Theorem mvmulfval 20167

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 20166	. . . 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 6113	. . . . 5 ⊢ (1st ‘𝑜) ∈ V
5		fvex 6113	. . . . 5 ⊢ (2nd ‘𝑜) ∈ V
6		xpeq12 5058	. . . . . . 7 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → (𝑚 × 𝑛) = ((1st ‘𝑜) × (2nd ‘𝑜)))
7	6	oveq2d 6565	. . . . . 6 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)) = ((Base‘𝑟) ↑𝑚 ((1st ‘𝑜) × (2nd ‘𝑜))))
8		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = (2nd ‘𝑜) → ((Base‘𝑟) ↑𝑚 𝑛) = ((Base‘𝑟) ↑𝑚 (2nd ‘𝑜)))
9	8	adantl 481	. . . . . 6 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → ((Base‘𝑟) ↑𝑚 𝑛) = ((Base‘𝑟) ↑𝑚 (2nd ‘𝑜)))
10		simpl 472	. . . . . . 7 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → 𝑚 = (1st ‘𝑜))
11		simpr 476	. . . . . . . . 9 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → 𝑛 = (2nd ‘𝑜))
12	11	mpteq1d 4666	. . . . . . . 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 6565	. . . . . . 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 4663	. . . . . 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	mpt2eq123dv 6615	. . . . 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 3529	. . . 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 790	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → 𝑟 = 𝑅)
18	17	fveq2d 6107	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = (Base‘𝑅))
19		mvmulfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
20	18, 19	syl6eqr 2662	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = 𝐵)
21		fveq2 6103	. . . . . . . . 9 ⊢ (𝑜 = ⟨𝑀, 𝑁⟩ → (1st ‘𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
22	21	ad2antll 761	. . . . . . . 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 7071	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
26	23, 24, 25	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
28	22, 27	eqtrd 2644	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘𝑜) = 𝑀)
29		fveq2 6103	. . . . . . . . 9 ⊢ (𝑜 = ⟨𝑀, 𝑁⟩ → (2nd ‘𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
30	29	ad2antll 761	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
31		op2ndg 7072	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
32	23, 24, 31	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
33	32	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
34	30, 33	eqtrd 2644	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘𝑜) = 𝑁)
35	28, 34	xpeq12d 5064	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((1st ‘𝑜) × (2nd ‘𝑜)) = (𝑀 × 𝑁))
36	20, 35	oveq12d 6567	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑𝑚 ((1st ‘𝑜) × (2nd ‘𝑜))) = (𝐵 ↑𝑚 (𝑀 × 𝑁)))
37	20, 34	oveq12d 6567	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑𝑚 (2nd ‘𝑜)) = (𝐵 ↑𝑚 𝑁))
38		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
39	38	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩) → (.r‘𝑟) = (.r‘𝑅))
40	39	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (.r‘𝑟) = (.r‘𝑅))
41		mvmulfval.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
42	40, 41	syl6eqr 2662	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (.r‘𝑟) = · )
43	42	oveqd 6566	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)) = ((𝑖𝑥𝑗) · (𝑦‘𝑗)))
44	34, 43	mpteq12dv 4663	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))
45	17, 44	oveq12d 6567	. . . . . 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 4663	. . . . 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	mpt2eq123dv 6615	. . . 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 2656	. . 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		elex 3185	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
51	49, 50	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
52		opex 4859	. . . 4 ⊢ ⟨𝑀, 𝑁⟩ ∈ V
53	52	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ V)
54		ovex 6577	. . . . 5 ⊢ (𝐵 ↑𝑚 (𝑀 × 𝑁)) ∈ V
55		ovex 6577	. . . . 5 ⊢ (𝐵 ↑𝑚 𝑁) ∈ V
56	54, 55	mpt2ex 7136	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑𝑚 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))) ∈ V
57	56	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑𝑚 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))) ∈ V)
58	3, 48, 51, 53, 57	ovmpt2d 6686	. 2 ⊢ (𝜑 → (𝑅 maVecMul ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑𝑚 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗)))))))
59	1, 58	syl5eq 2656	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 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  Fincfn 7841  Basecbs 15695  ._rcmulr 15769   Σ_g cgsu 15924   maVecMul cmvmul 20165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-mvmul 20166

This theorem is referenced by:  mvmulval  20168  mavmuldm  20175  mavmul0g  20178