Theorem pmatcollpw3fi1lem1 21935

Description: Lemma 1 for pmatcollpw3fi1 21937. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)

Hypotheses

Ref	Expression
pmatcollpw.p	⊢ 𝑃 = (Poly1‘𝑅)
pmatcollpw.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw.b	⊢ 𝐵 = (Base‘𝐶)
pmatcollpw.m	⊢ ∗ = ( ·𝑠 ‘𝐶)
pmatcollpw.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
pmatcollpw.x	⊢ 𝑋 = (var1‘𝑅)
pmatcollpw.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
pmatcollpw3.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pmatcollpw3.d	⊢ 𝐷 = (Base‘𝐴)
pmatcollpw3fi1lem1.0	⊢ 0 = (0g‘𝐴)
pmatcollpw3fi1lem1.h	⊢ 𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 ))

Assertion

Ref	Expression
pmatcollpw3fi1lem1	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑛,𝑋   ↑ ,𝑛   𝐶,𝑛   𝐵,𝑙   𝑀,𝑙   𝑁,𝑙   𝑅,𝑙   𝐷,𝑙,𝑛   𝐴,𝑙   𝐺,𝑙,𝑛

Allowed substitution hints:   𝐴(𝑛)   𝐶(𝑙)   𝑃(𝑙)   𝑇(𝑛,𝑙)   ↑ (𝑙)   𝐻(𝑛,𝑙)   ∗ (𝑛,𝑙)   𝑋(𝑙)   0 (𝑛,𝑙)

