Theorem chpscmat 21991

Description: The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)

Hypotheses

Ref	Expression
chp0mat.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
chp0mat.p	⊢ 𝑃 = (Poly1‘𝑅)
chp0mat.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
chp0mat.x	⊢ 𝑋 = (var1‘𝑅)
chp0mat.g	⊢ 𝐺 = (mulGrp‘𝑃)
chp0mat.m	⊢ ↑ = (.g‘𝐺)
chpscmat.d	⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))}
chpscmat.s	⊢ 𝑆 = (algSc‘𝑃)
chpscmat.m	⊢ − = (-g‘𝑃)

Assertion

Ref	Expression
chpscmat	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸))))

Distinct variable groups:   𝑖,𝑗,𝐴   𝑖,𝑁,𝑗   𝑃,𝑖,𝑗   𝑅,𝑖,𝑗   𝑖,𝑋,𝑗   𝐴,𝑐,𝑚   𝐷,𝑛   𝑛,𝐸   𝑛,𝐼   𝑀,𝑐,𝑖,𝑗,𝑚,𝑛   𝑁,𝑐,𝑚,𝑛   𝑃,𝑛   𝑅,𝑐,𝑚,𝑛   𝑆,𝑛

Allowed substitution hints:   𝐴(𝑛)   𝐶(𝑖,𝑗,𝑚,𝑛,𝑐)   𝐷(𝑖,𝑗,𝑚,𝑐)   𝑃(𝑚,𝑐)   𝑆(𝑖,𝑗,𝑚,𝑐)   𝐸(𝑖,𝑗,𝑚,𝑐)   ↑ (𝑖,𝑗,𝑚,𝑛,𝑐)   𝐺(𝑖,𝑗,𝑚,𝑛,𝑐)   𝐼(𝑖,𝑗,𝑚,𝑐)   − (𝑖,𝑗,𝑚,𝑛,𝑐)   𝑋(𝑚,𝑛,𝑐)

