Theorem pmatcollpw3fi1lem1 22808

Description: Lemma 1 for pmatcollpw3fi1 22810. (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 484	. . . . 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 22714	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5		ringmnd 20261	. . . . . . . . . 10 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Mnd)
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Mnd)
7	6	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐶 ∈ Mnd)
8		pmatcollpw.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
9		ringcmn 20296	. . . . . . . . . . 11 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
10	4, 9	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ CMnd)
11	10	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐶 ∈ CMnd)
12		snfi 9082	. . . . . . . . . 10 ⊢ {0} ∈ Fin
13	12	a1i 11	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → {0} ∈ Fin)
14		simplll 775	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑁 ∈ Fin)
15		simpllr 776	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑅 ∈ Ring)
16		elmapi 8888	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → 𝐺:{0}⟶𝐷)
17	16	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐺:{0}⟶𝐷)
18	17	ffvelcdmda 7104	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
19		elsni 4648	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {0} → 𝑛 = 0)
20		0nn0 12539	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
21	19, 20	eqeltrdi 2847	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {0} → 𝑛 ∈ ℕ0)
22	21	adantl 481	. . . . . . . . . . 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 22762	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐺‘𝑛) ∈ 𝐷 ∧ 𝑛 ∈ ℕ0)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
30	14, 15, 18, 22, 29	syl22anc 839	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
31	30	ralrimiva 3144	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → ∀𝑛 ∈ {0} ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
32	8, 11, 13, 31	gsummptcl 20000	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))) ∈ 𝐵)
33		eqid 2735	. . . . . . . . 9 ⊢ (+g‘𝐶) = (+g‘𝐶)
34		eqid 2735	. . . . . . . . 9 ⊢ (0g‘𝐶) = (0g‘𝐶)
35	8, 33, 34	mndrid 18781	. . . . . . . 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 13664	. . . . . . . . . . . 12 ⊢ (0...0) = {0}
38	37	eqcomi 2744	. . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 𝑛)
42	19	ad2antlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑛 = 0)
43	41, 42	eqtrd 2775	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
44	43	iftrued 4539	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
45		fveq2 6907	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0))
46	45	eqcomd 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (𝐺‘0) = (𝐺‘𝑛))
47	19, 46	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ {0} → (𝐺‘0) = (𝐺‘𝑛))
48	47	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → (𝐺‘0) = (𝐺‘𝑛))
49	44, 48	eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘𝑛))
50		1nn0 12540	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ0
51	50	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 0 → 1 ∈ ℕ0)
52		nn0uz 12918	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
53	51, 52	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → 1 ∈ (ℤ≥‘0))
54		eluzfz1 13568	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1))
55	53, 54	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → 0 ∈ (0...1))
56		eleq1 2827	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...1) ↔ 0 ∈ (0...1)))
57	55, 56	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → 𝑛 ∈ (0...1))
58	19, 57	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ {0} → 𝑛 ∈ (0...1))
59	58	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → 𝑛 ∈ (0...1))
60		ffvelcdm 7101	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺:{0}⟶𝐷 ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
61	60	ex 412	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝑛 ∈ {0} → (𝐺‘𝑛) ∈ 𝐷))
64	63	imp 406	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
65	40, 49, 59, 64	fvmptd2 7024	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐻‘𝑛) = (𝐺‘𝑛))
66	65	eqcomd 2741	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) = (𝐻‘𝑛))
67	66	fveq2d 6911	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺‘𝑛)) = (𝑇‘(𝐻‘𝑛)))
68	67	oveq2d 7447	. . . . . . . . . 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 5237	. . . . . . . . 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 7447	. . . . . . . 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 7466	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0 + 1) ∈ V)
72	8, 34	mndidcl 18775	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Mnd → (0g‘𝐶) ∈ 𝐵)
73	6, 72	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) ∈ 𝐵)
74	73	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0g‘𝐶) ∈ 𝐵)
75		0p1e1 12386	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 + 1) = 1
76	75	eqeq2i 2748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = (0 + 1) ↔ 𝑛 = 1)
77		ax-1ne0 11222	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≠ 0
78	77	neii 2940	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 1 = 0
79		eqeq1 2739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1 → (𝑛 = 0 ↔ 1 = 0))
80	78, 79	mtbiri 327	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → ¬ 𝑛 = 0)
81	76, 80	sylbi 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (0 + 1) → ¬ 𝑛 = 0)
82	81	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑛 = 0)
83		eqeq1 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
84	83	notbid 318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑛 → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
85	84	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → (¬ 𝑙 = 0 ↔ ¬ 𝑛 = 0))
86	82, 85	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) ∧ 𝑙 = 𝑛) → ¬ 𝑙 = 0)
87	86	iffalsed 4542	. . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . 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 13569	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ (ℤ≥‘0) → 1 ∈ (0...1))
