Theorem pmatcollpw3fi1lem1 23109

Description: Lemma 1 for pmatcollpw3fi1 23111. (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 23015	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
5		ringmnd 20475	. . . . . . . . . 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 20516	. . . . . . . . . . 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 9074	. . . . . . . . . 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 8872	. . . . . . . . . . . . 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 7084	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) ∈ 𝐷)
19		elsni 4597	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ {0} → 𝑛 = 0)
20		0nn0 12624	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
21	19, 20	eqeltrdi 2868	. . . . . . . . . . . 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 23063	. . . . . . . . . . 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 3154	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → ∀𝑛 ∈ {0} ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))) ∈ 𝐵)
32	8, 11, 13, 31	gsummptcl 20186	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛))))) ∈ 𝐵)
33		eqid 2760	. . . . . . . . 9 ⊢ (+g‘𝐶) = (+g‘𝐶)
34		eqid 2760	. . . . . . . . 9 ⊢ (0g‘𝐶) = (0g‘𝐶)
35	8, 33, 34	mndrid 18950	. . . . . . . 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 13764	. . . . . . . . . . . 12 ⊢ (0...0) = {0}
38	37	eqcomi 2769	. . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → 𝑙 = 0)
44	43	iftrued 4489	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘0))
45		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0))
46	45	eqcomd 2766	. . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, (𝐺‘0), 0 ) = (𝐺‘𝑛))
50		1nn0 12625	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ0
51	50	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 0 → 1 ∈ ℕ0)
52		nn0uz 13006	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
53	51, 52	eleqtrdi 2870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → 1 ∈ (ℤ≥‘0))
54		eluzfz1 13667	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ (ℤ≥‘0) → 0 ∈ (0...1))
55	53, 54	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → 0 ∈ (0...1))
56		eleq1 2848	. . . . . . . . . . . . . . . . 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 7081	. . . . . . . . . . . . . . . . . 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 7002	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐻‘𝑛) = (𝐺‘𝑛))
66	65	eqcomd 2766	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝐺‘𝑛) = (𝐻‘𝑛))
67	66	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ {0}) → (𝑇‘(𝐺‘𝑛)) = (𝑇‘(𝐻‘𝑛)))
68	67	oveq2d 7436	. . . . . . . . . 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 5190	. . . . . . . . 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 7436	. . . . . . . 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 7455	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (0 + 1) ∈ V)
72	8, 34	mndidcl 18944	. . . . . . . . . . . 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 12466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 + 1) = 1
76	75	eqeq2i 2773	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = (0 + 1) ↔ 𝑛 = 1)
77		ax-1ne0 11272	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≠ 0
78	77	neii 2957	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ 1 = 0
79		eqeq1 2764	. . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . 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 4492	. . . . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . . . 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 2870	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → 1 ∈ (ℤ≥‘0))
92		eluzfz2 13668	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ (ℤ≥‘0) → 1 ∈ (0...1))
93	91, 92	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → 1 ∈ (0...1))
94		eleq1 2848	. . . . . . . . . . . . . . . . . 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 6900	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (0g‘𝐴) ∈ V)
99	40, 89, 97, 98	fvmptd2 7002	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝐻‘𝑛) = (0g‘𝐴))
100	99	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (𝑇‘(0g‘𝐴)))
101	23	fveq2i 6888	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘(𝑁 Mat 𝑅))
102	3	fveq2i 6888	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐶) = (0g‘(𝑁 Mat 𝑃))
103	25, 2, 101, 102	0mat2pmat 23059	. . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 = (0 + 1)) → (𝑇‘(𝐻‘𝑛)) = (0g‘𝐶))
107	106	oveq2d 7436	. . . . . . . . . . 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 23016	. . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
116		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
117	2, 28, 115, 27, 116	ply1moncl 22595	. . . . . . . . . . . . . 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 22570	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
120	3	matsca2 22740	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
121	119, 120	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 = (Scalar‘𝐶))
122	121	eqcomd 2766	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐶) = 𝑃)
123	122	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
124	123	eleq2d 2846	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
128		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
129	127, 26, 128, 34	lmodvs0 21176	. . . . . . . . . . . 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 2795	. . . . . . . . . 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 20171	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → (𝐶 Σg (𝑛 ∈ {(0 + 1)} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))) = (0g‘𝐶))
133	132	eqcomd 2766	. . . . . . . 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 7438	. . . . . . 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 2797	. . . . . 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 2795	. . . 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 11303	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
144	143	snid 4619	. . . . . . . . . . . . . 14 ⊢ 0 ∈ {0}
145	144	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐺:{0}⟶𝐷 → 0 ∈ {0})
146	142, 145	ffvelcdmd 7085	. . . . . . . . . . . 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 22763	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
150	24, 88	ring0cl 20501	. . . . . . . . . . . 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 4528	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝐺‘0), 0 ) ∈ 𝐷)
154	153, 40	fmptd 7114	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) → 𝐻:(0...1)⟶𝐷)
155	75	oveq2i 7431	. . . . . . . . 9 ⊢ (0...(0 + 1)) = (0...1)
156	155	feq2i 6701	. . . . . . . 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 7084	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0})) ∧ 𝑛 ∈ (0...(0 + 1))) → (𝐻‘𝑛) ∈ 𝐷)
159		elfznn0 13757	. . . . . . 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 23063	. . . . . 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 20151	. . . 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 2798	. 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 5195	. . 3 ⊢ (𝑛 ∈ (0...(0 + 1)) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))
167	166	oveq2i 7431	. 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 2811	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 1569   ∈ wcel 2144  Vcvv 3450  ifcif 4481  {csn 4584   ↦ cmpt 5185  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8850  Fincfn 8976  0cc0 11203  1c1 11204   + caddc 11206  ℕ₀cn0 12609  ℤ_≥cuz 12968  ...cfz 13642  Basecbs 17392  +_gcplusg 17433  Scalarcsca 17436   ·_𝑠 cvsca 17437  0_gc0g 17615   Σ_g cgsu 17616  Mndcmnd 18928  ._gcmg 19282  CMndccmn 19999  mulGrpcmgp 20365  Ringcrg 20464  LModclmod 21140  var₁cv1 22499  Poly₁cpl1 22500   Mat cmat 22727   matToPolyMat cmat2pmat 23027

