Theorem chpscmat 22705

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 766	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑁 ∈ Fin)
2		simplr 768	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑅 ∈ CRing)
3		elrabi 3651	. . . . . 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 2846	. . . . 5 ⊢ (𝑀 ∈ 𝐷 → 𝑀 ∈ (Base‘𝐴))
6	5	3ad2ant1 1133	. . . 4 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → 𝑀 ∈ (Base‘𝐴))
7	6	adantl 481	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑀 ∈ (Base‘𝐴))
8		oveq 7375	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
9	8	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))))
10	9	2ralbidv 3199	. . . . . . . . 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 3157	. . . . . . . 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 3656	. . . . . . 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 4496	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ≠ 𝑗 → if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) = (0g‘𝑅))
14	13	eqeq2d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ≠ 𝑗 → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ (𝑖𝑀𝑗) = (0g‘𝑅)))
15	14	biimpcd 249	. . . . . . . . . . . . . 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 3145	. . . . . . . . . . . 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 3145	. . . . . . . . . . 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 412	. . . . . . . . . 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 3134	. . . . . . . 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 406	. . . . . . 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 217	. . . . . 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 2846	. . . . 5 ⊢ (𝑀 ∈ 𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
25	24	3ad2ant1 1133	. . . 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 407	. . 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 2729	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
32		chp0mat.x	. . . 4 ⊢ 𝑋 = (var1‘𝑅)
33		eqid 2729	. . . 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 22704	. . 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 1375	. 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 7387	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑛𝑀𝑛) = (𝑘𝑀𝑘))
40	39	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝑘𝑀𝑘) = 𝐸))
41	40	rspccv 3582	. . . . . . . . 9 ⊢ (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
42	41	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸) → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
43	42	adantl 481	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
44	43	imp 406	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑀𝑘) = 𝐸)
45	44	fveq2d 6844	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑆‘(𝑘𝑀𝑘)) = (𝑆‘𝐸))
46	45	oveq2d 7385	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑋 − (𝑆‘(𝑘𝑀𝑘))) = (𝑋 − (𝑆‘𝐸)))
47	46	mpteq2dva 5195	. . 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 7385	. 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 22059	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
50	34	crngmgp 20126	. . . . 5 ⊢ (𝑃 ∈ CRing → 𝐺 ∈ CMnd)
51		cmnmnd 19703	. . . . 5 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
52	49, 50, 51	3syl 18	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Mnd)
53	52	ad2antlr 727	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝐺 ∈ Mnd)
54		crngring 20130	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
55	28	ply1ring 22108	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
56	54, 55	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
57		ringgrp 20123	. . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
58	56, 57	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Grp)
59	58	ad2antlr 727	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑃 ∈ Grp)
60		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
61	32, 28, 60	vr1cl 22078	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
62	54, 61	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃))
63	62	ad2antlr 727	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑋 ∈ (Base‘𝑃))
64		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝐼 ∈ 𝑁)
65		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
66	56	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ Ring)
67	66	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑃 ∈ Ring)
68	28	ply1lmod 22112	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
69	54, 68	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ LMod)
70	69	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ LMod)
71	70	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑃 ∈ LMod)
72		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73	30, 65, 67, 71, 72, 60	asclf 21767	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑆:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
74	5	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑀 ∈ (Base‘𝐴))
75	74	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
76		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
77	29, 76	matecl 22288	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐼 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
78	64, 64, 75, 77	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
79	28	ply1sca 22113	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
80	79	ad2antll 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑅 = (Scalar‘𝑃))
81	80	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → 𝑅 = (Scalar‘𝑃))
82	81	eqcomd 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (Scalar‘𝑃) = 𝑅)
83	82	fveq2d 6844	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
84	78, 83	eleqtrrd 2831	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘(Scalar‘𝑃)))
85	73, 84	ffvelcdmd 7039	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃))
86		fveq2 6840	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 = (𝐼𝑀𝐼) → (𝑆‘𝐸) = (𝑆‘(𝐼𝑀𝐼)))
87	86	eqcoms 2737	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) = (𝑆‘(𝐼𝑀𝐼)))
88	87	eleq1d 2813	. . . . . . . . . . . . . . 15 ⊢ ((𝐼𝑀𝐼) = 𝐸 → ((𝑆‘𝐸) ∈ (Base‘𝑃) ↔ (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃)))
89	85, 88	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
90	89	adantr 480	. . . . . . . . . . . . 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 7387	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝐼 → (𝑛𝑀𝑛) = (𝐼𝑀𝐼))
93	92	eqeq1d 2731	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐼 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝐼𝑀𝐼) = 𝐸))
94	93	imbi1d 341	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐼 → (((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)) ↔ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃))))
95	94	adantl 481	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) ∧ 𝑛 = 𝐼) → ((𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
97	64, 96	rspcimdv 3575	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑆‘𝐸) ∈ (Base‘𝑃)))
98	97	ex 412	. . . . . . . . . 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 412	. . . . . . . 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 407	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑆‘𝐸) ∈ (Base‘𝑃))
104	60, 35	grpsubcl 18928	. . . . 5 ⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃) ∧ (𝑆‘𝐸) ∈ (Base‘𝑃)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃))
105	59, 63, 103, 104	syl3anc 1373	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃))
106	34, 60	mgpbas 20030	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝐺)
107	105, 106	eleqtrdi 2838	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝐺))
108		eqid 2729	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
109		chp0mat.m	. . . 4 ⊢ ↑ = (.g‘𝐺)
110	108, 109	gsumconst 19840	. . 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 1373	. 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 2768	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3402  ifcif 4484   ↦ cmpt 5183  ‘cfv 6499  (class class class)co 7369  Fincfn 8895  ♯chash 14271  Basecbs 17155  Scalarcsca 17199  0_gc0g 17378   Σ_g cgsu 17379  Mndcmnd 18637  Grpcgrp 18841  -_gcsg 18843  ._gcmg 18975  CMndccmn 19686  mulGrpcmgp 20025  Ringcrg 20118  CRingccrg 20119  LModclmod 20742  algSccascl 21737  var₁cv1 22036  Poly₁cpl1 22037   Mat cmat 22270   CharPlyMat cchpmat 22689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-addf 11123  ax-mulf 11124

