Theorem plypf1 25382

Description: Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.) (Proof shortened by AV, 29-Sep-2019.)

Hypotheses

Ref	Expression
plypf1.r	⊢ 𝑅 = (ℂfld ↾s 𝑆)
plypf1.p	⊢ 𝑃 = (Poly1‘𝑅)
plypf1.a	⊢ 𝐴 = (Base‘𝑃)
plypf1.e	⊢ 𝐸 = (eval1‘ℂfld)

Assertion

Ref	Expression
plypf1	⊢ (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸 “ 𝐴))

Proof of Theorem plypf1

Dummy variables 𝑓 𝑎 𝑘 𝑛 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elply 25365	. . . . 5 ⊢ (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
2	1	simprbi 497	. . . 4 ⊢ (𝑓 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
3		eqid 2739	. . . . . . . . 9 ⊢ (ℂfld ↑s ℂ) = (ℂfld ↑s ℂ)
4		cnfldbas 20610	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂfld)
5		eqid 2739	. . . . . . . . 9 ⊢ (0g‘(ℂfld ↑s ℂ)) = (0g‘(ℂfld ↑s ℂ))
6		cnex 10961	. . . . . . . . . 10 ⊢ ℂ ∈ V
7	6	a1i 11	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → ℂ ∈ V)
8		fzfid 13702	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (0...𝑛) ∈ Fin)
9		cnring 20629	. . . . . . . . . 10 ⊢ ℂfld ∈ Ring
10		ringcmn 19829	. . . . . . . . . 10 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd)
11	9, 10	mp1i 13	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → ℂfld ∈ CMnd)
12	4	subrgss 20034	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ)
13	12	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ⊆ ℂ)
14		elmapi 8646	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
15	14	ad2antll 726	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
16		subrgsubg 20039	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld))
17		cnfld0 20631	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 = (0g‘ℂfld)
18	17	subg0cl 18772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆)
19	16, 18	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 0 ∈ 𝑆)
20	19	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → 0 ∈ 𝑆)
21	20	snssd 4743	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → {0} ⊆ 𝑆)
22		ssequn2 4118	. . . . . . . . . . . . . . . 16 ⊢ ({0} ⊆ 𝑆 ↔ (𝑆 ∪ {0}) = 𝑆)
23	21, 22	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝑆 ∪ {0}) = 𝑆)
24	23	feq3d 6596	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝑎:ℕ0⟶(𝑆 ∪ {0}) ↔ 𝑎:ℕ0⟶𝑆))
25	15, 24	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → 𝑎:ℕ0⟶𝑆)
26		elfznn0 13358	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
27		ffvelrn 6968	. . . . . . . . . . . . 13 ⊢ ((𝑎:ℕ0⟶𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑎‘𝑘) ∈ 𝑆)
28	25, 26, 27	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ 𝑆)
29	13, 28	sseldd 3923	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℂ)
30	29	adantrl 713	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → (𝑎‘𝑘) ∈ ℂ)
31		simprl 768	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑧 ∈ ℂ)
32	26	ad2antll 726	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑘 ∈ ℕ0)
33		expcl 13809	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
34	31, 32, 33	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → (𝑧↑𝑘) ∈ ℂ)
35	30, 34	mulcld 11004	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → ((𝑎‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
36		eqid 2739	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))))
37	6	mptex 7108	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))) ∈ V
38	37	a1i 11	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))) ∈ V)
39		fvex 6796	. . . . . . . . . . 11 ⊢ (0g‘(ℂfld ↑s ℂ)) ∈ V
40	39	a1i 11	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (0g‘(ℂfld ↑s ℂ)) ∈ V)
41	36, 8, 38, 40	fsuppmptdm 9148	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘)))) finSupp (0g‘(ℂfld ↑s ℂ)))
