Theorem chfacfpmmulgsum2 23210

Description: Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)

Hypotheses

Ref	Expression
cayhamlem1.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cayhamlem1.b	⊢ 𝐵 = (Base‘𝐴)
cayhamlem1.p	⊢ 𝑃 = (Poly1‘𝑅)
cayhamlem1.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cayhamlem1.r	⊢ × = (.r‘𝑌)
cayhamlem1.s	⊢ − = (-g‘𝑌)
cayhamlem1.0	⊢ 0 = (0g‘𝑌)
cayhamlem1.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cayhamlem1.g	⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
cayhamlem1.e	⊢ ↑ = (.g‘(mulGrp‘𝑌))
chfacfpmmulgsum.p	⊢ + = (+g‘𝑌)

Assertion

Ref	Expression
chfacfpmmulgsum2	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑌   𝑛,𝑏   𝑛,𝑠   0 ,𝑛   𝐵,𝑖   𝑖,𝐺   𝑖,𝑀   𝑖,𝑁   𝑅,𝑖   𝑇,𝑖   × ,𝑖   ↑ ,𝑖   𝑖,𝑠   𝑖,𝑏   𝑇,𝑛,𝑖   𝑖,𝑌   × ,𝑛   − ,𝑛

Allowed substitution hints:   𝐴(𝑖, 𝑛, 𝑠, 𝑏)   𝐵(𝑠, 𝑏)   𝑃(𝑖, 𝑛, 𝑠, 𝑏)   + (𝑖, 𝑛, 𝑠, 𝑏)   𝑅(𝑠, 𝑏)   𝑇(𝑠, 𝑏)   × (𝑠, 𝑏)   ↑ (𝑛, 𝑠, 𝑏)   𝐺(𝑛, 𝑠, 𝑏)   𝑀(𝑠, 𝑏)   − (𝑖, 𝑠, 𝑏)   𝑁(𝑠, 𝑏)   𝑌(𝑠, 𝑏)   0 (𝑖, 𝑠, 𝑏)

