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Theorem mavmulass 22469

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 2725	. . . 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 22341	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐵 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
10	6, 5, 9	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → (𝐵 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
11	8, 10	eleqtrrd 2828	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
12		mavmulass.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴))
13	12, 10	eleqtrrd 2828	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
14	3, 5, 7, 6, 6, 6, 11, 13	mamucl 22319	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑍) ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
15	14, 10	eleqtrd 2827	. . . 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 22467	. . 3 ⊢ (𝜑 → ((𝑋 × 𝑍) · 𝑌) ∈ (𝐵 ↑_m 𝑁))
18		elmapi 8866	. . 3 ⊢ (((𝑋 × 𝑍) · 𝑌) ∈ (𝐵 ↑_m 𝑁) → ((𝑋 × 𝑍) · 𝑌):𝑁⟶𝐵)
19		ffn 6717	. . 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 22467	. . . 4 ⊢ (𝜑 → (𝑍 · 𝑌) ∈ (𝐵 ↑_m 𝑁))
22	1, 2, 3, 4, 5, 6, 8, 21	mavmulcl 22467	. . 3 ⊢ (𝜑 → (𝑋 · (𝑍 · 𝑌)) ∈ (𝐵 ↑_m 𝑁))
23		elmapi 8866	. . 3 ⊢ ((𝑋 · (𝑍 · 𝑌)) ∈ (𝐵 ↑_m 𝑁) → (𝑋 · (𝑍 · 𝑌)):𝑁⟶𝐵)
24		ffn 6717	. . 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 20224	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
27	26	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CMnd)
28	6	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin)
29	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → 𝑅 ∈ Ring)
30		elmapi 8866	. . . . . . . . 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 7590	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → (𝑖𝑋𝑘) ∈ 𝐵)
36		elmapi 8866	. . . . . . . . . 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 7590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → (𝑘𝑍𝑗) ∈ 𝐵)
41		elmapi 8866	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑_m 𝑁) → 𝑌:𝑁⟶𝐵)
42		ffvelcdm 7086	. . . . . . . . . . 11 ⊢ ((𝑌:𝑁⟶𝐵 ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
43	42	ex 411	. . . . . . . . . 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 405	. . . . . . . 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 20203	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)) ∈ 𝐵)
48	3, 4, 29, 35, 47	ringcld 20203	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ (𝑗 ∈ 𝑁 ∧ 𝑘 ∈ 𝑁)) → ((𝑖𝑋𝑘)(._r‘𝑅)((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))) ∈ 𝐵)
49	3, 27, 28, 28, 48	gsumcom3fi 19938	. . . 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 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
56	7, 3, 4, 50, 51, 51, 51, 52, 53, 54, 55	mamufv 22312	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑋 × 𝑍)𝑗) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗)))))
57	56	oveq1d 7431	. . . . . . 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 2725	. . . . . . . 8 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
59	45	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
60	5	adantr 479	. . . . . . . . . 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 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
65	62, 63, 64	fovcdmd 7590	. . . . . . . . . 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 479	. . . . . . . . . . 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 483	. . . . . . . . . 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 7590	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑍𝑗) ∈ 𝐵)
72	3, 4, 61, 66, 71	ringcld 20203	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗)) ∈ 𝐵)
73		eqid 2725	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗))) = (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗)))
74		ovexd 7451	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → ((𝑖𝑋𝑘)(._r‘𝑅)(𝑘𝑍𝑗)) ∈ V)
75		fvexd 6907	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (0_g‘𝑅) ∈ V)
76	73, 51, 74, 75	fsuppmptdm 9399	. . . . . . . 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 20256	. . . . . . 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 20197	. . . . . . . . . . 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 1369	. . . . . . . . . 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 466	. . . . . . . . 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 5243	. . . . . . . 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 7432	. . . . . . 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 2771	. . . . . 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 5243	. . . . 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 7432	. . . 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 22466	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → ((𝑍 · 𝑌)‘𝑘) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)))))
91	90	oveq2d 7432	. . . . . . 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 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
96	93, 94, 95	fovcdmd 7590	. . . . . . . . 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 405	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
99	3, 4, 92, 96, 98	ringcld 20203	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)) ∈ 𝐵)
100		eqid 2725	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑁 ↦ ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)))
101		ovexd 7451	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑘𝑍𝑗)(._r‘𝑅)(𝑌‘𝑗)) ∈ V)
102		fvexd 6907	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑘 ∈ 𝑁) → (0_g‘𝑅) ∈ V)
103	100, 87, 101, 102	fsuppmptdm 9399	. . . . . . . 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 20257	. . . . . . 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 2768	. . . . . 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 5243	. . . . 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 7432	. . . 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 2775	. . 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 479	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑋 × 𝑍) ∈ (Base‘𝐴))
110	16	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑌 ∈ (𝐵 ↑_m 𝑁))
111		simpr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁)
112	1, 2, 3, 4, 60, 28, 109, 110, 111	mavmulfv 22466	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (((𝑋 × 𝑍) · 𝑌)‘𝑖) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖(𝑋 × 𝑍)𝑗)(._r‘𝑅)(𝑌‘𝑗)))))
113	8	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴))
114	21	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑍 · 𝑌) ∈ (𝐵 ↑_m 𝑁))
115	1, 2, 3, 4, 60, 28, 113, 114, 111	mavmulfv 22466	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → ((𝑋 · (𝑍 · 𝑌))‘𝑖) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖𝑋𝑘)(._r‘𝑅)((𝑍 · 𝑌)‘𝑘)))))
116	108, 112, 115	3eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (((𝑋 × 𝑍) · 𝑌)‘𝑖) = ((𝑋 · (𝑍 · 𝑌))‘𝑖))
117	20, 25, 116	eqfnfvd 7038	1 ⊢ (𝜑 → ((𝑋 × 𝑍) · 𝑌) = (𝑋 · (𝑍 · 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⟨cop 4630 ⟨cotp 4632 ↦ cmpt 5226 × cxp 5670 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 ↑_m cmap 8843 Fincfn 8962 Basecbs 17179 ._rcmulr 17233 0_gc0g 17420 Σ_g cgsu 17421 CMndccmn 19739 Ringcrg 20177 maMul cmmul 22308 Mat cmat 22325 maVecMul cmvmul 22460

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-mulg 19028 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-ur 20126 df-ring 20179 df-sra 21062 df-rgmod 21063 df-dsmm 21670 df-frlm 21685 df-mamu 22309 df-mat 22326 df-mvmul 22461

This theorem is referenced by: slesolinv 22600 slesolinvbi 22601

W3C validator