 22040	. . . 4 ⊢ (𝜑 → (𝑍 · 𝑌) ∈ (𝐵 ↑_m 𝑁))
22	1, 2, 3, 4, 5, 6, 8, 21	mavmulcl 22040	. . 3 ⊢ (𝜑 → (𝑋 · (𝑍 · 𝑌)) ∈ (𝐵 ↑_m 𝑁))
23		elmapi 8839	. . 3 ⊢ ((𝑋 · (𝑍 · 𝑌)) ∈ (𝐵 ↑_m 𝑁) → (𝑋 · (𝑍 · 𝑌)):𝑁⟶𝐵)
24		ffn 6714	. . 3 ⊢ ((𝑋 · (𝑍 · 𝑌)):𝑁⟶𝐵 → (𝑋 · (𝑍 · 𝑌)) Fn 𝑁)
25	22, 23, 24	3syl 18	. 2 ⊢ (𝜑 → (𝑋 · (𝑍 · 𝑌)) Fn 𝑁)
26	5	ringcmnd 20094	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
27	26	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CMnd)
28	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin)
29	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → 𝑅 ∈ Ring)
30		elmapi 8839	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝐵 ↑_m (𝑁 × 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶𝐵)
31	11, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋:(𝑁 × 𝑁)⟶𝐵)
32	31	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶𝐵)
33		simplr 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
34		simprr 771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → 𝑘 ∈ 𝑁)
35	32, 33, 34	fovcdmd 7575	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → (𝑖𝑋𝑘) ∈ 𝐵)
36		elmapi 8839	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝐵 ↑_m (𝑁 × 𝑁)) → 𝑍:(𝑁 × 𝑁)⟶𝐵)
37	13, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍:(𝑁 × 𝑁)⟶𝐵)
38	37	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → 𝑍:(𝑁 × 𝑁)⟶𝐵)
39		simprl 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
40	38, 34, 39	fovcdmd 7575	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → (𝑘𝑍𝑗) ∈ 𝐵)
41		elmapi 8839	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑_m 𝑁) → 𝑌:𝑁⟶𝐵)
42		ffvelcdm 7080	. . . . . . . . . . 11 ⊢ ((𝑌:𝑁⟶𝐵 ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
43	42	ex 413	. . . . . . . . . 10 ⊢ (𝑌:𝑁⟶𝐵 → (𝑗 ∈ 𝑁 → (𝑌‘𝑗) ∈ 𝐵))
44	16, 41, 43	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ 𝑁 → (𝑌‘𝑗) ∈ 𝐵))
45	44	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
46	45	ad2ant2r 745	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → (𝑌‘𝑗) ∈ 𝐵)
47	3, 4, 29, 40, 46	ringcld 20073	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)) ∈ 𝐵)
48	3, 4, 29, 35, 47	ringcld 20073	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))) ∈ 𝐵)
49	3, 27, 28, 28, 48	gsumcom3fi 19841	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))))) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))))))
50	5	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
51	6	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin)
52	11	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
53	13	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑍 ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
54		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
55		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
56	7, 3, 4, 50, 51, 51, 51, 52, 53, 54, 55	mamufv 21880	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑋 × 𝑍)𝑗) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗)))))
57	56	oveq1d 7420	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑖(𝑋 × 𝑍)𝑗)(._r‘𝑅)(𝑌‘𝑗)) = ((𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))))(._r‘𝑅)(𝑌‘𝑗)))
58		eqid 2732	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
59	45	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
60	5	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ Ring)
61	60	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
62	31	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑋:(𝑁 × 𝑁)⟶𝐵)
63		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑖 ∈ 𝑁)
64		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
65	62, 63, 64	fovcdmd 7575	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (𝑖𝑋𝑘) ∈ 𝐵)
66	65	adantlr 713	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (𝑖𝑋𝑘) ∈ 𝐵)
67	37	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑍:(𝑁 × 𝑁)⟶𝐵)
68	67	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑍:(𝑁 × 𝑁)⟶𝐵)
69		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
70		simplr 767	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑗 ∈ 𝑁)
71	68, 69, 70	fovcdmd 7575	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑍𝑗) ∈ 𝐵)
72	3, 4, 61, 66, 71	ringcld 20073	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗)) ∈ 𝐵)
73		eqid 2732	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))) = (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗)))
74		ovexd 7440	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗)) ∈ V)
75		fvexd 6903	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (0_g‘𝑅) ∈ V)
76	73, 51, 74, 75	fsuppmptdm 9370	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))) finSupp (0_g‘𝑅))
77	3, 58, 4, 50, 51, 59, 72, 76	gsummulc1 20121	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))(._r‘𝑅)(𝑌‘𝑗)))) = ((𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))))(._r‘𝑅)(𝑌‘𝑗)))
78	3, 4	ringass 20069	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((𝑖𝑋𝑘) ∈ 𝐵 ∧ (𝑘𝑍𝑗) ∈ 𝐵 ∧ (𝑌‘𝑗) ∈ 𝐵)) → (((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))(._r‘𝑅)(𝑌‘𝑗)) = ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))
79	29, 35, 40, 46, 78	syl13anc 1372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → (((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))(._r‘𝑅)(𝑌‘𝑗)) = ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))
