Theorem chfacfpmmulgsum2 20670

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 20669	. 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 2622	. . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌)
14		crngring 18558	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
15	14	anim2i 593	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
16	3, 4	pmatring 20498	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
18	17	3adant3 1081	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
19	18	ad2antrr 762	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑌 ∈ Ring)
20		eqid 2622	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
21	20	ringmgp 18553	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mnd)
22		mndmgm 17300	. . . . . . . . . . 11 ⊢ ((mulGrp‘𝑌) ∈ Mnd → (mulGrp‘𝑌) ∈ Mgm)
23	21, 22	syl 17	. . . . . . . . . 10 ⊢ (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mgm)
24	18, 23	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝑌) ∈ Mgm)
25	24	ad2antrr 762	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (mulGrp‘𝑌) ∈ Mgm)
26		elfznn 12370	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
27	26	adantl 482	. . . . . . . 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 20531	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
29	14, 28	syl3an2 1360	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝑌))
30	29	ad2antrr 762	. . . . . . . 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 18495	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
32	31, 10	mulgnncl 17556	. . . . . . . 8 ⊢ (((mulGrp‘𝑌) ∈ Mgm ∧ 𝑖 ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘𝑌)) → (𝑖 ↑ (𝑇‘𝑀)) ∈ (Base‘𝑌))
33	25, 27, 30, 32	syl3anc 1326	. . . . . . 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 1081	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
35	34	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
36		elmapi 7879	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
37	36	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
38	37	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
39	38	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
40		1nn0 11308	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ0
41	40	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 1 ∈ ℕ0)
42		nnnn0 11299	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
43	42	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 𝑠 ∈ ℕ0)
44		nnge1 11046	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ℕ → 1 ≤ 𝑠)
45	44	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 1 ≤ 𝑠)
46		elfz2nn0 12431	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (0...𝑠) ↔ (1 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 1 ≤ 𝑠))
47	41, 43, 45, 46	syl3anbrc 1246	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 1 ∈ (0...𝑠))
48		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ (1...𝑠))
49		fz0fzdiffz0 12448	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ (0...𝑠) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
50	47, 48, 49	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
51	50	ex 450	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℕ → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
52	51	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
53	52	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
54	53	imp 445	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
55	39, 54	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏‘(𝑖 − 1)) ∈ 𝐵)
56		df-3an 1039	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵))
57	35, 55, 56	sylanbrc 698	. . . . . . . 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))
58	8, 1, 2, 3, 4	mat2pmatbas 20531	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑖 − 1))) ∈ (Base‘𝑌))
59	57, 58	syl 17	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘(𝑖 − 1))) ∈ (Base‘𝑌))
60	34, 16	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑌 ∈ Ring)
61	60	ad2antrr 762	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑌 ∈ Ring)
62		simpl1 1064	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑁 ∈ Fin)
63	14	3ad2ant2 1083	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
64	63	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑅 ∈ Ring)
65	42	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑠 ∈ ℕ0)
66	65	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑠 ∈ ℕ0)
67	62, 64, 66	3jca 1242	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
68	67	adantr 481	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
69		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠))) → 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))
70	69	adantl 482	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))
71		1eluzge0 11732	. . . . . . . . . . . 12 ⊢ 1 ∈ (ℤ≥‘0)
72		fzss1 12380	. . . . . . . . . . . 12 ⊢ (1 ∈ (ℤ≥‘0) → (1...𝑠) ⊆ (0...𝑠))
73	71, 72	ax-mp 5	. . . . . . . . . . 11 ⊢ (1...𝑠) ⊆ (0...𝑠)
74	73	sseli 3599	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑠) → 𝑖 ∈ (0...𝑠))
75	70, 74	anim12i 590	. . . . . . . . 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)))
76	1, 2, 3, 4, 8	m2pmfzmap 20552	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
77	68, 75, 76	syl2anc 693	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌))
78	13, 5	ringcl 18561	. . . . . . . 8 ⊢ ((𝑌 ∈ Ring ∧ (𝑇‘𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘𝑖)) ∈ (Base‘𝑌)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
79	61, 30, 77, 78	syl3anc 1326	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))) ∈ (Base‘𝑌))
80	13, 5, 6, 19, 33, 59, 79	ringsubdi 18599	. . . . . 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))))))
81	13, 5	ringass 18564	. . . . . . . . . 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)))))
82	61, 33, 30, 77, 81	syl13anc 1328	. . . . . . . . 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)))))
83	82	eqcomd 2628	. . . . . . . 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))))
84	29, 31	syl6eleq 2711	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘(mulGrp‘𝑌)))
85	84	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑇‘𝑀) ∈ (Base‘(mulGrp‘𝑌)))
86		eqid 2622	. . . . . . . . . . . . 13 ⊢ (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))
87		eqid 2622	. . . . . . . . . . . . 13 ⊢ (+g‘(mulGrp‘𝑌)) = (+g‘(mulGrp‘𝑌))
88	86, 10, 87	mulgnnp1 17549	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℕ ∧ (𝑇‘𝑀) ∈ (Base‘(mulGrp‘𝑌))) → ((𝑖 + 1) ↑ (𝑇‘𝑀)) = ((𝑖 ↑ (𝑇‘𝑀))(+g‘(mulGrp‘𝑌))(𝑇‘𝑀)))
89	26, 85, 88	syl2anr 495	. . . . . . . . . . 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)))
90	20, 5	mgpplusg 18493	. . . . . . . . . . . . . 14 ⊢ × = (+g‘(mulGrp‘𝑌))
91	90	eqcomi 2631	. . . . . . . . . . . . 13 ⊢ (+g‘(mulGrp‘𝑌)) = ×
92	91	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (+g‘(mulGrp‘𝑌)) = × )
93	92	oveqd 6667	. . . . . . . . . . 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)))
94	89, 93	eqtrd 2656	. . . . . . . . . 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)))
95	94	eqcomd 2628	. . . . . . . . 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)))
96	95	oveq1d 6665	. . . . . . . 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))))
97	83, 96	eqtrd 2656	. . . . . . 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))))
98	97	oveq2d 6666	. . . . . 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)))))
99	80, 98	eqtrd 2656	. . . . 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)))))
100	99	mpteq2dva 4744	. . . 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))))))
101	100	oveq2d 6666	. . 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)))))))
102	101	oveq1d 6665	. 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))))))
103	12, 102	eqtrd 2656	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))))))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ℤ_≥cuz 11687  ...cfz 12326  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  0_gc0g 16100   Σ_g cgsu 16101  Mgmcmgm 17240  Mndcmnd 17294  -_gcsg 17424  ._gcmg 17540  mulGrpcmgp 18489  Ringcrg 18547  CRingccrg 18548  Poly₁cpl1 19547   Mat cmat 20213   matToPolyMat cmat2pmat 20509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-ascl 19314  df-psr 19356  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-ply1 19552  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-mat2pmat 20512

This theorem is referenced by:  cayhamlem1  20671