Theorem cpmadugsumfi 23034

Description: The product of the characteristic matrix of a given matrix and its adjunct represented as finite sum. (Contributed by AV, 7-Nov-2019.) (Proof shortened by AV, 29-Nov-2019.)

Hypotheses

Ref	Expression
cpmadugsum.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cpmadugsum.b	⊢ 𝐵 = (Base‘𝐴)
cpmadugsum.p	⊢ 𝑃 = (Poly1‘𝑅)
cpmadugsum.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cpmadugsum.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmadugsum.x	⊢ 𝑋 = (var1‘𝑅)
cpmadugsum.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
cpmadugsum.m	⊢ · = ( ·𝑠 ‘𝑌)
cpmadugsum.r	⊢ × = (.r‘𝑌)
cpmadugsum.1	⊢ 1 = (1r‘𝑌)
cpmadugsum.g	⊢ + = (+g‘𝑌)
cpmadugsum.s	⊢ − = (-g‘𝑌)
cpmadugsum.i	⊢ 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀))
cpmadugsum.j	⊢ 𝐽 = (𝑁 maAdju 𝑃)

Assertion

Ref	Expression
cpmadugsumfi	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐼 × (𝐽‘𝐼)) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Distinct variable groups:   𝐵,𝑖   𝑖,𝑀   𝑖,𝑁   𝑅,𝑖   𝑖,𝑋   𝑖,𝑌   × ,𝑖   · ,𝑖   1 ,𝑖   𝑖,𝑏,𝑠,𝑇   ↑ ,𝑖   − ,𝑖   𝐴,𝑏,𝑠   𝐵,𝑏,𝑠   𝐼,𝑏,𝑖,𝑠   𝐽,𝑏,𝑖,𝑠   𝑀,𝑏,𝑠   𝑁,𝑏,𝑠   𝑃,𝑖   𝑅,𝑏,𝑠   𝑇,𝑏,𝑠   𝑋,𝑏,𝑠   𝑌,𝑏,𝑠   ↑ ,𝑠,𝑏   · ,𝑏,𝑠

Allowed substitution hints:   𝐴(𝑖)   𝑃(𝑠,𝑏)   + (𝑖,𝑠,𝑏)   × (𝑠,𝑏)   1 (𝑠,𝑏)   − (𝑠,𝑏)