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8180  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-er 8720  df-map 8852  df-pm 8853  df-ixp 8929  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-fsupp 9357  df-sup 9437  df-oi 9507  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-z 12697  df-dec 12818  df-uz 12969  df-fz 13643  df-fzo 13792  df-seq 14148  df-hash 14478  df-struct 17330  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-mulr 17447  df-sca 17449  df-vsca 17450  df-ip 17451  df-tset 17452  df-ple 17453  df-ds 17455  df-hom 17457  df-cco 17458  df-0g 17617  df-gsum 17618  df-prds 17623  df-pws 17625  df-mre 17761  df-mrc 17762  df-acs 17764  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-mhm 18983  df-submnd 18984  df-grp 19152  df-minusg 19153  df-sbg 19154  df-mulg 19283  df-subg 19338  df-ghm 19433  df-cntz 19536  df-cmn 20001  df-abl 20002  df-mgp 20366  df-rng 20380  df-ur 20413  df-ring 20466  df-subrng 20803  df-subrg 20827  df-lmod 21142  df-lss 21212  df-sra 21453  df-rgmod 21454  df-dsmm 22043  df-frlm 22058  df-ascl 22168  df-psr 22222  df-mvr 22223  df-mpl 22224  df-opsr 22226  df-psr1 22503  df-vr1 22504  df-ply1 22505  df-mamu 22711  df-mat 22728  df-mat2pmat 23030

This theorem is used by:  pmatcollpw3fi1lem2  23110