Proof of Theorem chpscmat

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 764	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑁 ∈ Fin)
2		simplr 766	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑅 ∈ CRing)
3		elrabi 3618	. . . . . 6 ⊢ (𝑀 ∈ {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} → 𝑀 ∈ (Base‘𝐴))
4		chpscmat.d	. . . . . 6 ⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))}
5	3, 4	eleq2s 2857	. . . . 5 ⊢ (𝑀 ∈ 𝐷 → 𝑀 ∈ (Base‘𝐴))
6	5	3ad2ant1 1132	. . . 4 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → 𝑀 ∈ (Base‘𝐴))
7	6	adantl 482	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑀 ∈ (Base‘𝐴))
8		oveq 7281	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
9	8	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))))
10	9	2ralbidv 3129	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))))
11	10	rexbidv 3226	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))))
12	11	elrab 3624	. . . . . . 7 ⊢ (𝑀 ∈ {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} ↔ (𝑀 ∈ (Base‘𝐴) ∧ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))))
13		ifnefalse 4471	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ≠ 𝑗 → if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) = (0g‘𝑅))
14	13	eqeq2d 2749	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ≠ 𝑗 → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ (𝑖𝑀𝑗) = (0g‘𝑅)))
15	14	biimpcd 248	. . . . . . . . . . . . . 14 ⊢ ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))
16	15	a1i 11	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
17	16	ralimdva 3108	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝑖 ∈ 𝑁) → (∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
18	17	ralimdva 3108	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
19	18	ex 413	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))))
20	19	com23 86	. . . . . . . . 9 ⊢ ((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))))
21	20	rexlimdva 3213	. . . . . . . 8 ⊢ (𝑀 ∈ (Base‘𝐴) → (∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))))
22	21	imp 407	. . . . . . 7 ⊢ ((𝑀 ∈ (Base‘𝐴) ∧ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
23	12, 22	sylbi 216	. . . . . 6 ⊢ (𝑀 ∈ {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
24	23, 4	eleq2s 2857	. . . . 5 ⊢ (𝑀 ∈ 𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
25	24	3ad2ant1 1132	. . . 4 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
26	25	impcom 408	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅)))
27		chp0mat.c	. . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
28		chp0mat.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
29		chp0mat.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
30		chpscmat.s	. . . 4 ⊢ 𝑆 = (algSc‘𝑃)
31		eqid 2738	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
32		chp0mat.x	. . . 4 ⊢ 𝑋 = (var1‘𝑅)
33		eqid 2738	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
34		chp0mat.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑃)
35		chpscmat.m	. . . 4 ⊢ − = (-g‘𝑃)
36	27, 28, 29, 30, 31, 32, 33, 34, 35	chpdmat 21990	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘𝐴)) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))) → (𝐶‘𝑀) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘))))))
37	1, 2, 7, 26, 36	syl31anc 1372	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘))))))
38		id 22	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘)
39	38, 38	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑛𝑀𝑛) = (𝑘𝑀𝑘))
40	39	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝑘𝑀𝑘) = 𝐸))
41	40	rspccv 3558	. . . . . . . . 9 ⊢ (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
42	41	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
43	42	adantl 482	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
44	43	imp 407	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑀𝑘) = 𝐸)
45	44	fveq2d 6778	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑆‘(𝑘𝑀𝑘)) = (𝑆‘𝐸))
46	45	oveq2d 7291	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑋 − (𝑆‘(𝑘𝑀𝑘))) = (𝑋 − (𝑆‘𝐸)))
47	46	mpteq2dva 5174	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘)))) = (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘𝐸))))
48	47	oveq2d 7291	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘))))) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘𝐸)))))
49	28	ply1crng 21369	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
50	34	crngmgp 19791	. . . . 5 ⊢ (𝑃 ∈ CRing → 𝐺 ∈ CMnd)
51		cmnmnd 19402	. . . . 5 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
52	49, 50, 51	3syl 18	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Mnd)
53	52	ad2antlr 724	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝐺 ∈ Mnd)
54		crngring 19795	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
55	28	ply1ring 21419	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
56	54, 55	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
57		ringgrp 19788	. . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
58	56, 57	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Grp)
59	58	ad2antlr 724	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑃 ∈ Grp)
60		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
61	32, 28, 60	vr1cl 21388	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
62	54, 61	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃))
63	62	ad2antlr 724	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑋 ∈ (Base‘𝑃))
64		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝐼 ∈ 𝑁)
65		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
66	56	ad2antll 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ Ring)
67	66	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑃 ∈ Ring)
68	28	ply1lmod 21423	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
69	54, 68	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ LMod)
70	69	ad2antll 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ LMod)
71	70	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑃 ∈ LMod)
72		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73	30, 65, 67, 71, 72, 60	asclf 21086	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑆:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
74	5	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑀 ∈ (Base‘𝐴))
75	74	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
76		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
77	29, 76	matecl 21574	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐼 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
78	64, 64, 75, 77	syl3anc 1370	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
79	28	ply1sca 21424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
80	79	ad2antll 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑅 = (Scalar‘𝑃))
81	80	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑅 = (Scalar‘𝑃))
82	81	eqcomd 2744	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (Scalar‘𝑃) = 𝑅)
83	82	fveq2d 6778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
84	78, 83	eleqtrrd 2842	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘(Scalar‘𝑃)))
85	73, 84	ffvelrnd 6962	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃))
86		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 = (𝐼𝑀𝐼) → (𝑆‘𝐸) = (𝑆‘(𝐼𝑀𝐼)))
87	86	eqcoms 2746	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) = (𝑆‘(𝐼𝑀𝐼)))
88	87	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ ((𝐼𝑀𝐼) = 𝐸 → ((𝑆‘𝐸) ∈ (Base‘𝑃) ↔ (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃)))
89	85, 88	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
90	89	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) ∧ 𝑛 = 𝐼) → ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
91		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝐼 → 𝑛 = 𝐼)
92	91, 91	oveq12d 7293	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝐼 → (𝑛𝑀𝑛) = (𝐼𝑀𝐼))
93	92	eqeq1d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐼 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝐼𝑀𝐼) = 𝐸))
94	93	imbi1d 342	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐼 → (((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)) ↔ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))))
95	94	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) ∧ 𝑛 = 𝐼) → (((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)) ↔ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))))
96	90, 95	mpbird 256	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) ∧ 𝑛 = 𝐼) → ((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
97	64, 96	rspcimdv 3551	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
98	97	ex 413	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (𝐼 ∈ 𝑁 → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))))
99	98	com23 86	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝐼 ∈ 𝑁 → (𝑆‘𝐸) ∈ (Base‘𝑃))))
100	99	ex 413	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝐼 ∈ 𝑁 → (𝑆‘𝐸) ∈ (Base‘𝑃)))))
101	100	com24 95	. . . . . . 7 ⊢ (𝑀 ∈ 𝐷 → (𝐼 ∈ 𝑁 → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑆‘𝐸) ∈ (Base‘𝑃)))))
102	101	3imp 1110	. . . . . 6 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑆‘𝐸) ∈ (Base‘𝑃)))
103	102	impcom 408	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑆‘𝐸) ∈ (Base‘𝑃))
104	60, 35	grpsubcl 18655	. . . . 5 ⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃) ∧ (𝑆‘𝐸) ∈ (Base‘𝑃)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃))
105	59, 63, 103, 104	syl3anc 1370	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃))
106	34, 60	mgpbas 19726	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝐺)
107	105, 106	eleqtrdi 2849	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝐺))
108		eqid 2738	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
109		chp0mat.m	. . . 4 ⊢ ↑ = (.g‘𝐺)
110	108, 109	gsumconst 19535	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ Fin ∧ (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝐺)) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘𝐸)))) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸))))
111	53, 1, 107, 110	syl3anc 1370	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘𝐸)))) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸))))
112	37, 48, 111	3eqtrd 2782	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068  ifcif 4459   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  ♯chash 14044  Basecbs 16912  Scalarcsca 16965  0_gc0g 17150   Σ_g cgsu 17151  Mndcmnd 18385  Grpcgrp 18577  -_gcsg 18579  ._gcmg 18700  CMndccmn 19386  mulGrpcmgp 19720  Ringcrg 19783  CRingccrg 19784  LModclmod 20123  algSccascl 21059  var₁cv1 21347  Poly₁cpl1 21348   Mat cmat 21554   CharPlyMat cchpmat 21975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-addf 10950  ax-mulf 10951

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1507  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-ofr 7534  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-dec 12438  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-word 14218  df-lsw 14266  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384  df-splice 14463  df-reverse 14472  df-s2 14561  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-0g 17152  df-gsum 17153  df-prds 17158  df-pws 17160  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-efmnd 18508  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-gim 18875  df-cntz 18923  df-oppg 18950  df-symg 18975  df-pmtr 19050  df-psgn 19099  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-cring 19786  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-rnghom 19959  df-drng 19993  df-subrg 20022  df-lmod 20125  df-lss 20194  df-sra 20434  df-rgmod 20435  df-cnfld 20598  df-zring 20671  df-zrh 20705  df-dsmm 20939  df-frlm 20954  df-ascl 21062  df-psr 21112  df-mvr 21113  df-mpl 21114  df-opsr 21116  df-psr1 21351  df-vr1 21352  df-ply1 21353  df-mamu 21533  df-mat 21555  df-mdet 21734  df-mat2pmat 21856  df-chpmat 21976

This theorem is referenced by:  chpscmat0  21992