Theorem pmatcollpw3fi1lem2 22289

Description: Lemma 2 for pmatcollpw3fi1 22290. (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)

Assertion

Ref	Expression
pmatcollpw3fi1lem2	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑓 ∈ (𝐷 ↑m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))

Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑛,𝑋   ↑ ,𝑛   𝐵,𝑠,𝑛   𝐶,𝑛   𝑀,𝑠   𝑁,𝑠   𝑅,𝑠   𝐵,𝑓   𝐶,𝑓,𝑛   𝐷,𝑓   𝑓,𝑀   𝑓,𝑁   𝑅,𝑓   𝑇,𝑓   𝑓,𝑋   ↑ ,𝑓   ∗ ,𝑓,𝑠   𝐷,𝑛   𝐴,𝑓,𝑛,𝑠   𝐶,𝑠   𝐷,𝑠   𝑇,𝑠   𝑋,𝑠   ↑ ,𝑠   ∗ ,𝑠

Allowed substitution hints:   𝑃(𝑓,𝑠)   𝑇(𝑛)   ∗ (𝑛)

Proof of Theorem pmatcollpw3fi1lem2

Dummy variables 𝑙 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6891	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
2	1	fveq2d 6896	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑇‘(𝑓‘𝑛)) = (𝑇‘(𝑔‘𝑛)))
3	2	oveq2d 7425	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛))))
4	3	mpteq2dv 5251	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))) = (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))
5	4	oveq2d 7425	. . . 4 ⊢ (𝑓 = 𝑔 → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛))))))
6	5	eqeq2d 2744	. . 3 ⊢ (𝑓 = 𝑔 → (𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))))
7	6	cbvrexvw 3236	. 2 ⊢ (∃𝑓 ∈ (𝐷 ↑m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) ↔ ∃𝑔 ∈ (𝐷 ↑m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛))))))
8		crngring 20068	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	8	anim2i 618	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
10	9	3adant3 1133	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
11	10	ad2antrr 725	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
12		simplr 768	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) → 𝑔 ∈ (𝐷 ↑m {0}))
13		simpr 486	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛))))))
14		pmatcollpw.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅)
15		pmatcollpw.c	. . . . . 6 ⊢ 𝐶 = (𝑁 Mat 𝑃)
16		pmatcollpw.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
17		pmatcollpw.m	. . . . . 6 ⊢ ∗ = ( ·𝑠 ‘𝐶)
18		pmatcollpw.e	. . . . . 6 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
19		pmatcollpw.x	. . . . . 6 ⊢ 𝑋 = (var1‘𝑅)
20		pmatcollpw.t	. . . . . 6 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
21		pmatcollpw3.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
22		pmatcollpw3.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐴)
23		eqid 2733	. . . . . 6 ⊢ (0g‘𝐴) = (0g‘𝐴)
24		eqid 2733	. . . . . 6 ⊢ (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))
25	14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24	pmatcollpw3fi1lem1 22288	. . . . 5 ⊢ (((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‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛))))))
26	11, 12, 13, 25	syl3anc 1372	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛))))))
27		1nn 12223	. . . . . 6 ⊢ 1 ∈ ℕ
28	27	a1i 11	. . . . 5 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛)))))) → 1 ∈ ℕ)
29		oveq2 7417	. . . . . . . 8 ⊢ (𝑠 = 1 → (0...𝑠) = (0...1))
30	29	oveq2d 7425	. . . . . . 7 ⊢ (𝑠 = 1 → (𝐷 ↑m (0...𝑠)) = (𝐷 ↑m (0...1)))
31	29	mpteq1d 5244	. . . . . . . . 9 ⊢ (𝑠 = 1 → (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))
32	31	oveq2d 7425	. . . . . . . 8 ⊢ (𝑠 = 1 → (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))
33	32	eqeq2d 2744	. . . . . . 7 ⊢ (𝑠 = 1 → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
34	30, 33	rexeqbidv 3344	. . . . . 6 ⊢ (𝑠 = 1 → (∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) ↔ ∃𝑓 ∈ (𝐷 ↑m (0...1))𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
35	34	adantl 483	. . . . 5 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛)))))) ∧ 𝑠 = 1) → (∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) ↔ ∃𝑓 ∈ (𝐷 ↑m (0...1))𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
36		elmapi 8843	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝐷 ↑m {0}) → 𝑔:{0}⟶𝐷)
37		c0ex 11208	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
38	37	snid 4665	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0}
39	38	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...1) → 0 ∈ {0})
40		ffvelcdm 7084	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:{0}⟶𝐷 ∧ 0 ∈ {0}) → (𝑔‘0) ∈ 𝐷)
41	39, 40	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:{0}⟶𝐷 ∧ 𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
42	41	ex 414	. . . . . . . . . . . . . 14 ⊢ (𝑔:{0}⟶𝐷 → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
43	36, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝐷 ↑m {0}) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
44	43	adantl 483	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
45	44	imp 408	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