Proof of Theorem pmatcollpw3fi1lem1

Step	Hyp	Ref	Expression
1		simpr 485	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))))
2		pmatcollpw.p	. . . . . . . . . . 11 ⊢ 𝑃 = (Poly1‘𝑅)
3		pmatcollpw.c	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑁 Mat 𝑃)
4	2, 3	pmatring 21841	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5		ringmnd 19793	. . . . . . . . . 10 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Mnd)
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Mnd)
7	6	adantr 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐶 ∈ Mnd)
8		pmatcollpw.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
9		ringcmn 19820	. . . . . . . . . . 11 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
10	4, 9	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ CMnd)
11	10	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐶 ∈ CMnd)
12		snfi 8834	. . . . . . . . . 10 ⊢ {0} ∈ Fin
13	12	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → {0} ∈ Fin)
14		simplll 772	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑁 ∈ Fin)
15		simpllr 773	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑅 ∈ Ring)
16		elmapi 8637	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → 𝐺:{0}⟶𝐷)
17	16	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐺:{0}⟶𝐷)
18	17	ffvelrnda 6961	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
19		elsni 4578	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {0} → 𝑛 = 0)
20		0nn0 12248	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
21	19, 20	eqeltrdi 2847	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {0} → 𝑛 ∈ ℕ0)
22	21	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ ℕ0)
23		pmatcollpw3.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (𝑁 Mat 𝑅)
24		pmatcollpw3.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (Base‘𝐴)
25		pmatcollpw.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
26		pmatcollpw.m	. . . . . . . . . . . 12 ⊢ ∗ = ( ·𝑠 ‘𝐶)
27		pmatcollpw.e	. . . . . . . . . . . 12 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
28		pmatcollpw.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (var1‘𝑅)
29	23, 24, 25, 2, 3, 8, 26, 27, 28	mat2pmatscmxcl 21889	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐺‘𝑛) ∈ 𝐷 ∧ 𝑛 ∈ ℕ0)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
30	14, 15, 18, 22, 29	syl22anc 836	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
31	30	ralrimiva 3103	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → ∀𝑛 ∈ {0} ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
32	8, 11, 13, 31	gsummptcl 19568	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))) ∈ 𝐵)
33		eqid 2738	. . . . . . . . 9 ⊢ (+g‘𝐶) = (+g‘𝐶)
34		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝐶) = (0g‘𝐶)
35	8, 33, 34	mndrid 18406	. . . . . . . 8 ⊢ ((𝐶 ∈ Mnd ∧ (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))) ∈ 𝐵) → ((𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))(+g‘𝐶)(0g‘𝐶)) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))))
36	7, 32, 35	syl2anc 584	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → ((𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))(+g‘𝐶)(0g‘𝐶)) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))))
37		fz0sn 13356	. . . . . . . . . . . 12 ⊢ (0...0) = {0}
38	37	eqcomi 2747	. . . . . . . . . . 11 ⊢ {0} = (0...0)
39	38	a1i 11	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → {0} = (0...0))
40		pmatcollpw3fi1lem1.h	. . . . . . . . . . . . . 14 ⊢ 𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 ))
41		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 𝑛)
42	19	ad2antlr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑛 = 0)
43	41, 42	eqtrd 2778	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
44	43	iftrued 4467	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
45		fveq2 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0))
46	45	eqcomd 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (𝐺‘0) = (𝐺‘𝑛))
47	19, 46	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ {0} → (𝐺‘0) = (𝐺‘𝑛))
48	47	ad2antlr 724	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → (𝐺‘0) = (𝐺‘𝑛))
49	44, 48	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘𝑛))
50		1nn0 12249	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ0
51	50	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 0 → 1 ∈ ℕ0)
52		nn0uz 12620	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
53	51, 52	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → 1 ∈ (ℤ≥‘0))
54		eluzfz1 13263	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1))
55	53, 54	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → 0 ∈ (0...1))
56		eleq1 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...1) ↔ 0 ∈ (0...1)))
57	55, 56	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → 𝑛 ∈ (0...1))
58	19, 57	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ {0} → 𝑛 ∈ (0...1))
59	58	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ (0...1))
60		ffvelrn 6959	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺:{0}⟶𝐷 ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
61	60	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:{0}⟶𝐷 → (𝑛 ∈ {0} → (𝐺‘𝑛) ∈ 𝐷))
62	16, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → (𝑛 ∈ {0} → (𝐺‘𝑛) ∈ 𝐷))
63	62	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝑛 ∈ {0} → (𝐺‘𝑛) ∈ 𝐷))
64	63	imp 407	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
65	40, 49, 59, 64	fvmptd2 6883	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐻‘𝑛) = (𝐺‘𝑛))
66	65	eqcomd 2744	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) = (𝐻‘𝑛))
67	66	fveq2d 6778	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺‘𝑛)) = (𝑇‘(𝐻‘𝑛)))
68	67	oveq2d 7291	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))
69	39, 68	mpteq12dva 5163	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))) = (𝑛 ∈ (0...0) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))
70	69	oveq2d 7291	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))))
71		ovexd 7310	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0 + 1) ∈ V)
72	8, 34	mndidcl 18400	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Mnd → (0g‘𝐶) ∈ 𝐵)
73	6, 72	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) ∈ 𝐵)
74	73	adantr 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0g‘𝐶) ∈ 𝐵)
75		0p1e1 12095	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 + 1) = 1
76	75	eqeq2i 2751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = (0 + 1) ↔ 𝑛 = 1)
77		ax-1ne0 10940	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≠ 0
78	77	neii 2945	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 1 = 0
79		eqeq1 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1 → (𝑛 = 0 ↔ 1 = 0))
80	78, 79	mtbiri 327	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → ¬ 𝑛 = 0)
81	76, 80	sylbi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (0 + 1) → ¬ 𝑛 = 0)
82	81	ad2antlr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑛 = 0)
83		eqeq1 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
84	83	notbid 318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑛 → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
85	84	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
86	82, 85	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑙 = 0)
87	86	iffalsed 4470	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = 0 )
88		pmatcollpw3fi1lem1.0	. . . . . . . . . . . . . . . 16 ⊢ 0 = (0g‘𝐴)
89	87, 88	eqtrdi 2794	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (0g‘𝐴))
90	50	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → 1 ∈ ℕ0)
91	90, 52	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → 1 ∈ (ℤ≥‘0))
92		eluzfz2 13264	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ (ℤ≥‘0) → 1 ∈ (0...1))
93	91, 92	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → 1 ∈ (0...1))
94		eleq1 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝑛 ∈ (0...1) ↔ 1 ∈ (0...1)))