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-ofr 7634  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-sup 9369  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-xnn0 12492  df-z 12506  df-dec 12626  df-uz 12770  df-rp 12928  df-fz 13445  df-fzo 13592  df-seq 13943  df-exp 14003  df-hash 14272  df-word 14455  df-lsw 14504  df-concat 14512  df-s1 14537  df-substr 14582  df-pfx 14612  df-splice 14691  df-reverse 14700  df-s2 14790  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-0g 17380  df-gsum 17381  df-prds 17386  df-pws 17388  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-efmnd 18772  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-ghm 19121  df-gim 19167  df-cntz 19225  df-oppg 19254  df-symg 19276  df-pmtr 19348  df-psgn 19397  df-cmn 19688  df-abl 19689  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-cring 20121  df-oppr 20222  df-dvdsr 20242  df-unit 20243  df-invr 20273  df-dvr 20286  df-rhm 20357  df-subrng 20431  df-subrg 20455  df-drng 20616  df-lmod 20744  df-lss 20814  df-sra 21056  df-rgmod 21057  df-cnfld 21241  df-zring 21333  df-zrh 21389  df-dsmm 21617  df-frlm 21632  df-ascl 21740  df-psr 21794  df-mvr 21795  df-mpl 21796  df-opsr 21798  df-psr1 22040  df-vr1 22041  df-ply1 22042  df-mamu 22254  df-mat 22271  df-mdet 22448  df-mat2pmat 22570  df-chpmat 22690

This theorem is referenced by:  chpscmat0  22706