Theorem chpscmat 23153

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 779	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑁 ∈ Fin)
2		simplr 781	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑅 ∈ CRing)
3		elrabi 3641	. . . . . 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 2879	. . . . 5 ⊢ (𝑀 ∈ 𝐷 → 𝑀 ∈ (Base‘𝐴))
6	5	3ad2ant1 1151	. . . 4 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → 𝑀 ∈ (Base‘𝐴))
7	6	adantl 487	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑀 ∈ (Base‘𝐴))
8		oveq 7424	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
9	8	eqeq1d 2763	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))))
10	9	2ralbidv 3227	. . . . . . . . 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 3187	. . . . . . . 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 3645	. . . . . . 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 4494	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ≠ 𝑗 → if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) = (0g‘𝑅))
14	13	eqeq2d 2772	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ≠ 𝑗 → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ (𝑖𝑀𝑗) = (0g‘𝑅)))
15	14	biimpcd 252	. . . . . . . . . . . . . 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 3175	. . . . . . . . . . . 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 3175	. . . . . . . . . . 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 418	. . . . . . . . . 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 87	. . . . . . . . 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 3164	. . . . . . . 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 412	. . . . . . 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 220	. . . . . 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 2879	. . . . 5 ⊢ (𝑀 ∈ 𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
25	24	3ad2ant1 1151	. . . 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 413	. . 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 2761	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
32		chp0mat.x	. . . 4 ⊢ 𝑋 = (var1‘𝑅)
33		eqid 2761	. . . 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 23152	. . 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 1400	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘))))))
38		id 23	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘)
39	38, 38	oveq12d 7436	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑛𝑀𝑛) = (𝑘𝑀𝑘))
40	39	eqeq1d 2763	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝑘𝑀𝑘) = 𝐸))
41	40	rspccv 3574	. . . . . . . . 9 ⊢ (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
42	41	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
43	42	adantl 487	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
44	43	imp 412	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑀𝑘) = 𝐸)
45	44	fveq2d 6887	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑆‘(𝑘𝑀𝑘)) = (𝑆‘𝐸))
46	45	oveq2d 7434	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑋 − (𝑆‘(𝑘𝑀𝑘))) = (𝑋 − (𝑆‘𝐸)))
47	46	mpteq2dva 5198	. . 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 7434	. 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 22509	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
50	34	crngmgp 20460	. . . . 5 ⊢ (𝑃 ∈ CRing → 𝐺 ∈ CMnd)
51		cmnmnd 20004	. . . . 5 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
52	49, 50, 51	3syl 19	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Mnd)
53	52	ad2antlr 740	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝐺 ∈ Mnd)
54		crngring 20465	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
55	28	ply1ring 22558	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
56	54, 55	syl 18	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
57		ringgrp 20457	. . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
58	56, 57	syl 18	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Grp)
59	58	ad2antlr 740	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑃 ∈ Grp)
60		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
61	32, 28, 60	vr1cl 22528	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
62	54, 61	syl 18	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃))
63	62	ad2antlr 740	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑋 ∈ (Base‘𝑃))
64		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝐼 ∈ 𝑁)
65		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
66	56	ad2antll 742	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ Ring)
67	66	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑃 ∈ Ring)
68	28	ply1lmod 22562	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
69	54, 68	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ LMod)
70	69	ad2antll 742	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ LMod)
71	70	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑃 ∈ LMod)
72		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73	30, 65, 67, 71, 72, 60	asclf 22182	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑆:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
74	5	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑀 ∈ (Base‘𝐴))
75	74	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
76		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
77	29, 76	matecl 22733	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐼 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
78	64, 64, 75, 77	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
79	28	ply1sca 22563	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
80	79	ad2antll 742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑅 = (Scalar‘𝑃))
81	80	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑅 = (Scalar‘𝑃))
82	81	eqcomd 2767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (Scalar‘𝑃) = 𝑅)
83	82	fveq2d 6887	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
84	78, 83	eleqtrrd 2864	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘(Scalar‘𝑃)))
85	73, 84	ffvelcdmd 7083	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃))
86		fveq2 6883	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 = (𝐼𝑀𝐼) → (𝑆‘𝐸) = (𝑆‘(𝐼𝑀𝐼)))
87	86	eqcoms 2769	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) = (𝑆‘(𝐼𝑀𝐼)))
88	87	eleq1d 2846	. . . . . . . . . . . . . . 15 ⊢ ((𝐼𝑀𝐼) = 𝐸 → ((𝑆‘𝐸) ∈ (Base‘𝑃) ↔ (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃)))
89	85, 88	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
90	89	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) ∧ 𝑛 = 𝐼) → ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
91		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝐼 → 𝑛 = 𝐼)
92	91, 91	oveq12d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝐼 → (𝑛𝑀𝑛) = (𝐼𝑀𝐼))
93	92	eqeq1d 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐼 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝐼𝑀𝐼) = 𝐸))
94	93	imbi1d 344	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐼 → (((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)) ↔ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))))
95	94	adantl 487	. . . . . . . . . . . . 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 260	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) ∧ 𝑛 = 𝐼) → ((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
97	64, 96	rspcimdv 3567	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
98	97	ex 418	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (𝐼 ∈ 𝑁 → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))))
99	98	com23 87	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝐼 ∈ 𝑁 → (𝑆‘𝐸) ∈ (Base‘𝑃))))
100	99	ex 418	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝐼 ∈ 𝑁 → (𝑆‘𝐸) ∈ (Base‘𝑃)))))
101	100	com24 96	. . . . . . 7 ⊢ (𝑀 ∈ 𝐷 → (𝐼 ∈ 𝑁 → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑆‘𝐸) ∈ (Base‘𝑃)))))
102	101	3imp 1128	. . . . . 6 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑆‘𝐸) ∈ (Base‘𝑃)))
103	102	impcom 413	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑆‘𝐸) ∈ (Base‘𝑃))
104	60, 35	grpsubcl 19223	. . . . 5 ⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃) ∧ (𝑆‘𝐸) ∈ (Base‘𝑃)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃))
105	59, 63, 103, 104	syl3anc 1398	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃))
106	34, 60	mgpbas 20358	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝐺)
107	105, 106	eleqtrdi 2871	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝐺))
108		eqid 2761	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
109		chp0mat.m	. . . 4 ⊢ ↑ = (.g‘𝐺)
110	108, 109	gsumconst 20141	. . 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 1398	. 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 2800	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413  ifcif 4482   ↦ cmpt 5186  ‘cfv 6537  (class class class)co 7418  Fincfn 8966  ♯chash 14467  Basecbs 17380  Scalarcsca 17424  0_gc0g 17603   Σ_g cgsu 17604  Mndcmnd 18916  Grpcgrp 19137  -_gcsg 19139  ._gcmg 19270  CMndccmn 19987  mulGrpcmgp 20353  Ringcrg 20452  CRingccrg 20453  LModclmod 21128  algSccascl 22153  var₁cv1 22487  Poly₁cpl1 22488   Mat cmat 22715   CharPlyMat cchpmat 23137