Proof of Theorem cpmadugsumfi

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7418	. . 3 ⊢ ((𝐽‘𝐼) = (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))))) → (𝐼 × (𝐽‘𝐼)) = (𝐼 × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))))
2		cpmadugsum.i	. . . . . 6 ⊢ 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀))
3	2	a1i 11	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)))
4	3	oveq1d 7425	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐼 × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))) = (((𝑋 · 1 ) − (𝑇‘𝑀)) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))))
5		eqid 2763	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
6		cpmadugsum.r	. . . . 5 ⊢ × = (.r‘𝑌)
7		cpmadugsum.s	. . . . 5 ⊢ − = (-g‘𝑌)
8		crngring 20322	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	8	anim2i 628	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
10	9	3adant3 1150	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
11	10	ad2antrr 738	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
12		cpmadugsum.p	. . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅)
13		cpmadugsum.y	. . . . . . 7 ⊢ 𝑌 = (𝑁 Mat 𝑃)
14	12, 13	pmatring 22849	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
15	11, 14	syl 18	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑌 ∈ Ring)
16	12, 13	pmatlmod 22850	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod)
17	8, 16	sylan2 604	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
18	8	adantl 486	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
19		cpmadugsum.x	. . . . . . . . . . 11 ⊢ 𝑋 = (var1‘𝑅)
20		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
21	19, 12, 20	vr1cl 22377	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
22	18, 21	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘𝑃))
23	12	ply1crng 22358	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
24	13	matsca2 22577	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
25	23, 24	sylan2 604	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
26	25	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘𝑃) = (Base‘(Scalar‘𝑌)))
27	22, 26	eleqtrd 2865	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
28	8, 14	sylan2 604	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
29		cpmadugsum.1	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑌)
30	5, 29	ringidcl 20344	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 1 ∈ (Base‘𝑌))
31	28, 30	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 1 ∈ (Base‘𝑌))
32		eqid 2763	. . . . . . . . 9 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
33		cpmadugsum.m	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑌)
34		eqid 2763	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
35	5, 32, 33, 34	lmodvscl 20999	. . . . . . . 8 ⊢ ((𝑌 ∈ LMod ∧ 𝑋 ∈ (Base‘(Scalar‘𝑌)) ∧ 1 ∈ (Base‘𝑌)) → (𝑋 · 1 ) ∈ (Base‘𝑌))
36	17, 27, 31, 35	syl3anc 1398	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑋 · 1 ) ∈ (Base‘𝑌))
37	36	3adant3 1150	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑋 · 1 ) ∈ (Base‘𝑌))
38	37	ad2antrr 738	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑋 · 1 ) ∈ (Base‘𝑌))
39		cpmadugsum.t	. . . . . . . 8 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
40		cpmadugsum.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
41		cpmadugsum.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴)
42	39, 40, 41, 12, 13	mat2pmatbas 22883	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
43	8, 42	syl3an2 1182	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
44	43	ad2antrr 738	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
45		ringcmn 20361	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
46	28, 45	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ CMnd)
47	46	3adant3 1150	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
48	47	ad2antrr 738	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑌 ∈ CMnd)
49		fzfid 14005	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0...𝑠) ∈ Fin)
50	10	ad3antrrr 742	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ (0...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
51		elmapi 8842	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
52		ffvelcdm 7076	. . . . . . . . . . . 12 ⊢ ((𝑏:(0...𝑠)⟶𝐵 ∧ 𝑛 ∈ (0...𝑠)) → (𝑏‘𝑛) ∈ 𝐵)
53	52	ex 417	. . . . . . . . . . 11 ⊢ (𝑏:(0...𝑠)⟶𝐵 → (𝑛 ∈ (0...𝑠) → (𝑏‘𝑛) ∈ 𝐵))
54	51, 53	syl 18	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → (𝑛 ∈ (0...𝑠) → (𝑏‘𝑛) ∈ 𝐵))
55	54	adantl 486	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑛 ∈ (0...𝑠) → (𝑏‘𝑛) ∈ 𝐵))
56	55	imp 411	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ (0...𝑠)) → (𝑏‘𝑛) ∈ 𝐵)
57		elfznn0 13644	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...𝑠) → 𝑛 ∈ ℕ0)
58	57	adantl 486	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ (0...𝑠)) → 𝑛 ∈ ℕ0)
59		cpmadugsum.e	. . . . . . . . 9 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
60	40, 41, 39, 12, 13, 5, 33, 59, 19	mat2pmatscmxcl 22897	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑏‘𝑛) ∈ 𝐵 ∧ 𝑛 ∈ ℕ0)) → ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌))
61	50, 56, 58, 60	syl12anc 849	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ (0...𝑠)) → ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌))
62	61	ralrimiva 3157	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → ∀𝑛 ∈ (0...𝑠)((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌))
63	5, 48, 49, 62	gsummptcl 20032	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))))) ∈ (Base‘𝑌))
64	5, 6, 7, 15, 38, 44, 63	ringsubdir 20387	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (((𝑋 · 1 ) − (𝑇‘𝑀)) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))) = (((𝑋 · 1 ) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))) − ((𝑇‘𝑀) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))))))))
65		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝑛 ↑ 𝑋) = (𝑖 ↑ 𝑋))
66		2fveq3 6886	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝑇‘(𝑏‘𝑛)) = (𝑇‘(𝑏‘𝑖)))
67	65, 66	oveq12d 7428	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))) = ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))
68	67	cbvmptv 5215	. . . . . . . 8 ⊢ (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))
69	68	oveq2i 7421	. . . . . . 7 ⊢ (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))
70	69	oveq2i 7421	. . . . . 6 ⊢ ((𝑋 · 1 ) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))) = ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))
71	69	oveq2i 7421	. . . . . 6 ⊢ ((𝑇‘𝑀) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))) = ((𝑇‘𝑀) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))
72	70, 71	oveq12i 7422	. . . . 5 ⊢ (((𝑋 · 1 ) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))) − ((𝑇‘𝑀) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))))))) = (((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑇‘𝑀) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))))
73		cpmadugsum.g	. . . . . . 7 ⊢ + = (+g‘𝑌)
74	40, 41, 12, 13, 39, 19, 59, 33, 6, 29, 73, 7	cpmadugsumlemF 23033	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑇‘𝑀) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
75	74	anassrs 472	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑇‘𝑀) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
76	72, 75	eqtrid 2810	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (((𝑋 · 1 ) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))) − ((𝑇‘𝑀) × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
77	4, 64, 76	3eqtrd 2802	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝐼 × (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
78	1, 77	sylan9eqr 2820	. 2 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ (𝐽‘𝐼) = (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))))) → (𝐼 × (𝐽‘𝐼)) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
79		cpmadugsum.j	. . . . . . 7 ⊢ 𝐽 = (𝑁 maAdju 𝑃)
80	13, 79, 5	maduf 22798	. . . . . 6 ⊢ (𝑃 ∈ CRing → 𝐽:(Base‘𝑌)⟶(Base‘𝑌))
81	23, 80	syl 18	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝐽:(Base‘𝑌)⟶(Base‘𝑌))
82	81	3ad2ant2 1152	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐽:(Base‘𝑌)⟶(Base‘𝑌))
83	40, 41, 12, 13, 19, 39, 7, 33, 29, 2	chmatcl 22985	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌))
84	8, 83	syl3an2 1182	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌))
85	82, 84	ffvelcdmd 7080	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘𝐼) ∈ (Base‘𝑌))
86	12, 13, 5, 33, 59, 19, 39, 40, 41	pmatcollpw3fi1 22945	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ (𝐽‘𝐼) ∈ (Base‘𝑌)) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐽‘𝐼) = (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))))))
87	85, 86	syld3an3 1436	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐽‘𝐼) = (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))))))
88	78, 87	reximddv2 3224	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐼 × (𝐽‘𝐼)) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ↦ cmpt 5192  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  Fincfn 8939  0cc0 11095  1c1 11096   + caddc 11098   − cmin 11436  ℕcn 12228  ℕ₀cn0 12499  ...cfz 13530  Basecbs 17264  +_gcplusg 17305  ._rcmulr 17306  Scalarcsca 17308   ·_𝑠 cvsca 17309   Σ_g cgsu 17488  -_gcsg 18997  ._gcmg 19128  CMndccmn 19845  mulGrpcmgp 20211  1_rcur 20258  Ringcrg 20310  CRingccrg 20311  LModclmod 20981  var₁cv1 22336  Poly₁cpl1 22337   Mat cmat 22564   maAdju cmadu 22789   matToPolyMat cmat2pmat 22861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-addf 11174  ax-mulf 11175

