Theorem pmatcollpw3fi1lem1 22973

Description: Lemma 1 for pmatcollpw3fi1 22975. (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 490	. . . . 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 22879	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5		ringmnd 20350	. . . . . . . . . 10 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Mnd)
6	4, 5	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Mnd)
7	6	adantr 486	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐶 ∈ Mnd)
8		pmatcollpw.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
9		ringcmn 20391	. . . . . . . . . . 11 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
10	4, 9	syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ CMnd)
11	10	adantr 486	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐶 ∈ CMnd)
12		snfi 9050	. . . . . . . . . 10 ⊢ {0} ∈ Fin
13	12	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → {0} ∈ Fin)
14		simplll 787	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑁 ∈ Fin)
15		simpllr 788	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑅 ∈ Ring)
16		elmapi 8855	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → 𝐺:{0}⟶𝐷)
17	16	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐺:{0}⟶𝐷)
18	17	ffvelcdmda 7086	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
19		elsni 4611	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {0} → 𝑛 = 0)
20		0nn0 12537	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
21	19, 20	eqeltrdi 2874	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {0} → 𝑛 ∈ ℕ0)
22	21	adantl 487	. . . . . . . . . . 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 22927	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐺‘𝑛) ∈ 𝐷 ∧ 𝑛 ∈ ℕ0)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
30	14, 15, 18, 22, 29	syl22anc 852	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
31	30	ralrimiva 3160	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → ∀𝑛 ∈ {0} ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
32	8, 11, 13, 31	gsummptcl 20062	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))) ∈ 𝐵)
33		eqid 2766	. . . . . . . . 9 ⊢ (+g‘𝐶) = (+g‘𝐶)
34		eqid 2766	. . . . . . . . 9 ⊢ (0g‘𝐶) = (0g‘𝐶)
35	8, 33, 34	mndrid 18838	. . . . . . . 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 596	. . . . . . 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 13674	. . . . . . . . . . . 12 ⊢ (0...0) = {0}
38	37	eqcomi 2775	. . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 𝑛)
42	19	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑛 = 0)
43	41, 42	eqtrd 2801	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
44	43	iftrued 4500	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
45		fveq2 6888	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0))
46	45	eqcomd 2772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (𝐺‘0) = (𝐺‘𝑛))
47	19, 46	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ {0} → (𝐺‘0) = (𝐺‘𝑛))
48	47	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → (𝐺‘0) = (𝐺‘𝑛))
49	44, 48	eqtrd 2801	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘𝑛))
50		1nn0 12538	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ0
51	50	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 0 → 1 ∈ ℕ0)
52		nn0uz 12918	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
53	51, 52	eleqtrdi 2876	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → 1 ∈ (ℤ≥‘0))
54		eluzfz1 13577	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1))
55	53, 54	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → 0 ∈ (0...1))
56		eleq1 2854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...1) ↔ 0 ∈ (0...1)))
57	55, 56	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → 𝑛 ∈ (0...1))
58	19, 57	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ {0} → 𝑛 ∈ (0...1))
59	58	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ (0...1))
60		ffvelcdm 7083	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺:{0}⟶𝐷 ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
61	60	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:{0}⟶𝐷 → (𝑛 ∈ {0} → (𝐺‘𝑛) ∈ 𝐷))
62	16, 61	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → (𝑛 ∈ {0} → (𝐺‘𝑛) ∈ 𝐷))
63	62	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝑛 ∈ {0} → (𝐺‘𝑛) ∈ 𝐷))
64	63	imp 412	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
65	40, 49, 59, 64	fvmptd2 7005	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐻‘𝑛) = (𝐺‘𝑛))
66	65	eqcomd 2772	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) = (𝐻‘𝑛))
67	66	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺‘𝑛)) = (𝑇‘(𝐻‘𝑛)))