42	3, 4, 5, 7, 8, 11, 35, 41	pwsgsum 19592	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → ((ℂfld ↑s ℂ) Σg (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))))) = (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))))))
43		fzfid 13702	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑧 ∈ ℂ) → (0...𝑛) ∈ Fin)
44	35	anassrs 468	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
45	43, 44	gsumfsum 20674	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑎‘𝑘) · (𝑧↑𝑘)))) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))
46	45	mpteq2dva 5175	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
47	42, 46	eqtrd 2779	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → ((ℂfld ↑s ℂ) Σg (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
48	3	pwsring 19863	. . . . . . . . . 10 ⊢ ((ℂfld ∈ Ring ∧ ℂ ∈ V) → (ℂfld ↑s ℂ) ∈ Ring)
49	9, 6, 48	mp2an 689	. . . . . . . . 9 ⊢ (ℂfld ↑s ℂ) ∈ Ring
50		ringcmn 19829	. . . . . . . . 9 ⊢ ((ℂfld ↑s ℂ) ∈ Ring → (ℂfld ↑s ℂ) ∈ CMnd)
51	49, 50	mp1i 13	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (ℂfld ↑s ℂ) ∈ CMnd)
52		cncrng 20628	. . . . . . . . . . 11 ⊢ ℂfld ∈ CRing
53		plypf1.e	. . . . . . . . . . . 12 ⊢ 𝐸 = (eval1‘ℂfld)
54		eqid 2739	. . . . . . . . . . . 12 ⊢ (Poly1‘ℂfld) = (Poly1‘ℂfld)
55	53, 54, 3, 4	evl1rhm 21507	. . . . . . . . . . 11 ⊢ (ℂfld ∈ CRing → 𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂfld ↑s ℂ)))
56	52, 55	ax-mp 5	. . . . . . . . . 10 ⊢ 𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂfld ↑s ℂ))
57		plypf1.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (ℂfld ↾s 𝑆)
58		plypf1.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
59		plypf1.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Base‘𝑃)
60	54, 57, 58, 59	subrgply1 21413	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
61	60	adantr 481	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
62		rhmima 20064	. . . . . . . . . 10 ⊢ ((𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂfld ↑s ℂ)) ∧ 𝐴 ∈ (SubRing‘(Poly1‘ℂfld))) → (𝐸 “ 𝐴) ∈ (SubRing‘(ℂfld ↑s ℂ)))
63	56, 61, 62	sylancr 587	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝐸 “ 𝐴) ∈ (SubRing‘(ℂfld ↑s ℂ)))
64		subrgsubg 20039	. . . . . . . . 9 ⊢ ((𝐸 “ 𝐴) ∈ (SubRing‘(ℂfld ↑s ℂ)) → (𝐸 “ 𝐴) ∈ (SubGrp‘(ℂfld ↑s ℂ)))
65		subgsubm 18786	. . . . . . . . 9 ⊢ ((𝐸 “ 𝐴) ∈ (SubGrp‘(ℂfld ↑s ℂ)) → (𝐸 “ 𝐴) ∈ (SubMnd‘(ℂfld ↑s ℂ)))
66	63, 64, 65	3syl 18	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝐸 “ 𝐴) ∈ (SubMnd‘(ℂfld ↑s ℂ)))
67		eqid 2739	. . . . . . . . . . . 12 ⊢ (Base‘(ℂfld ↑s ℂ)) = (Base‘(ℂfld ↑s ℂ))
68	9	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → ℂfld ∈ Ring)
69	6	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → ℂ ∈ V)
70		fconst6g 6672	. . . . . . . . . . . . . 14 ⊢ ((𝑎‘𝑘) ∈ ℂ → (ℂ × {(𝑎‘𝑘)}):ℂ⟶ℂ)
71	29, 70	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎‘𝑘)}):ℂ⟶ℂ)
72	3, 4, 67	pwselbasb 17208	. . . . . . . . . . . . . 14 ⊢ ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((ℂ × {(𝑎‘𝑘)}) ∈ (Base‘(ℂfld ↑s ℂ)) ↔ (ℂ × {(𝑎‘𝑘)}):ℂ⟶ℂ))
73	9, 6, 72	mp2an 689	. . . . . . . . . . . . 13 ⊢ ((ℂ × {(𝑎‘𝑘)}) ∈ (Base‘(ℂfld ↑s ℂ)) ↔ (ℂ × {(𝑎‘𝑘)}):ℂ⟶ℂ)
74	71, 73	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎‘𝑘)}) ∈ (Base‘(ℂfld ↑s ℂ)))
75	34	anass1rs 652	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑧↑𝑘) ∈ ℂ)
76	75	fmpttd 6998	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)):ℂ⟶ℂ)
77	3, 4, 67	pwselbasb 17208	. . . . . . . . . . . . . 14 ⊢ ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((𝑧 ∈ ℂ ↦ (𝑧↑𝑘)) ∈ (Base‘(ℂfld ↑s ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)):ℂ⟶ℂ))