80	79	anassrs 468	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))(._r‘𝑅)(𝑌‘𝑗)) = ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))
81	80	mpteq2dva 5247	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑘 ∈ 𝑁 ↦ (((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))(._r‘𝑅)(𝑌‘𝑗))) = (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)))))
82	81	oveq2d 7421	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))(._r‘𝑅)(𝑌‘𝑗)))) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))))
83	57, 77, 82	3eqtr2d 2778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑖(𝑋 × 𝑍)𝑗)(._r‘𝑅)(𝑌‘𝑗)) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))))
84	83	mpteq2dva 5247	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑗 ∈ 𝑁 ↦ ((𝑖(𝑋 × 𝑍)𝑗)(._r‘𝑅)(𝑌‘𝑗))) = (𝑗 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)))))))
85	84	oveq2d 7421	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖(𝑋 × 𝑍)𝑗)(._r‘𝑅)(𝑌‘𝑗)))) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))))))
86	5	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
87	6	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑁 ∈ Fin)
88	12	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑍 ∈ (Base‘𝐴))
89	16	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑌 ∈ (𝐵 ↑_m 𝑁))
90	1, 2, 3, 4, 86, 87, 88, 89, 64	mavmulfv 22039	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → ((𝑍 · 𝑌)‘𝑘) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)))))
91	90	oveq2d 7421	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → ((𝑖𝑋𝑘)(._r‘𝑅)((𝑍 · 𝑌)‘𝑘)) = ((𝑖𝑋𝑘)(._r‘𝑅)(𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))))
92	60	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
93	67	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑍:(𝑁 × 𝑁)⟶𝐵)
94		simplr 767	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑘 ∈ 𝑁)
95		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
96	93, 94, 95	fovcdmd 7575	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑘𝑍𝑗) ∈ 𝐵)
97	44	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (𝑗 ∈ 𝑁 → (𝑌‘𝑗) ∈ 𝐵))
98	97	imp 407	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
99	3, 4, 92, 96, 98	ringcld 20073	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)) ∈ 𝐵)
100		eqid 2732	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑁 ↦ ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)))
101		ovexd 7440	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)) ∈ V)
102		fvexd 6903	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (0_g‘𝑅) ∈ V)
103	100, 87, 101, 102	fsuppmptdm 9370	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (𝑗 ∈ 𝑁 ↦ ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))) finSupp (0_g‘𝑅))
104	3, 58, 4, 86, 87, 65, 99, 103	gsummulc2 20122	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))) = ((𝑖𝑋𝑘)(._r‘𝑅)(𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))))
105	91, 104	eqtr4d 2775	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → ((𝑖𝑋𝑘)(._r‘𝑅)((𝑍 · 𝑌)‘𝑘)) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))))
106	105	mpteq2dva 5247	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑍 · 𝑌)‘𝑘))) = (𝑘 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)))))))
107	106	oveq2d 7421	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑍 · 𝑌)‘𝑘)))) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))))))))
108	49, 85, 107	3eqtr4d 2782	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖(𝑋 × 𝑍)𝑗)(._r‘𝑅)(𝑌‘𝑗)))) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑍 · 𝑌)‘𝑘)))))
109	15	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑋 × 𝑍) ∈ (Base‘𝐴))
110	16	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑌 ∈ (𝐵 ↑_m 𝑁))
111		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁)
112	1, 2, 3, 4, 60, 28, 109, 110, 111	mavmulfv 22039	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (((𝑋 × 𝑍) · 𝑌)‘𝑖) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖(𝑋 × 𝑍)𝑗)(._r‘𝑅)(𝑌‘𝑗)))))
113	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
114	21	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑍 · 𝑌) ∈ (𝐵 ↑_m 𝑁))
115	1, 2, 3, 4, 60, 28, 113, 114, 111	mavmulfv 22039	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → ((𝑋 · (𝑍 · 𝑌))‘𝑖) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑍 · 𝑌)‘𝑘)))))
116	108, 112, 115	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (((𝑋 × 𝑍) · 𝑌)‘𝑖) = ((𝑋 · (𝑍 · 𝑌))‘𝑖))
117	20, 25, 116	eqfnfvd 7032	1 ⊢ (𝜑 → ((𝑋 × 𝑍) · 𝑌) = (𝑋 · (𝑍 · 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟨cop 4633 ⟨cotp 4635 ↦ cmpt 5230 × cxp 5673 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 Fincfn 8935 Basecbs 17140 ._rcmulr 17194 0_gc0g 17381 Σ_g cgsu 17382 CMndccmn 19642 Ringcrg 20049 maMul cmmul 21876 Mat cmat 21898 maVecMul cmvmul 22033

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-mulg 18945 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-sra 20777 df-rgmod 20778 df-dsmm 21278 df-frlm 21293 df-mamu 21877 df-mat 21899 df-mvmul 22034

This theorem is referenced by: slesolinv 22173 slesolinvbi 22174

W3C validator