68	67	oveq2d 7439	. . . . . . . . . 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 5202	. . . . . . . . 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 7439	. . . . . . . 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 7458	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0 + 1) ∈ V)
72	8, 34	mndidcl 18832	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Mnd → (0g‘𝐶) ∈ 𝐵)
73	6, 72	syl 18	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) ∈ 𝐵)
74	73	adantr 486	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0g‘𝐶) ∈ 𝐵)
75		0p1e1 12379	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 + 1) = 1
76	75	eqeq2i 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = (0 + 1) ↔ 𝑛 = 1)
77		ax-1ne0 11187	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≠ 0
78	77	neii 2963	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 1 = 0
79		eqeq1 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1 → (𝑛 = 0 ↔ 1 = 0))
80	78, 79	mtbiri 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → ¬ 𝑛 = 0)
81	76, 80	sylbi 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (0 + 1) → ¬ 𝑛 = 0)
82	81	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑛 = 0)
83		eqeq1 2770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
84	83	notbid 321	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑛 → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
85	84	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
86	82, 85	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑙 = 0)
87	86	iffalsed 4503	. . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . 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 2876	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → 1 ∈ (ℤ≥‘0))
92		eluzfz2 13578	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ (ℤ≥‘0) → 1 ∈ (0...1))
93	91, 92	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → 1 ∈ (0...1))
94		eleq1 2854	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝑛 ∈ (0...1) ↔ 1 ∈ (0...1)))
95	93, 94	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → 𝑛 ∈ (0...1))
96	76, 95	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (0 + 1) → 𝑛 ∈ (0...1))
97	96	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ (0...1))
98		fvexd 6903	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (0g‘𝐴) ∈ V)
99	40, 89, 97, 98	fvmptd2 7005	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝐻‘𝑛) = (0g‘𝐴))
100	99	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (𝑇‘(0g‘𝐴)))
101	23	fveq2i 6891	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘(𝑁 Mat 𝑅))
102	3	fveq2i 6891	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐶) = (0g‘(𝑁 Mat 𝑃))
103	25, 2, 101, 102	0mat2pmat 22923	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
104	103	ancoms 464	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
105	104	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
106	100, 105	eqtrd 2801	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (0g‘𝐶))
107	106	oveq2d 7439	. . . . . . . . . . 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 22880	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
109	108	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝐶 ∈ LMod)
110		simpllr 788	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑅 ∈ Ring)
111		eleq1 2854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑛 ∈ ℕ0 ↔ 1 ∈ ℕ0))
112	90, 111	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → 𝑛 ∈ ℕ0)
113	76, 112	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (0 + 1) → 𝑛 ∈ ℕ0)
114	113	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ ℕ0)
115		eqid 2766	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
116		eqid 2766	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
117	2, 28, 115, 27, 116	ply1moncl 22462	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
118	110, 114, 117	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
119	2	ply1ring 22437	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
120	3	matsca2 22607	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
121	119, 120	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
122	121	eqcomd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
123	122	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
124	123	eleq2d 2852	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 ↑ 𝑋) ∈ (Base‘𝑃)))
125	124	ad2antrr 739	. . . . . . . . . . . . 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 260	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)))
127		eqid 2766	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
128		eqid 2766	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
129	127, 26, 128, 34	lmodvs0 21047	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ LMod ∧ (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶))) → ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶)) = (0g‘𝐶))