95	93, 94	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → 𝑛 ∈ (0...1))
96	76, 95	sylbi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (0 + 1) → 𝑛 ∈ (0...1))
97	96	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ (0...1))
98		fvexd 6789	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (0g‘𝐴) ∈ V)
99	40, 89, 97, 98	fvmptd2 6883	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝐻‘𝑛) = (0g‘𝐴))
100	99	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (𝑇‘(0g‘𝐴)))
101	23	fveq2i 6777	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘(𝑁 Mat 𝑅))
102	3	fveq2i 6777	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐶) = (0g‘(𝑁 Mat 𝑃))
103	25, 2, 101, 102	0mat2pmat 21885	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
104	103	ancoms 459	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
105	104	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
106	100, 105	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (0g‘𝐶))
107	106	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))) = ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶)))
108	2, 3	pmatlmod 21842	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
109	108	ad2antrr 723	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝐶 ∈ LMod)
110		simpllr 773	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑅 ∈ Ring)
111		eleq1 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑛 ∈ ℕ0 ↔ 1 ∈ ℕ0))
112	90, 111	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → 𝑛 ∈ ℕ0)
113	76, 112	sylbi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (0 + 1) → 𝑛 ∈ ℕ0)
114	113	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ ℕ0)
115		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
116		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
117	2, 28, 115, 27, 116	ply1moncl 21442	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
118	110, 114, 117	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
119	2	ply1ring 21419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
120	3	matsca2 21569	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
121	119, 120	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
122	121	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
123	122	fveq2d 6778	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
124	123	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 ↑ 𝑋) ∈ (Base‘𝑃)))
125	124	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 ↑ 𝑋) ∈ (Base‘𝑃)))
126	118, 125	mpbird 256	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)))
127		eqid 2738	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
128		eqid 2738	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
129	127, 26, 128, 34	lmodvs0 20157	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶)) = (0g‘𝐶))
130	109, 126, 129	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶)) = (0g‘𝐶))
131	107, 130	eqtrd 2778	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))) = (0g‘𝐶))
132	8, 7, 71, 74, 131	gsumsnd 19553	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))) = (0g‘𝐶))
133	132	eqcomd 2744	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0g‘𝐶) = (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))))
134	70, 133	oveq12d 7293	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → ((𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))(+g‘𝐶)(0g‘𝐶)) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))(+g‘𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))))
135	36, 134	eqtr3d 2780	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))(+g‘𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))))
136	135	adantr 481	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))(+g‘𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))))
137	1, 136	eqtrd 2778	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → 𝑀 = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))(+g‘𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))))
138	137	3impa 1109	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → 𝑀 = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))(+g‘𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))))
139	20	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 0 ∈ ℕ0)
140		simplll 772	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑁 ∈ Fin)
141		simpllr 773	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑅 ∈ Ring)
142		id 22	. . . . . . . . . . . . 13 ⊢ (𝐺:{0}⟶𝐷 → 𝐺:{0}⟶𝐷)
143		c0ex 10969	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
144	143	snid 4597	. . . . . . . . . . . . . 14 ⊢ 0 ∈ {0}
145	144	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐺:{0}⟶𝐷 → 0 ∈ {0})
146	142, 145	ffvelrnd 6962	. . . . . . . . . . . 12 ⊢ (𝐺:{0}⟶𝐷 → (𝐺‘0) ∈ 𝐷)
147	16, 146	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → (𝐺‘0) ∈ 𝐷)
148	147	ad2antlr 724	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → (𝐺‘0) ∈ 𝐷)
149	23	matring 21592	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
150	24, 88	ring0cl 19808	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 0 ∈ 𝐷)
151	149, 150	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ 𝐷)
152	151	ad2antrr 723	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → 0 ∈ 𝐷)
153	148, 152	ifcld 4505	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
154	153, 40	fmptd 6988	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐻:(0...1)⟶𝐷)
155	75	oveq2i 7286	. . . . . . . . 9 ⊢ (0...(0 + 1)) = (0...1)
156	155	feq2i 6592	. . . . . . . 8 ⊢ (𝐻:(0...(0 + 1))⟶𝐷 ↔ 𝐻:(0...1)⟶𝐷)
157	154, 156	sylibr 233	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐻:(0...(0 + 1))⟶𝐷)
158	157	ffvelrnda 6961	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻‘𝑛) ∈ 𝐷)
159		elfznn0 13349	. . . . . . 7 ⊢ (𝑛 ∈ (0...(0 + 1)) → 𝑛 ∈ ℕ0)
160	159	adantl 482	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑛 ∈ ℕ0)
161	23, 24, 25, 2, 3, 8, 26, 27, 28	mat2pmatscmxcl 21889	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐻‘𝑛) ∈ 𝐷 ∧ 𝑛 ∈ ℕ0)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))) ∈ 𝐵)
162	140, 141, 158, 160, 161	syl22anc 836	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))) ∈ 𝐵)
163	8, 33, 11, 139, 162	gsummptfzsplit 19533	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))(+g‘𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))))
164	163	3adant3 1131	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))) = ((𝐶 Σg (𝑛 ∈ (0...0) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))(+g‘𝐶)(𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))))
165	138, 164	eqtr4d 2781	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))))
166	155	mpteq1i 5170	. . 3 ⊢ (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))
167	166	oveq2i 7286	. 2 ⊢ (𝐶 Σg (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))
168	165, 167	eqtrdi 2794	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ifcif 4459  {csn 4561   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  Fincfn 8733  0cc0 10871  1c1 10872   + caddc 10874  ℕ₀cn0 12233  ℤ_≥cuz 12582  ...cfz 13239  Basecbs 16912  +_gcplusg 16962  Scalarcsca 16965   ·_𝑠 cvsca 16966  0_gc0g 17150   Σ_g cgsu 17151  Mndcmnd 18385  ._gcmg 18700  CMndccmn 19386  mulGrpcmgp 19720  Ringcrg 19783  LModclmod 20123  var₁cv1 21347  Poly₁cpl1 21348   Mat cmat 21554   matToPolyMat cmat2pmat 21853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-ofr 7534  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-hom 16986  df-cco 16987  df-0g 17152  df-gsum 17153  df-prds 17158  df-pws 17160  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-cntz 18923  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-subrg 20022  df-lmod 20125  df-lss 20194  df-sra 20434  df-rgmod 20435  df-dsmm 20939  df-frlm 20954  df-ascl 21062  df-psr 21112  df-mvr 21113  df-mpl 21114  df-opsr 21116  df-psr1 21351  df-vr1 21352  df-ply1 21353  df-mamu 21533  df-mat 21555  df-mat2pmat 21856

This theorem is referenced by:  pmatcollpw3fi1lem2  21936