Theorem mvmulfval 21145

Description: Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.)

Hypotheses

Ref	Expression
mvmulfval.x	⊢ × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
mvmulfval.b	⊢ 𝐵 = (Base‘𝑅)
mvmulfval.t	⊢ · = (.r‘𝑅)
mvmulfval.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
mvmulfval.m	⊢ (𝜑 → 𝑀 ∈ Fin)
mvmulfval.n	⊢ (𝜑 → 𝑁 ∈ Fin)

Assertion

Ref	Expression
mvmulfval	⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))

Distinct variable groups:   𝑖,𝑗,𝑥,𝑦,𝜑   𝑖,𝑀,𝑗,𝑥,𝑦   𝑖,𝑁,𝑗,𝑥,𝑦   𝑅,𝑖,𝑗,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥, · ,𝑦,𝑖

Allowed substitution hints:   𝐵(𝑖,𝑗)   · (𝑗)   × (𝑥,𝑦,𝑖,𝑗)   𝑉(𝑥,𝑦,𝑖,𝑗)

Proof of Theorem mvmulfval

Dummy variables 𝑚 𝑛 𝑜 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvmulfval.x	. 2 ⊢ × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
2		df-mvmul 21144	. . . 4 ⊢ maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘𝑜) / 𝑚⦌⦋(2nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))))))
3	2	a1i 11	. . 3 ⊢ (𝜑 → maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘𝑜) / 𝑚⦌⦋(2nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))))))))
4		fvex 6677	. . . . 5 ⊢ (1st ‘𝑜) ∈ V
5		fvex 6677	. . . . 5 ⊢ (2nd ‘𝑜) ∈ V
6		xpeq12 5574	. . . . . . 7 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → (𝑚 × 𝑛) = ((1st ‘𝑜) × (2nd ‘𝑜)))
7	6	oveq2d 7166	. . . . . 6 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → ((Base‘𝑟) ↑m (𝑚 × 𝑛)) = ((Base‘𝑟) ↑m ((1st ‘𝑜) × (2nd ‘𝑜))))
8		oveq2 7158	. . . . . . 7 ⊢ (𝑛 = (2nd ‘𝑜) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑟) ↑m (2nd ‘𝑜)))
9	8	adantl 484	. . . . . 6 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑟) ↑m (2nd ‘𝑜)))
10		simpl 485	. . . . . . 7 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → 𝑚 = (1st ‘𝑜))
11		simpr 487	. . . . . . . . 9 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → 𝑛 = (2nd ‘𝑜))
12	11	mpteq1d 5147	. . . . . . . 8 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))) = (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))))
13	12	oveq2d 7166	. . . . . . 7 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))) = (𝑟 Σg (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))))
14	10, 13	mpteq12dv 5143	. . . . . 6 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))))) = (𝑖 ∈ (1st ‘𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))))))
15	7, 9, 14	mpoeq123dv 7223	. . . . 5 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → (𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m ((1st ‘𝑜) × (2nd ‘𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑m (2nd ‘𝑜)) ↦ (𝑖 ∈ (1st ‘𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))))))
16	4, 5, 15	csbie2 3921	. . . 4 ⊢ ⦋(1st ‘𝑜) / 𝑚⦌⦋(2nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m ((1st ‘𝑜) × (2nd ‘𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑m (2nd ‘𝑜)) ↦ (𝑖 ∈ (1st ‘𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))))))
17		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → 𝑟 = 𝑅)
18	17	fveq2d 6668	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = (Base‘𝑅))
19		mvmulfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
20	18, 19	syl6eqr 2874	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = 𝐵)
21		fveq2 6664	. . . . . . . . 9 ⊢ (𝑜 = ⟨𝑀, 𝑁⟩ → (1st ‘𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
22	21	ad2antll 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
23		mvmulfval.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ Fin)
24		mvmulfval.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ Fin)
25		op1stg 7695	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
26	23, 24, 25	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
27	26	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
28	22, 27	eqtrd 2856	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘𝑜) = 𝑀)
29		fveq2 6664	. . . . . . . . 9 ⊢ (𝑜 = ⟨𝑀, 𝑁⟩ → (2nd ‘𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
30	29	ad2antll 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
31		op2ndg 7696	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
32	23, 24, 31	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
33	32	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
34	30, 33	eqtrd 2856	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘𝑜) = 𝑁)
35	28, 34	xpeq12d 5580	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((1st ‘𝑜) × (2nd ‘𝑜)) = (𝑀 × 𝑁))
36	20, 35	oveq12d 7168	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑m ((1st ‘𝑜) × (2nd ‘𝑜))) = (𝐵 ↑m (𝑀 × 𝑁)))
37	20, 34	oveq12d 7168	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑m (2nd ‘𝑜)) = (𝐵 ↑m 𝑁))
38		fveq2 6664	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
39	38	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩) → (.r‘𝑟) = (.r‘𝑅))
40	39	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (.r‘𝑟) = (.r‘𝑅))
41		mvmulfval.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
42	40, 41	syl6eqr 2874	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (.r‘𝑟) = · )
43	42	oveqd 7167	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)) = ((𝑖𝑥𝑗) · (𝑦‘𝑗)))
44	34, 43	mpteq12dv 5143	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))
45	17, 44	oveq12d 7168	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑟 Σg (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))
46	28, 45	mpteq12dv 5143	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑖 ∈ (1st ‘𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))))) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))))
47	36, 37, 46	mpoeq123dv 7223	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑥 ∈ ((Base‘𝑟) ↑m ((1st ‘𝑜) × (2nd ‘𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑m (2nd ‘𝑜)) ↦ (𝑖 ∈ (1st ‘𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))))) = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))
48	16, 47	syl5eq 2868	. . 3 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ⦋(1st ‘𝑜) / 𝑚⦌⦋(2nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))))) = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))
49		mvmulfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
50	49	elexd 3514	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
51		opex 5348	. . . 4 ⊢ ⟨𝑀, 𝑁⟩ ∈ V
52	51	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ V)
53		ovex 7183	. . . . 5 ⊢ (𝐵 ↑m (𝑀 × 𝑁)) ∈ V
54		ovex 7183	. . . . 5 ⊢ (𝐵 ↑m 𝑁) ∈ V
55	53, 54	mpoex 7771	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))) ∈ V
56	55	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))) ∈ V)
57	3, 48, 50, 52, 56	ovmpod 7296	. 2 ⊢ (𝜑 → (𝑅 maVecMul ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))
58	1, 57	syl5eq 2868	1 ⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ⦋csb 3882  ⟨cop 4566   ↦ cmpt 5138   × cxp 5547  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  1^st c1st 7681  2^nd c2nd 7682   ↑_m cmap 8400  Fincfn 8503  Basecbs 16477  ._rcmulr 16560   Σ_g cgsu 16708   maVecMul cmvmul 21143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-mvmul 21144

This theorem is referenced by:  mvmulval  21146  mavmuldm  21153  mavmul0g  21156