93	91, 92	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → 1 ∈ (0...1))
94		eleq1 2827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝑛 ∈ (0...1) ↔ 1 ∈ (0...1)))
95	93, 94	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → 𝑛 ∈ (0...1))
96	76, 95	sylbi 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (0 + 1) → 𝑛 ∈ (0...1))
97	96	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ (0...1))
98		fvexd 6922	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (0g‘𝐴) ∈ V)
99	40, 89, 97, 98	fvmptd2 7024	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝐻‘𝑛) = (0g‘𝐴))
100	99	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (𝑇‘(0g‘𝐴)))
101	23	fveq2i 6910	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘(𝑁 Mat 𝑅))
102	3	fveq2i 6910	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐶) = (0g‘(𝑁 Mat 𝑃))
103	25, 2, 101, 102	0mat2pmat 22758	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
104	103	ancoms 458	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
105	104	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
106	100, 105	eqtrd 2775	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (0g‘𝐶))
107	106	oveq2d 7447	. . . . . . . . . . 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 22715	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
109	108	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝐶 ∈ LMod)
110		simpllr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑅 ∈ Ring)
111		eleq1 2827	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑛 ∈ ℕ0 ↔ 1 ∈ ℕ0))
112	90, 111	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → 𝑛 ∈ ℕ0)
113	76, 112	sylbi 217	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (0 + 1) → 𝑛 ∈ ℕ0)
114	113	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → 𝑛 ∈ ℕ0)
115		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
116		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
117	2, 28, 115, 27, 116	ply1moncl 22290	. . . . . . . . . . . . . 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 22265	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
120	3	matsca2 22442	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
121	119, 120	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
122	121	eqcomd 2741	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
123	122	fveq2d 6911	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
124	123	eleq2d 2825	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)) ↔ (𝑛 ↑ 𝑋) ∈ (Base‘𝑃)))
125	124	ad2antrr 726	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑛 ↑ 𝑋) ∈ (Base‘(Scalar‘𝐶)))
127		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
128		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
129	127, 26, 128, 34	lmodvs0 20911	. . . . . . . . . . . 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 2775	. . . . . . . . . 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 19985	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))) = (0g‘𝐶))
133	132	eqcomd 2741	. . . . . . . 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 7449	. . . . . . 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 2777	. . . . . 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 480	. . . . 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 2775	. . . 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 775	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → 𝑁 ∈ Fin)
141		simpllr 776	. . . . . 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 11253	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
144	143	snid 4667	. . . . . . . . . . . . . 14 ⊢ 0 ∈ {0}
145	144	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐺:{0}⟶𝐷 → 0 ∈ {0})
146	142, 145	ffvelcdmd 7105	. . . . . . . . . . . 12 ⊢ (𝐺:{0}⟶𝐷 → (𝐺‘0) ∈ 𝐷)
147	16, 146	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝐷 ↑m {0}) → (𝐺‘0) ∈ 𝐷)
148	147	ad2antlr 727	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → (𝐺‘0) ∈ 𝐷)
149	23	matring 22465	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
150	24, 88	ring0cl 20281	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 0 ∈ 𝐷)
151	149, 150	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ 𝐷)
152	151	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → 0 ∈ 𝐷)
153	148, 152	ifcld 4577	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
154	153, 40	fmptd 7134	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐻:(0...1)⟶𝐷)
155	75	oveq2i 7442	. . . . . . . . 9 ⊢ (0...(0 + 1)) = (0...1)
156	155	feq2i 6729	. . . . . . . 8 ⊢ (𝐻:(0...(0 + 1))⟶𝐷 ↔ 𝐻:(0...1)⟶𝐷)
157	154, 156	sylibr 234	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐻:(0...(0 + 1))⟶𝐷)
158	157	ffvelcdmda 7104	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻‘𝑛) ∈ 𝐷)
159		elfznn0 13657	. . . . . . 7 ⊢ (𝑛 ∈ (0...(0 + 1)) → 𝑛 ∈ ℕ0)
160	159	adantl 481	. . . . . 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 22762	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝐻‘𝑛) ∈ 𝐷 ∧ 𝑛 ∈ ℕ0)) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))) ∈ 𝐵)
162	140, 141, 158, 160, 161	syl22anc 839	. . . . 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 19965	. . . 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 2778	. 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 5244	. . 3 ⊢ (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))
167	166	oveq2i 7442	. 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 2791	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ifcif 4531  {csn 4631   ↦ cmpt 5231  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865  Fincfn 8984  0cc0 11153  1c1 11154   + caddc 11156  ℕ₀cn0 12524  ℤ_≥cuz 12876  ...cfz 13544  Basecbs 17245  +_gcplusg 17298  Scalarcsca 17301   ·_𝑠 cvsca 17302  0_gc0g 17486   Σ_g cgsu 17487  Mndcmnd 18760  ._gcmg 19098  CMndccmn 19813  mulGrpcmgp 20152  Ringcrg 20251  LModclmod 20875  var₁cv1 22193  Poly₁cpl1 22194   Mat cmat 22427   matToPolyMat cmat2pmat 22726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-mamu 22411  df-mat 22428  df-mat2pmat 22729

This theorem is referenced by:  pmatcollpw3fi1lem2  22809