Theorem chpscmat 22807

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 767	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑁 ∈ Fin)
2		simplr 769	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑅 ∈ CRing)
3		elrabi 3630	. . . . . 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 2854	. . . . 5 ⊢ (𝑀 ∈ 𝐷 → 𝑀 ∈ (Base‘𝐴))
6	5	3ad2ant1 1134	. . . 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 7373	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
9	8	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) ↔ (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))))
10	9	2ralbidv 3201	. . . . . . . . 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 3161	. . . . . . . 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 3634	. . . . . . 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 4478	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ≠ 𝑗 → if(𝑖 = 𝑗, 𝑐, (0g‘𝑅)) = (0g‘𝑅))
14	13	eqeq2d 2747	. . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . 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 3149	. . . . . . . . . . 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 3138	. . . . . . . 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 2854	. . . . 5 ⊢ (𝑀 ∈ 𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = (0g‘𝑅))))
25	24	3ad2ant1 1134	. . . 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 2736	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
32		chp0mat.x	. . . 4 ⊢ 𝑋 = (var1‘𝑅)
33		eqid 2736	. . . 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 22806	. . 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 1376	. 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 7385	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑛𝑀𝑛) = (𝑘𝑀𝑘))
40	39	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝑘𝑀𝑘) = 𝐸))
41	40	rspccv 3561	. . . . . . . . 9 ⊢ (∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑘 ∈ 𝑁 → (𝑘𝑀𝑘) = 𝐸))
42	41	3ad2ant3 1136	. . . . . . . 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 7383	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘 ∈ 𝑁) → (𝑋 − (𝑆‘(𝑘𝑀𝑘))) = (𝑋 − (𝑆‘𝐸)))
47	46	mpteq2dva 5178	. . 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 7383	. 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 22162	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
50	34	crngmgp 20222	. . . . 5 ⊢ (𝑃 ∈ CRing → 𝐺 ∈ CMnd)
51		cmnmnd 19772	. . . . 5 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
52	49, 50, 51	3syl 18	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Mnd)
53	52	ad2antlr 728	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝐺 ∈ Mnd)
54		crngring 20226	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
55	28	ply1ring 22211	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
56	54, 55	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
57		ringgrp 20219	. . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
58	56, 57	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Grp)
59	58	ad2antlr 728	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑃 ∈ Grp)
60		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
61	32, 28, 60	vr1cl 22181	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
62	54, 61	syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃))
63	62	ad2antlr 728	. . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
66	56	ad2antll 730	. . . . . . . . . . . . . . . . . 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 22215	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
69	54, 68	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ LMod)
70	69	ad2antll 730	. . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73	30, 65, 67, 71, 72, 60	asclf 21861	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
77	29, 76	matecl 22390	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐼 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
78	64, 64, 75, 77	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
79	28	ply1sca 22216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
80	79	ad2antll 730	. . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ 𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼 ∈ 𝑁) → (𝐼𝑀𝐼) ∈ (Base‘(Scalar‘𝑃)))
85	73, 84	ffvelcdmd 7037	. . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼𝑀𝐼) = 𝐸 → (𝑆‘𝐸) = (𝑆‘(𝐼𝑀𝐼)))
88	87	eleq1d 2821	. . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝐼 → (𝑛𝑀𝑛) = (𝐼𝑀𝐼))
93	92	eqeq1d 2738	. . . . . . . . . . . . . . 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 3554	. . . . . . . . . . 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 1111	. . . . . 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 18996	. . . . 5 ⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃) ∧ (𝑆‘𝐸) ∈ (Base‘𝑃)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃))
105	59, 63, 103, 104	syl3anc 1374	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝑃))
106	34, 60	mgpbas 20126	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝐺)
107	105, 106	eleqtrdi 2846	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 − (𝑆‘𝐸)) ∈ (Base‘𝐺))
108		eqid 2736	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
109		chp0mat.m	. . . 4 ⊢ ↑ = (.g‘𝐺)
110	108, 109	gsumconst 19909	. . 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 1374	. 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 2775	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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  {crab 3389  ifcif 4466   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367  Fincfn 8893  ♯chash 14292  Basecbs 17179  Scalarcsca 17223  0_gc0g 17402   Σ_g cgsu 17403  Mndcmnd 18702  Grpcgrp 18909  -_gcsg 18911  ._gcmg 19043  CMndccmn 19755  mulGrpcmgp 20121  Ringcrg 20214  CRingccrg 20215  LModclmod 20855  algSccascl 21832  var₁cv1 22139  Poly₁cpl1 22140   Mat cmat 22372   CharPlyMat cchpmat 22791

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-addf 11117  ax-mulf 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1514  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-word 14476  df-lsw 14525  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-splice 14712  df-reverse 14721  df-s2 14810  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-efmnd 18837  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-gim 19234  df-cntz 19292  df-oppg 19321  df-symg 19345  df-pmtr 19417  df-psgn 19466  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-oppr 20317  df-dvdsr 20337  df-unit 20338  df-invr 20368  df-dvr 20381  df-rhm 20452  df-subrng 20523  df-subrg 20547  df-drng 20708  df-lmod 20857  df-lss 20927  df-sra 21168  df-rgmod 21169  df-cnfld 21353  df-zring 21427  df-zrh 21483  df-dsmm 21712  df-frlm 21727  df-ascl 21835  df-psr 21889  df-mvr 21890  df-mpl 21891  df-opsr 21893  df-psr1 22143  df-vr1 22144  df-ply1 22145  df-mamu 22356  df-mat 22373  df-mdet 22550  df-mat2pmat 22672  df-chpmat 22792

This theorem is referenced by:  chpscmat0  22808