Theorem pmatcollpw3fi1lem1 21396

Description: Lemma 1 for pmatcollpw3fi1 21398. (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 487	. . . . 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 21303	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5		ringmnd 19308	. . . . . . . . . 10 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Mnd)
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Mnd)
7	6	adantr 483	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐶 ∈ Mnd)
8		pmatcollpw.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
9		ringcmn 19333	. . . . . . . . . . 11 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
10	4, 9	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ CMnd)
11	10	adantr 483	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐶 ∈ CMnd)
12		snfi 8596	. . . . . . . . . 10 ⊢ {0} ∈ Fin
13	12	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → {0} ∈ Fin)
14		simplll 773	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑁 ∈ Fin)
15		simpllr 774	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑅 ∈ Ring)
16		elmapi 8430	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → 𝐺:{0}⟶𝐷)
17	16	adantl 484	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐺:{0}⟶𝐷)
18	17	ffvelrnda 6853	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
19		elsni 4586	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {0} → 𝑛 = 0)
20		0nn0 11915	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
21	19, 20	eqeltrdi 2923	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {0} → 𝑛 ∈ ℕ0)
22	21	adantl 484	. . . . . . . . . . 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 21350	. . . . . . . . . . 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 3184	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → ∀𝑛 ∈ {0} ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
32	8, 11, 13, 31	gsummptcl 19089	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))) ∈ 𝐵)
33		eqid 2823	. . . . . . . . 9 ⊢ (+g‘𝐶) = (+g‘𝐶)
34		eqid 2823	. . . . . . . . 9 ⊢ (0g‘𝐶) = (0g‘𝐶)
35	8, 33, 34	mndrid 17934	. . . . . . . 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 586	. . . . . . 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 13010	. . . . . . . . . . . 12 ⊢ (0...0) = {0}
38	37	eqcomi 2832	. . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 𝑛)
42	19	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑛 = 0)
43	41, 42	eqtrd 2858	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
44	43	iftrued 4477	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
45		fveq2 6672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0))
46	45	eqcomd 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (𝐺‘0) = (𝐺‘𝑛))
47	19, 46	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ {0} → (𝐺‘0) = (𝐺‘𝑛))
48	47	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → (𝐺‘0) = (𝐺‘𝑛))
49	44, 48	eqtrd 2858	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘𝑛))
50		1nn0 11916	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ0
51	50	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 0 → 1 ∈ ℕ0)
52		nn0uz 12283	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
53	51, 52	eleqtrdi 2925	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → 1 ∈ (ℤ≥‘0))
54		eluzfz1 12917	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1))
55	53, 54	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → 0 ∈ (0...1))
56		eleq1 2902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...1) ↔ 0 ∈ (0...1)))
57	55, 56	mpbird 259	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → 𝑛 ∈ (0...1))
58	19, 57	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ {0} → 𝑛 ∈ (0...1))
59	58	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ (0...1))
60		ffvelrn 6851	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺:{0}⟶𝐷 ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
61	60	ex 415	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝑛 ∈ {0} → (𝐺‘𝑛) ∈ 𝐷))
64	63	imp 409	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
65	40, 49, 59, 64	fvmptd2 6778	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐻‘𝑛) = (𝐺‘𝑛))
66	65	eqcomd 2829	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) = (𝐻‘𝑛))
67	66	fveq2d 6676	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺‘𝑛)) = (𝑇‘(𝐻‘𝑛)))
68	67	oveq2d 7174	. . . . . . . . . 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 5152	. . . . . . . . 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 7174	. . . . . . . 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 7193	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0 + 1) ∈ V)
72	8, 34	mndidcl 17928	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Mnd → (0g‘𝐶) ∈ 𝐵)
73	6, 72	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) ∈ 𝐵)
74	73	adantr 483	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0g‘𝐶) ∈ 𝐵)
75		0p1e1 11762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 + 1) = 1
76	75	eqeq2i 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = (0 + 1) ↔ 𝑛 = 1)
77		ax-1ne0 10608	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≠ 0
78	77	neii 3020	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 1 = 0
79		eqeq1 2827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1 → (𝑛 = 0 ↔ 1 = 0))
80	78, 79	mtbiri 329	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → ¬ 𝑛 = 0)
81	76, 80	sylbi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (0 + 1) → ¬ 𝑛 = 0)
82	81	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑛 = 0)
83		eqeq1 2827	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
84	83	notbid 320	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑛 → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
85	84	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
86	82, 85	mpbird 259	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑙 = 0)
87	86	iffalsed 4480	. . . . . . . . . . . . . . . 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	syl6eq 2874	. . . . . . . . . . . . . . 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 2925	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → 1 ∈ (ℤ≥‘0))
92		eluzfz2 12918	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ (ℤ≥‘0) → 1 ∈ (0...1))
93	91, 92	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → 1 ∈ (0...1))
94		eleq1 2902	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝑛 ∈ (0...1) ↔ 1 ∈ (0...1)))
95	93, 94	mpbird 259	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → 𝑛 ∈ (0...1))
96	76, 95	sylbi 219	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (0 + 1) → 𝑛 ∈ (0...1))