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-addf 11272  ax-mulf 11273

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-ofr 7692  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-tpos 8236  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-map 8842  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-sup 9427  df-oi 9497  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-xnn0 12673  df-z 12687  df-dec 12808  df-uz 12959  df-rp 13114  df-fz 13633  df-fzo 13782  df-seq 14138  df-exp 14198  df-hash 14468  df-word 14652  df-lsw 14701  df-concat 14709  df-s1 14736  df-substr 14782  df-pfx 14814  df-splice 14892  df-reverse 14901  df-s2 14992  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-mulr 17435  df-starv 17436  df-sca 17437  df-vsca 17438  df-ip 17439  df-tset 17440  df-ple 17441  df-ds 17443  df-unif 17444  df-hom 17445  df-cco 17446  df-0g 17605  df-gsum 17606  df-prds 17611  df-pws 17613  df-mre 17749  df-mrc 17750  df-acs 17752  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-mhm 18971  df-submnd 18972  df-efmnd 19058  df-grp 19140  df-minusg 19141  df-sbg 19142  df-mulg 19271  df-subg 19326  df-ghm 19421  df-gim 19466  df-cntz 19524  df-oppg 19553  df-symg 19577  df-pmtr 19649  df-psgn 19698  df-cmn 19989  df-abl 19990  df-mgp 20354  df-rng 20368  df-ur 20401  df-ring 20454  df-cring 20455  df-oppr 20560  df-dvdsr 20580  df-unit 20581  df-invr 20611  df-dvr 20624  df-rhm 20695  df-subrng 20791  df-subrg 20815  df-drng 20975  df-lmod 21130  df-lss 21200  df-sra 21441  df-rgmod 21442  df-cnfld 21672  df-zring 21746  df-zrh 21802  df-dsmm 22031  df-frlm 22046  df-ascl 22156  df-psr 22210  df-mvr 22211  df-mpl 22212  df-opsr 22214  df-psr1 22491  df-vr1 22492  df-ply1 22493  df-mamu 22699  df-mat 22716  df-mdet 22893  df-mat2pmat 23018  df-chpmat 23138

This theorem is used by:  chpscmat0  23154