Theorem pmatcollpw3fi1lem2 23106

Description: Lemma 2 for pmatcollpw3fi1 23107. (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 6885	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
2	1	fveq2d 6890	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑇‘(𝑓‘𝑛)) = (𝑇‘(𝑔‘𝑛)))
3	2	oveq2d 7437	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛))))
4	3	mpteq2dv 5199	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))) = (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛)))))
5	4	oveq2d 7437	. . . 4 ⊢ (𝑓 = 𝑔 → (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑔‘𝑛))))))
6	5	eqeq2d 2772	. . 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 3242	. 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 20473	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	8	anim2i 629	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
10	9	3adant3 1150	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
11	10	ad2antrr 739	. . . . 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 781	. . . . 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 490	. . . . 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 2761	. . . . . 6 ⊢ (0g‘𝐴) = (0g‘𝐴)
24		eqid 2761	. . . . . 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 23105	. . . . 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 1398	. . . 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 12347	. . . . . 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 7429	. . . . . . . 8 ⊢ (𝑠 = 1 → (0...𝑠) = (0...1))
30	29	oveq2d 7437	. . . . . . 7 ⊢ (𝑠 = 1 → (𝐷 ↑m (0...𝑠)) = (𝐷 ↑m (0...1)))
31	29	mpteq1d 5195	. . . . . . . . 9 ⊢ (𝑠 = 1 → (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))) = (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))
32	31	oveq2d 7437	. . . . . . . 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 2772	. . . . . . 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 3336	. . . . . 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 487	. . . . 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 8870	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝐷 ↑m {0}) → 𝑔:{0}⟶𝐷)
37		c0ex 11301	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
38	37	snid 4623	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0}
39	38	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...1) → 0 ∈ {0})
40		ffvelcdm 7082	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:{0}⟶𝐷 ∧ 0 ∈ {0}) → (𝑔‘0) ∈ 𝐷)
41	39, 40	sylan2 605	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:{0}⟶𝐷 ∧ 𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
42	41	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑔:{0}⟶𝐷 → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
43	36, 42	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝐷 ↑m {0}) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
44	43	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) → (𝑙 ∈ (0...1) → (𝑔‘0) ∈ 𝐷))
45	44	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → (𝑔‘0) ∈ 𝐷)
46	21	matring 22759	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
47	8, 46	sylan2 605	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
48	47	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
49	22, 23	ring0cl 20497	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → (0g‘𝐴) ∈ 𝐷)
50	48, 49	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (0g‘𝐴) ∈ 𝐷)
51	50	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑔 ∈ (𝐷 ↑m {0})) ∧ 𝑙 ∈ (0...1)) → (0g‘𝐴) ∈ 𝐷)
52	45, 51	ifcld 4529	. . . . . . . . . 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 7116	. . . . . . . . 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 6900	. . . . . . . . . . 11 ⊢ 𝐷 ∈ V
55		ovex 7454	. . . . . . . . . . 11 ⊢ (0...1) ∈ V
56	54, 55	pm3.2i 476	. . . . . . . . . 10 ⊢ (𝐷 ∈ V ∧ (0...1) ∈ V)
57		elmapg 8860	. . . . . . . . . 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 14	. . . . . . . . 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 260	. . . . . . . 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 486	. . . . . . 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 6885	. . . . . . . . . . . . 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 6890	. . . . . . . . . . . 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 7437	. . . . . . . . . . 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 5199	. . . . . . . . . 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 7437	. . . . . . . . 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 2772	. . . . . . . 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 487	. . . . . . 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 3570	. . . . . 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 412	. . . . 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 3579	. . . 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 700	. . 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 3166	. 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 245	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)))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451  ifcif 4482  {csn 4584   ↦ cmpt 5186  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ↑_m cmap 8848  Fincfn 8974  0cc0 11201  1c1 11202  ℕcn 12336  ...cfz 13640  Basecbs 17388   ·_𝑠 cvsca 17433  0_gc0g 17611   Σ_g cgsu 17612  ._gcmg 19278  mulGrpcmgp 20361  Ringcrg 20460  CRingccrg 20461  var₁cv1 22495  Poly₁cpl1 22496   Mat cmat 22723   matToPolyMat cmat2pmat 23023

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7752  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-pss 4035  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-of 7694  df-ofr 7695  df-om 7879  df-1st 8002  df-2nd 8003  df-supp 8179  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-2o 8478  df-er 8718  df-map 8850  df-pm 8851  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9355  df-sup 9435  df-oi 9505  df-card 10021  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-nn 12337  df-2 12406  df-3 12407  df-4 12408  df-5 12409  df-6 12410  df-7 12411  df-8 12412  df-9 12413  df-n0 12608  df-z 12695  df-dec 12816  df-uz 12967  df-fz 13641  df-fzo 13790  df-seq 14146  df-hash 14476  df-struct 17326  df-sets 17343  df-slot 17361  df-ndx 17373  df-base 17389  df-ress 17410  df-plusg 17442  df-mulr 17443  df-sca 17445  df-vsca 17446  df-ip 17447  df-tset 17448  df-ple 17449  df-ds 17451  df-hom 17453  df-cco 17454  df-0g 17613  df-gsum 17614  df-prds 17619  df-pws 17621  df-mre 17757  df-mrc 17758  df-acs 17760  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-mhm 18979  df-submnd 18980  df-grp 19148  df-minusg 19149  df-sbg 19150  df-mulg 19279  df-subg 19334  df-ghm 19429  df-cntz 19532  df-cmn 19997  df-abl 19998  df-mgp 20362  df-rng 20376  df-ur 20409  df-ring 20462  df-cring 20463  df-subrng 20799  df-subrg 20823  df-lmod 21138  df-lss 21208  df-sra 21449  df-rgmod 21450  df-dsmm 22039  df-frlm 22054  df-ascl 22164  df-psr 22218  df-mvr 22219  df-mpl 22220  df-opsr 22222  df-psr1 22499  df-vr1 22500  df-ply1 22501  df-mamu 22707  df-mat 22724  df-mat2pmat 23026

This theorem is used by:  pmatcollpw3fi1  23107