97	96	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ (0...1))
98		fvexd 6687	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (0g‘𝐴) ∈ V)
99	40, 89, 97, 98	fvmptd2 6778	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝐻‘𝑛) = (0g‘𝐴))
100	99	fveq2d 6676	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (𝑇‘(0g‘𝐴)))
101	23	fveq2i 6675	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘(𝑁 Mat 𝑅))
102	3	fveq2i 6675	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐶) = (0g‘(𝑁 Mat 𝑃))
103	25, 2, 101, 102	0mat2pmat 21346	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
104	103	ancoms 461	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
105	104	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
106	100, 105	eqtrd 2858	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (0g‘𝐶))
107	106	oveq2d 7174	. . . . . . . . . . 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 21304	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
109	108	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝐶 ∈ LMod)
110		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑅 ∈ Ring)
111		eleq1 2902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑛 ∈ ℕ0 ↔ 1 ∈ ℕ0))
112	90, 111	mpbird 259	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → 𝑛 ∈ ℕ0)
113	76, 112	sylbi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (0 + 1) → 𝑛 ∈ ℕ0)
114	113	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ ℕ0)
115		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
116		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
117	2, 28, 115, 27, 116	ply1moncl 20441	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
118	110, 114, 117	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
119	2	ply1ring 20418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
120	3	matsca2 21031	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
121	119, 120	sylan2 594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
122	121	eqcomd 2829	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
123	122	fveq2d 6676	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
124	123	eleq2d 2900	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 ↑ 𝑋) ∈ (Base‘𝑃)))
125	124	ad2antrr 724	. . . . . . . . . . . . 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 259	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)))
127		eqid 2823	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
128		eqid 2823	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
129	127, 26, 128, 34	lmodvs0 19670	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶)) = (0g‘𝐶))
130	109, 126, 129	syl2anc 586	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶)) = (0g‘𝐶))
131	107, 130	eqtrd 2858	. . . . . . . . . 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 19074	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))) = (0g‘𝐶))
133	132	eqcomd 2829	. . . . . . . 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 7176	. . . . . . 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 2860	. . . . . 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 483	. . . . 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 2858	. . . 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 1106	. . 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 773	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑁 ∈ Fin)
141		simpllr 774	. . . . . 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 10637	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
144	143	snid 4603	. . . . . . . . . . . . . 14 ⊢ 0 ∈ {0}
145	144	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐺:{0}⟶𝐷 → 0 ∈ {0})
146	142, 145	ffvelrnd 6854	. . . . . . . . . . . 12 ⊢ (𝐺:{0}⟶𝐷 → (𝐺‘0) ∈ 𝐷)
147	16, 146	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → (𝐺‘0) ∈ 𝐷)
148	147	ad2antlr 725	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → (𝐺‘0) ∈ 𝐷)
149	23	matring 21054	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
150	24, 88	ring0cl 19321	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 0 ∈ 𝐷)
151	149, 150	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ 𝐷)
152	151	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → 0 ∈ 𝐷)
153	148, 152	ifcld 4514	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
154	153, 40	fmptd 6880	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐻:(0...1)⟶𝐷)
155	75	oveq2i 7169	. . . . . . . . 9 ⊢ (0...(0 + 1)) = (0...1)
156	155	feq2i 6508	. . . . . . . 8 ⊢ (𝐻:(0...(0 + 1))⟶𝐷 ↔ 𝐻:(0...1)⟶𝐷)
157	154, 156	sylibr 236	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐻:(0...(0 + 1))⟶𝐷)
158	157	ffvelrnda 6853	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻‘𝑛) ∈ 𝐷)
159		elfznn0 13003	. . . . . . 7 ⊢ (𝑛 ∈ (0...(0 + 1)) → 𝑛 ∈ ℕ0)
160	159	adantl 484	. . . . . 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 21350	. . . . . 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 19054	. . . 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 1128	. . 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 2861	. 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 5158	. . 3 ⊢ (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))
167	166	oveq2i 7169	. 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	syl6eq 2874	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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ifcif 4469  {csn 4569   ↦ cmpt 5148  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ↑_m cmap 8408  Fincfn 8511  0cc0 10539  1c1 10540   + caddc 10542  ℕ₀cn0 11900  ℤ_≥cuz 12246  ...cfz 12895  Basecbs 16485  +_gcplusg 16567  Scalarcsca 16570   ·_𝑠 cvsca 16571  0_gc0g 16715   Σ_g cgsu 16716  Mndcmnd 17913  ._gcmg 18226  CMndccmn 18908  mulGrpcmgp 19241  Ringcrg 19299  LModclmod 19636  var₁cv1 20346  Poly₁cpl1 20347   Mat cmat 21018   matToPolyMat cmat2pmat 21314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-ot 4578  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-ofr 7412  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-hom 16591  df-cco 16592  df-0g 16717  df-gsum 16718  df-prds 16723  df-pws 16725  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-grp 18108  df-minusg 18109  df-sbg 18110  df-mulg 18227  df-subg 18278  df-ghm 18358  df-cntz 18449  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-ring 19301  df-subrg 19535  df-lmod 19638  df-lss 19706  df-sra 19946  df-rgmod 19947  df-ascl 20089  df-psr 20138  df-mvr 20139  df-mpl 20140  df-opsr 20142  df-psr1 20350  df-vr1 20351  df-ply1 20352  df-dsmm 20878  df-frlm 20893  df-mamu 20997  df-mat 21019  df-mat2pmat 21317

This theorem is referenced by:  pmatcollpw3fi1lem2  21397