Theorem cpmadugsumfi 22934

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 7404	. . 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 7411	. . . 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 2762	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
6		cpmadugsum.r	. . . . 5 ⊢ × = (.r‘𝑌)
7		cpmadugsum.s	. . . . 5 ⊢ − = (-g‘𝑌)
8		crngring 20291	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	8	anim2i 626	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
10	9	3adant3 1145	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
11	10	ad2antrr 736	. . . . . 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 22749	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
15	11, 14	syl 17	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑌 ∈ Ring)
16	12, 13	pmatlmod 22750	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod)
17	8, 16	sylan2 602	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
18	8	adantl 485	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
19		cpmadugsum.x	. . . . . . . . . . 11 ⊢ 𝑋 = (var1‘𝑅)
20		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
21	19, 12, 20	vr1cl 22276	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
22	18, 21	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘𝑃))
23	12	ply1crng 22257	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
24	13	matsca2 22477	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
25	23, 24	sylan2 602	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
26	25	fveq2d 6871	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘𝑃) = (Base‘(Scalar‘𝑌)))
27	22, 26	eleqtrd 2864	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
28	8, 14	sylan2 602	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
29		cpmadugsum.1	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑌)
30	5, 29	ringidcl 20311	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 1 ∈ (Base‘𝑌))
31	28, 30	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 1 ∈ (Base‘𝑌))
32		eqid 2762	. . . . . . . . 9 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
33		cpmadugsum.m	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑌)
34		eqid 2762	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
35	5, 32, 33, 34	lmodvscl 20942	. . . . . . . 8 ⊢ ((𝑌 ∈ LMod ∧ 𝑋 ∈ (Base‘(Scalar‘𝑌)) ∧ 1 ∈ (Base‘𝑌)) → (𝑋 · 1 ) ∈ (Base‘𝑌))
36	17, 27, 31, 35	syl3anc 1390	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑋 · 1 ) ∈ (Base‘𝑌))
37	36	3adant3 1145	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑋 · 1 ) ∈ (Base‘𝑌))
38	37	ad2antrr 736	. . . . 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 22783	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
43	8, 42	syl3an2 1177	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
44	43	ad2antrr 736	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑇‘𝑀) ∈ (Base‘𝑌))
45		ringcmn 20328	. . . . . . . . 9 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
46	28, 45	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ CMnd)
47	46	3adant3 1145	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ CMnd)
48	47	ad2antrr 736	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑌 ∈ CMnd)
49		fzfid 13986	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (0...𝑠) ∈ Fin)
50	10	ad3antrrr 740	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ (0...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
51		elmapi 8830	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
52		ffvelcdm 7062	. . . . . . . . . . . 12 ⊢ ((𝑏:(0...𝑠)⟶𝐵 ∧ 𝑛 ∈ (0...𝑠)) → (𝑏‘𝑛) ∈ 𝐵)
53	52	ex 416	. . . . . . . . . . 11 ⊢ (𝑏:(0...𝑠)⟶𝐵 → (𝑛 ∈ (0...𝑠) → (𝑏‘𝑛) ∈ 𝐵))
54	51, 53	syl 17	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → (𝑛 ∈ (0...𝑠) → (𝑏‘𝑛) ∈ 𝐵))
55	54	adantl 485	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑛 ∈ (0...𝑠) → (𝑏‘𝑛) ∈ 𝐵))
56	55	imp 410	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝑛 ∈ (0...𝑠)) → (𝑏‘𝑛) ∈ 𝐵)
57		elfznn0 13625	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...𝑠) → 𝑛 ∈ ℕ0)
58	57	adantl 485	. . . . . . . 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 22797	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑏‘𝑛) ∈ 𝐵 ∧ 𝑛 ∈ ℕ0)) → ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))) ∈ (Base‘𝑌))
61	50, 56, 58, 60	syl12anc 847	. . . . . . 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 3154	. . . . . 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 20007	. . . . 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 20354	. . . 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 7403	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝑛 ↑ 𝑋) = (𝑖 ↑ 𝑋))