46	21	matring 21945	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
47	8, 46	sylan2 594	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
48	47	3adant3 1133	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
49	22, 23	ring0cl 20084	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → (0g‘𝐴) ∈ 𝐷)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0g‘𝐴) ∈ 𝐷)
51	50	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → (0g‘𝐴) ∈ 𝐷)
52	45, 51	ifcld 4575	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)) ∈ 𝐷)
53	52	fmpttd 7115	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))):(0...1)⟶𝐷)
54	22	fvexi 6906	. . . . . . . . . . 11 ⊢ 𝐷 ∈ V
55		ovex 7442	. . . . . . . . . . 11 ⊢ (0...1) ∈ V
56	54, 55	pm3.2i 472	. . . . . . . . . 10 ⊢ (𝐷 ∈ V ∧ (0...1) ∈ V)
57		elmapg 8833	. . . . . . . . . 10 ⊢ ((𝐷 ∈ V ∧ (0...1) ∈ V) → ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) ∈ (𝐷 ↑m (0...1)) ↔ (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))):(0...1)⟶𝐷))
58	56, 57	mp1i 13	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) → ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) ∈ (𝐷 ↑m (0...1)) ↔ (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))):(0...1)⟶𝐷))
59	53, 58	mpbird 257	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) ∈ (𝐷 ↑m (0...1)))
60	59	adantr 482	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) → (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) ∈ (𝐷 ↑m (0...1)))
61		fveq1 6891	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) → (𝑓‘𝑛) = ((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛))
62	61	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) → (𝑇‘(𝑓‘𝑛)) = (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛)))
63	62	oveq2d 7425	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛))))
64	63	mpteq2dv 5251	. . . . . . . . . 10 ⊢ (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) → (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛)))))
65	64	oveq2d 7425	. . . . . . . . 9 ⊢ (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) → (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛))))))
66	65	eqeq2d 2744	. . . . . . . 8 ⊢ (𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴))) → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛)))))))
67	66	adantl 483	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) ∧ 𝑓 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))) → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) ↔ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛)))))))
68	60, 67	rspcedv 3606	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) → (𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛))))) → ∃𝑓 ∈ (𝐷 ↑m (0...1))𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
69	68	imp 408	. . . . 5 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛)))))) → ∃𝑓 ∈ (𝐷 ↑m (0...1))𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))
70	28, 35, 69	rspcedvd 3615	. . . 4 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘((𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝑔‘0), (0g‘𝐴)))‘𝑛)))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))
71	26, 70	mpdan 686	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))
72	71	rexlimdva2 3158	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑔 ∈ (𝐷 ↑m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
73	7, 72	biimtrid 241	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑓 ∈ (𝐷 ↑m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  Vcvv 3475  ifcif 4529  {csn 4629   ↦ cmpt 5232  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409   ↑_m cmap 8820  Fincfn 8939  0cc0 11110  1c1 11111  ℕcn 12212  ...cfz 13484  Basecbs 17144   ·_𝑠 cvsca 17201  0_gc0g 17385   Σ_g cgsu 17386  ._gcmg 18950  mulGrpcmgp 19987  Ringcrg 20056  CRingccrg 20057  var₁cv1 21700  Poly₁cpl1 21701   Mat cmat 21907   matToPolyMat cmat2pmat 22206

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-ofr 7671  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-sup 9437  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-fzo 13628  df-seq 13967  df-hash 14291  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-hom 17221  df-cco 17222  df-0g 17387  df-gsum 17388  df-prds 17393  df-pws 17395  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mulg 18951  df-subg 19003  df-ghm 19090  df-cntz 19181  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-cring 20059  df-subrg 20317  df-lmod 20473  df-lss 20543  df-sra 20785  df-rgmod 20786  df-dsmm 21287  df-frlm 21302  df-ascl 21410  df-psr 21462  df-mvr 21463  df-mpl 21464  df-opsr 21466  df-psr1 21704  df-vr1 21705  df-ply1 21706  df-mamu 21886  df-mat 21908  df-mat2pmat 22209

This theorem is referenced by:  pmatcollpw3fi1  22290