Proof of Theorem chfacfpmmulgsum2

Step	Hyp	Ref	Expression
1		cayhamlem1.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		cayhamlem1.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
3		cayhamlem1.p	. . 3 ⊢ 𝑃 = (Poly1‘𝑅)
4		cayhamlem1.y	. . 3 ⊢ 𝑌 = (𝑁 Mat 𝑃)
5		cayhamlem1.r	. . 3 ⊢ × = (.r‘𝑌)
6		cayhamlem1.s	. . 3 ⊢ − = (-g‘𝑌)
7		cayhamlem1.0	. . 3 ⊢ 0 = (0g‘𝑌)
8		cayhamlem1.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
9		cayhamlem1.g	. . 3 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
10		cayhamlem1.e	. . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑌))
11		chfacfpmmulgsum.p	. . 3 ⊢ + = (+g‘𝑌)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	chfacfpmmulgsum 23209	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
13		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌)
14		crngring 20499	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
15	14	anim2i 629	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
16	3, 4	pmatring 23037	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
17	15, 16	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
18	17	3adant3 1150	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
19	18	ad2antrr 739	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑌 ∈ Ring)
20		eqid 2763	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
21	20	ringmgp 20492	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mnd)
22		mndmgm 18956	. . . . . . . . . . 11 ⊢ ((mulGrp‘𝑌) ∈ Mnd → (mulGrp‘𝑌) ∈ Mgm)
23	21, 22	syl 18	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mgm)
24	18, 23	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑌) ∈ Mgm)
25	24	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (mulGrp‘𝑌) ∈ Mgm)
26		elfznn 13711	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
27	26	adantl 487	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ)
28	8, 1, 2, 3, 4	mat2pmatbas 23071	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
29	14, 28	syl3an2 1182	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
30	29	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘𝑀) ∈ (Base‘𝑌))
31	20, 13	mgpbas 20392	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
32	31, 10	mulgnncl 19325	. . . . . . . 8 ⊢ (((mulGrp‘𝑌) ∈ Mgm ∧ 𝑖 ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → (𝑖 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
33	25, 27, 30, 32	syl3anc 1399	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
34	15	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
35	34	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
36		elmapi 8894	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
37	36	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
38	37	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
39	38	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
40		1nn0 12646	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ0
41	40	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 1 ∈ ℕ0)
42		nnnn0 12637	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
43	42	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 𝑠 ∈ ℕ0)
44		nnge1 12390	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ℕ → 1 ≤ 𝑠)
45	44	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 1 ≤ 𝑠)
46		elfz2nn0 13776	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (0...𝑠) ↔ (1 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 1 ≤ 𝑠))
47	41, 43, 45, 46	syl3anbrc 1362	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 1 ∈ (0...𝑠))
48		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ (1...𝑠))
49		fz0fzdiffz0 13795	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ (0...𝑠) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
50	47, 48, 49	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
51	50	ex 418	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
52	51	ad2antrl 741	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
53	52	imp 412	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
54	39, 53	ffvelcdmd 7106	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏‘(𝑖 − 1)) ∈ 𝐵)
55		df-3an 1105	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵))
56	35, 54, 55	sylanbrc 595	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵))
57	8, 1, 2, 3, 4	mat2pmatbas 23071	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑖 − 1))) ∈ (Base‘𝑌))
58	56, 57	syl 18	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘(𝑖 − 1))) ∈ (Base‘𝑌))
59	34, 16	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
60	59	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑌 ∈ Ring)
61		simpl1 1210	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑁 ∈ Fin)
62	14	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
63	62	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑅 ∈ Ring)
64	42	ad2antrl 741	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑠 ∈ ℕ0)
65	61, 63, 64	3jca 1146	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
66	65	adantr 486	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
67		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → 𝑏 ∈ (𝐵 ↑m (0...𝑠)))
68	67	adantl 487	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑m (0...𝑠)))
69		fz1ssfz0 13781	. . . . . . . . . . 11 ⊢ (1...𝑠) ⊆ (0...𝑠)
70	69	sseli 3909	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ (0...𝑠))
71	68, 70	anim12i 625	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠)))
72	1, 2, 3, 4, 8	m2pmfzmap 23092	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
73	66, 71, 72	syl2anc 596	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
74	13, 5	ringcl 20504	. . . . . . . 8 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
75	60, 30, 73, 74	syl3anc 1399	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
76	13, 5, 6, 19, 33, 58, 75	ringsubdi 20565	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) = (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))
77	13, 5	ringass 20507	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Ring ∧ ((𝑖 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌) ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))) → (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))) = ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
78	60, 33, 30, 73, 77	syl13anc 1400	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))) = ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))
79	78	eqcomd 2769	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) = (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))
80	29, 31	eleqtrdi 2874	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘(mulGrp‘𝑌)))
81	80	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘(mulGrp‘𝑌)))
82		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))
83		eqid 2763	. . . . . . . . . . . . 13 ⊢ (+g‘(mulGrp‘𝑌)) = (+g‘(mulGrp‘𝑌))
84	82, 10, 83	mulgnnp1 19318	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘(mulGrp‘𝑌))) → ((𝑖 + 1) ↑ (𝑇‘𝑀)) = ((𝑖 ↑ (𝑇‘𝑀))(+g‘(mulGrp‘𝑌))(𝑇‘𝑀)))
85	26, 81, 84	syl2anr 609	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 + 1) ↑ (𝑇‘𝑀)) = ((𝑖 ↑ (𝑇‘𝑀))(+g‘(mulGrp‘𝑌))(𝑇‘𝑀)))
86	20, 5	mgpplusg 20391	. . . . . . . . . . . . . 14 ⊢ × = (+g‘(mulGrp‘𝑌))
87	86	eqcomi 2772	. . . . . . . . . . . . 13 ⊢ (+g‘(mulGrp‘𝑌)) = ×
88	87	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (+g‘(mulGrp‘𝑌)) = × )
89	88	oveqd 7457	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ (𝑇‘𝑀))(+g‘(mulGrp‘𝑌))(𝑇‘𝑀)) = ((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘𝑀)))
90	85, 89	eqtrd 2798	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 + 1) ↑ (𝑇‘𝑀)) = ((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘𝑀)))
91	90	eqcomd 2769	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘𝑀)) = ((𝑖 + 1) ↑ (𝑇‘𝑀)))
92	91	oveq1d 7455	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))) = (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))
93	79, 92	eqtrd 2798	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))) = (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))
94	93	oveq2d 7456	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) = (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))
95	76, 94	eqtrd 2798	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))) = (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))
96	95	mpteq2dva 5201	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))) = (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖))))))
97	96	oveq2d 7456	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))))
98	97	oveq1d 7455	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
99	12, 98	eqtrd 2798	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ifcif 4485   class class class wbr 5106   ↦ cmpt 5189  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872  Fincfn 8998  0cc0 11224  1c1 11225   + caddc 11227   < clt 11367   ≤ cle 11368   − cmin 11565  ℕcn 12359  ℕ₀cn0 12630  ...cfz 13663  Basecbs 17413  +_gcplusg 17454  ._rcmulr 17455  0_gc0g 17636   Σ_g cgsu 17637  Mgmcmgm 18840  Mndcmnd 18949  -_gcsg 19172  ._gcmg 19303  mulGrpcmgp 20387  Ringcrg 20486  CRingccrg 20487  Poly₁cpl1 22522   Mat cmat 22749   matToPolyMat cmat2pmat 23049

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-ofr 7715  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-er 8742  df-map 8874  df-pm 8875  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-sup 9458  df-oi 9528  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-rp 13145  df-fz 13664  df-fzo 13813  df-seq 14169  df-hash 14499  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-hom 17478  df-cco 17479  df-0g 17638  df-gsum 17639  df-prds 17644  df-pws 17646  df-mre 17782  df-mrc 17783  df-acs 17785  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-submnd 19005  df-grp 19173  df-minusg 19174  df-sbg 19175  df-mulg 19304  df-subg 19359  df-ghm 19454  df-cntz 19558  df-cmn 20023  df-abl 20024  df-mgp 20388  df-rng 20402  df-ur 20435  df-ring 20488  df-cring 20489  df-subrng 20825  df-subrg 20849  df-lmod 21164  df-lss 21234  df-sra 21475  df-rgmod 21476  df-dsmm 22065  df-frlm 22080  df-ascl 22190  df-psr 22244  df-mpl 22246  df-opsr 22248  df-psr1 22525  df-ply1 22527  df-mamu 22733  df-mat 22750  df-mat2pmat 23052

This theorem is used by:  cayhamlem1  23211