130	109, 126, 129	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → ((𝑛 ↑ 𝑋) ∗ (0g‘𝐶)) = (0g‘𝐶))
131	107, 130	eqtrd 2801	. . . . . . . . . 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 20047	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))) = (0g‘𝐶))
133	132	eqcomd 2772	. . . . . . . 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 7441	. . . . . . 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 2803	. . . . . 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 486	. . . . 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 2801	. . . 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 1127	. . 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 787	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑁 ∈ Fin)
141		simpllr 788	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑅 ∈ Ring)
142		id 23	. . . . . . . . . . . . 13 ⊢ (𝐺:{0}⟶𝐷 → 𝐺:{0}⟶𝐷)
143		c0ex 11218	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
144	143	snid 4633	. . . . . . . . . . . . . 14 ⊢ 0 ∈ {0}
145	144	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐺:{0}⟶𝐷 → 0 ∈ {0})
146	142, 145	ffvelcdmd 7087	. . . . . . . . . . . 12 ⊢ (𝐺:{0}⟶𝐷 → (𝐺‘0) ∈ 𝐷)
147	16, 146	syl 18	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → (𝐺‘0) ∈ 𝐷)
148	147	ad2antlr 740	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → (𝐺‘0) ∈ 𝐷)
149	23	matring 22630	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
150	24, 88	ring0cl 20376	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 0 ∈ 𝐷)
151	149, 150	syl 18	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ 𝐷)
152	151	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → 0 ∈ 𝐷)
153	148, 152	ifcld 4539	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
154	153, 40	fmptd 7116	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐻:(0...1)⟶𝐷)
155	75	oveq2i 7434	. . . . . . . . 9 ⊢ (0...(0 + 1)) = (0...1)
156	155	feq2i 6704	. . . . . . . 8 ⊢ (𝐻:(0...(0 + 1))⟶𝐷 ↔ 𝐻:(0...1)⟶𝐷)
157	154, 156	sylibr 237	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐻:(0...(0 + 1))⟶𝐷)
158	157	ffvelcdmda 7086	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻‘𝑛) ∈ 𝐷)
159		elfznn0 13667	. . . . . . 7 ⊢ (𝑛 ∈ (0...(0 + 1)) → 𝑛 ∈ ℕ0)
160	159	adantl 487	. . . . . 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 22927	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐻‘𝑛) ∈ 𝐷 ∧ 𝑛 ∈ ℕ0)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))) ∈ 𝐵)
162	140, 141, 158, 160, 161	syl22anc 852	. . . . 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 20027	. . . 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 1150	. . 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 2804	. 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 5207	. . 3 ⊢ (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))
167	166	oveq2i 7434	. 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 2817	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))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3458  ifcif 4492  {csn 4594   ↦ cmpt 5197  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423   ↑_m cmap 8833  Fincfn 8952  0cc0 11118  1c1 11119   + caddc 11121  ℕ₀cn0 12522  ℤ_≥cuz 12880  ...cfz 13553  Basecbs 17294  +_gcplusg 17335  Scalarcsca 17338   ·_𝑠 cvsca 17339  0_gc0g 17517   Σ_g cgsu 17518  Mndcmnd 18817  ._gcmg 19158  CMndccmn 19875  mulGrpcmgp 20241  Ringcrg 20340  LModclmod 21011  var₁cv1 22366  Poly₁cpl1 22367   Mat cmat 22594   matToPolyMat cmat2pmat 22891

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-of 7687  df-ofr 7688  df-om 7872  df-1st 7995  df-2nd 7996  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8703  df-map 8835  df-pm 8836  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-sup 9412  df-oi 9482  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12881  df-fz 13554  df-fzo 13702  df-seq 14058  df-hash 14387  df-struct 17232  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-mulr 17349  df-sca 17351  df-vsca 17352  df-ip 17353  df-tset 17354  df-ple 17355  df-ds 17357  df-hom 17359  df-cco 17360  df-0g 17519  df-gsum 17520  df-prds 17525  df-pws 17527  df-mre 17663  df-mrc 17664  df-acs 17666  df-mgm 18723  df-sgrp 18802  df-mnd 18818  df-mhm 18866  df-submnd 18867  df-grp 19028  df-minusg 19029  df-sbg 19030  df-mulg 19159  df-subg 19214  df-ghm 19309  df-cntz 19412  df-cmn 19877  df-abl 19878  df-mgp 20242  df-rng 20256  df-ur 20289  df-ring 20342  df-subrng 20675  df-subrg 20699  df-lmod 21013  df-lss 21083  df-sra 21324  df-rgmod 21325  df-dsmm 21912  df-frlm 21927  df-ascl 22035  df-psr 22089  df-mvr 22090  df-mpl 22091  df-opsr 22093  df-psr1 22370  df-vr1 22371  df-ply1 22372  df-mamu 22578  df-mat 22595  df-mat2pmat 22894

This theorem is used by:  pmatcollpw3fi1lem2  22974