66		2fveq3 6872	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝑇‘(𝑏‘𝑛)) = (𝑇‘(𝑏‘𝑖)))
67	65, 66	oveq12d 7414	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))) = ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))
68	67	cbvmptv 5204	. . . . . . . 8 ⊢ (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))
69	68	oveq2i 7407	. . . . . . 7 ⊢ (𝑌 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(𝑏‘𝑛))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))
70	69	oveq2i 7407	. . . . . 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 7407	. . . . . 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 7408	. . . . 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 22933	. . . . . 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 471	. . . . 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 2809	. . . 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 2801	. . 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 2819	. 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 22698	. . . . . 6 ⊢ (𝑃 ∈ CRing → 𝐽:(Base‘𝑌)⟶(Base‘𝑌))
81	23, 80	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝐽:(Base‘𝑌)⟶(Base‘𝑌))
82	81	3ad2ant2 1147	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐽:(Base‘𝑌)⟶(Base‘𝑌))
83	40, 41, 12, 13, 19, 39, 7, 33, 29, 2	chmatcl 22885	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌))
84	8, 83	syl3an2 1177	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌))
85	82, 84	ffvelcdmd 7066	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘𝐼) ∈ (Base‘𝑌))
86	12, 13, 5, 33, 59, 19, 39, 40, 41	pmatcollpw3fi1 22845	. . 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 1428	. 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 3221	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 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∃wrex 3086   ↦ cmpt 5181  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808  Fincfn 8927  0cc0 11073  1c1 11074   + caddc 11076   − cmin 11414  ℕcn 12210  ℕ₀cn0 12481  ...cfz 13512  Basecbs 17245  +_gcplusg 17286  ._rcmulr 17287  Scalarcsca 17289   ·_𝑠 cvsca 17290   Σ_g cgsu 17469  -_gcsg 18977  ._gcmg 19109  CMndccmn 19820  mulGrpcmgp 20186  1_rcur 20227  Ringcrg 20279  CRingccrg 20280  LModclmod 20924  var₁cv1 22235  Poly₁cpl1 22236   Mat cmat 22464   maAdju cmadu 22689   matToPolyMat cmat2pmat 22761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-addf 11152  ax-mulf 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-xor 1532  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-ofr 7661  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-tpos 8206  df-cur 8247  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-sup 9388  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-xnn0 12555  df-z 12569  df-dec 12689  df-uz 12840  df-rp 12994  df-fz 13513  df-fzo 13660  df-seq 14015  df-exp 14075  df-hash 14344  df-word 14527  df-lsw 14576  df-concat 14584  df-s1 14610  df-substr 14655  df-pfx 14685  df-splice 14763  df-reverse 14772  df-s2 14861  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-starv 17301  df-sca 17302  df-vsca 17303  df-ip 17304  df-tset 17305  df-ple 17306  df-ds 17308  df-unif 17309  df-hom 17310  df-cco 17311  df-0g 17470  df-gsum 17471  df-prds 17476  df-pws 17478  df-mre 17614  df-mrc 17615  df-acs 17617  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-mhm 18817  df-submnd 18818  df-efmnd 18903  df-grp 18978  df-minusg 18979  df-sbg 18980  df-mulg 19110  df-subg 19165  df-ghm 19254  df-gim 19299  df-cntz 19357  df-oppg 19386  df-symg 19410  df-pmtr 19482  df-psgn 19531  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20228  df-srg 20233  df-ring 20281  df-cring 20282  df-oppr 20382  df-dvdsr 20402  df-unit 20403  df-invr 20433  df-dvr 20446  df-rhm 20517  df-subrng 20592  df-subrg 20616  df-drng 20777  df-lmod 20926  df-lss 20996  df-sra 21237  df-rgmod 21238  df-cnfld 21422  df-zring 21496  df-zrh 21552  df-dsmm 21781  df-frlm 21796  df-assa 21902  df-ascl 21904  df-psr 21958  df-mvr 21959  df-mpl 21960  df-opsr 21962  df-psr1 22239  df-vr1 22240  df-ply1 22241  df-coe1 22242  df-mamu 22448  df-mat 22465  df-mdet 22642  df-madu 22691  df-mat2pmat 22764  df-decpmat 22820

This theorem is referenced by:  cpmadugsum  22935