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-ofr 7675  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-tpos 8218  df-cur 8259  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-xnn0 12573  df-z 12587  df-dec 12707  df-uz 12858  df-rp 13012  df-fz 13531  df-fzo 13679  df-seq 14034  df-exp 14094  df-hash 14363  df-word 14547  df-lsw 14596  df-concat 14604  df-s1 14630  df-substr 14675  df-pfx 14705  df-splice 14783  df-reverse 14792  df-s2 14881  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-unif 17328  df-hom 17329  df-cco 17330  df-0g 17489  df-gsum 17490  df-prds 17495  df-pws 17497  df-mre 17633  df-mrc 17634  df-acs 17636  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-submnd 18837  df-efmnd 18923  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-ghm 19279  df-gim 19324  df-cntz 19382  df-oppg 19411  df-symg 19435  df-pmtr 19507  df-psgn 19556  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-srg 20264  df-ring 20312  df-cring 20313  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-invr 20466  df-dvr 20479  df-rhm 20550  df-subrng 20645  df-subrg 20669  df-drng 20829  df-lmod 20983  df-lss 21053  df-sra 21294  df-rgmod 21295  df-cnfld 21523  df-zring 21597  df-zrh 21653  df-dsmm 21882  df-frlm 21897  df-assa 22003  df-ascl 22005  df-psr 22059  df-mvr 22060  df-mpl 22061  df-opsr 22063  df-psr1 22340  df-vr1 22341  df-ply1 22342  df-coe1 22343  df-mamu 22548  df-mat 22565  df-mdet 22742  df-madu 22791  df-mat2pmat 22864  df-decpmat 22920

This theorem is referenced by:  cpmadugsum  23035