Theorem mvmulfval 21255

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 21254	. . . 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 6676	. . . . 5 ⊢ (1st ‘𝑜) ∈ V
5		fvex 6676	. . . . 5 ⊢ (2nd ‘𝑜) ∈ V
6		xpeq12 5553	. . . . . . 7 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → (𝑚 × 𝑛) = ((1st ‘𝑜) × (2nd ‘𝑜)))
7	6	oveq2d 7172	. . . . . 6 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → ((Base‘𝑟) ↑m (𝑚 × 𝑛)) = ((Base‘𝑟) ↑m ((1st ‘𝑜) × (2nd ‘𝑜))))
8		oveq2 7164	. . . . . . 7 ⊢ (𝑛 = (2nd ‘𝑜) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑟) ↑m (2nd ‘𝑜)))
9	8	adantl 485	. . . . . 6 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑟) ↑m (2nd ‘𝑜)))
10		simpl 486	. . . . . . 7 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → 𝑚 = (1st ‘𝑜))
11		simpr 488	. . . . . . . . 9 ⊢ ((𝑚 = (1st ‘𝑜) ∧ 𝑛 = (2nd ‘𝑜)) → 𝑛 = (2nd ‘𝑜))
12	11	mpteq1d 5125	. . . . . . . 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 7172	. . . . . . 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 5121	. . . . . 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 7229	. . . . 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 3846	. . . 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 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → 𝑟 = 𝑅)
18	17	fveq2d 6667	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = (Base‘𝑅))
19		mvmulfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
20	18, 19	eqtr4di 2811	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = 𝐵)
21		fveq2 6663	. . . . . . . . 9 ⊢ (𝑜 = ⟨𝑀, 𝑁⟩ → (1st ‘𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
22	21	ad2antll 728	. . . . . . . 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 7711	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
26	23, 24, 25	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
27	26	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
28	22, 27	eqtrd 2793	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘𝑜) = 𝑀)
29		fveq2 6663	. . . . . . . . 9 ⊢ (𝑜 = ⟨𝑀, 𝑁⟩ → (2nd ‘𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
30	29	ad2antll 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
31		op2ndg 7712	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
32	23, 24, 31	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
33	32	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
34	30, 33	eqtrd 2793	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘𝑜) = 𝑁)
35	28, 34	xpeq12d 5559	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((1st ‘𝑜) × (2nd ‘𝑜)) = (𝑀 × 𝑁))
36	20, 35	oveq12d 7174	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑m ((1st ‘𝑜) × (2nd ‘𝑜))) = (𝐵 ↑m (𝑀 × 𝑁)))
37	20, 34	oveq12d 7174	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑m (2nd ‘𝑜)) = (𝐵 ↑m 𝑁))
38		fveq2 6663	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
39	38	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩) → (.r‘𝑟) = (.r‘𝑅))
40	39	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (.r‘𝑟) = (.r‘𝑅))
41		mvmulfval.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
42	40, 41	eqtr4di 2811	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (.r‘𝑟) = · )
43	42	oveqd 7173	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)) = ((𝑖𝑥𝑗) · (𝑦‘𝑗)))
44	34, 43	mpteq12dv 5121	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝑅 ∧ 𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑗 ∈ (2nd ‘𝑜) ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))
45	17, 44	oveq12d 7174	. . . . . 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 5121	. . . . 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 7229	. . . 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 2805	. . 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 3430	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
51		opex 5328	. . . 4 ⊢ ⟨𝑀, 𝑁⟩ ∈ V
52	51	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ V)
53		ovex 7189	. . . . 5 ⊢ (𝐵 ↑m (𝑀 × 𝑁)) ∈ V
54		ovex 7189	. . . . 5 ⊢ (𝐵 ↑m 𝑁) ∈ V
55	53, 54	mpoex 7788	. . . 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 7303	. 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 2805	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 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⦋csb 3807  ⟨cop 4531   ↦ cmpt 5116   × cxp 5526  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  1^st c1st 7697  2^nd c2nd 7698   ↑_m cmap 8422  Fincfn 8540  Basecbs 16554  ._rcmulr 16637   Σ_g cgsu 16785   maVecMul cmvmul 21253

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-mvmul 21254

This theorem is referenced by:  mvmulval  21256  mavmuldm  21263  mavmul0g  21266