78	9, 6, 77	mp2an 689	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ↦ (𝑧↑𝑘)) ∈ (Base‘(ℂfld ↑s ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)):ℂ⟶ℂ)
79	76, 78	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)) ∈ (Base‘(ℂfld ↑s ℂ)))
80		cnfldmul 20612	. . . . . . . . . . . 12 ⊢ · = (.r‘ℂfld)
81		eqid 2739	. . . . . . . . . . . 12 ⊢ (.r‘(ℂfld ↑s ℂ)) = (.r‘(ℂfld ↑s ℂ))
82	3, 67, 68, 69, 74, 79, 80, 81	pwsmulrval 17211	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎‘𝑘)})(.r‘(ℂfld ↑s ℂ))(𝑧 ∈ ℂ ↦ (𝑧↑𝑘))) = ((ℂ × {(𝑎‘𝑘)}) ∘f · (𝑧 ∈ ℂ ↦ (𝑧↑𝑘))))
83	29	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑎‘𝑘) ∈ ℂ)
84		fconstmpt 5650	. . . . . . . . . . . . 13 ⊢ (ℂ × {(𝑎‘𝑘)}) = (𝑧 ∈ ℂ ↦ (𝑎‘𝑘))
85	84	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎‘𝑘)}) = (𝑧 ∈ ℂ ↦ (𝑎‘𝑘)))
86		eqidd 2740	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)) = (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)))
87	69, 83, 75, 85, 86	offval2 7562	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎‘𝑘)}) ∘f · (𝑧 ∈ ℂ ↦ (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))))
88	82, 87	eqtrd 2779	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎‘𝑘)})(.r‘(ℂfld ↑s ℂ))(𝑧 ∈ ℂ ↦ (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))))
89	63	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸 “ 𝐴) ∈ (SubRing‘(ℂfld ↑s ℂ)))
90		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (algSc‘(Poly1‘ℂfld)) = (algSc‘(Poly1‘ℂfld))
91	53, 54, 4, 90	evl1sca 21509	. . . . . . . . . . . . 13 ⊢ ((ℂfld ∈ CRing ∧ (𝑎‘𝑘) ∈ ℂ) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎‘𝑘))) = (ℂ × {(𝑎‘𝑘)}))
92	52, 29, 91	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎‘𝑘))) = (ℂ × {(𝑎‘𝑘)}))
93		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(Poly1‘ℂfld)) = (Base‘(Poly1‘ℂfld))
94	93, 67	rhmf 19979	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂfld ↑s ℂ)) → 𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂfld ↑s ℂ)))
95	56, 94	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂfld ↑s ℂ))
96		ffn 6609	. . . . . . . . . . . . . 14 ⊢ (𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂfld ↑s ℂ)) → 𝐸 Fn (Base‘(Poly1‘ℂfld)))
97	95, 96	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐸 Fn (Base‘(Poly1‘ℂfld)))
98	93	subrgss 20034	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (SubRing‘(Poly1‘ℂfld)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
99	60, 98	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
100	99	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
101		simpll 764	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ∈ (SubRing‘ℂfld))
102	54, 90, 57, 58, 101, 59, 4, 29	subrg1asclcl 21440	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (((algSc‘(Poly1‘ℂfld))‘(𝑎‘𝑘)) ∈ 𝐴 ↔ (𝑎‘𝑘) ∈ 𝑆))
103	28, 102	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → ((algSc‘(Poly1‘ℂfld))‘(𝑎‘𝑘)) ∈ 𝐴)
104		fnfvima 7118	. . . . . . . . . . . . 13 ⊢ ((𝐸 Fn (Base‘(Poly1‘ℂfld)) ∧ 𝐴 ⊆ (Base‘(Poly1‘ℂfld)) ∧ ((algSc‘(Poly1‘ℂfld))‘(𝑎‘𝑘)) ∈ 𝐴) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎‘𝑘))) ∈ (𝐸 “ 𝐴))
105	97, 100, 103, 104	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎‘𝑘))) ∈ (𝐸 “ 𝐴))
106	92, 105	eqeltrrd 2841	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎‘𝑘)}) ∈ (𝐸 “ 𝐴))
107	67	subrgss 20034	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 “ 𝐴) ∈ (SubRing‘(ℂfld ↑s ℂ)) → (𝐸 “ 𝐴) ⊆ (Base‘(ℂfld ↑s ℂ)))
108	89, 107	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸 “ 𝐴) ⊆ (Base‘(ℂfld ↑s ℂ)))
109	60	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
110		eqid 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘(Poly1‘ℂfld))
111	110	subrgsubm 20046	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ (SubRing‘(Poly1‘ℂfld)) → 𝐴 ∈ (SubMnd‘(mulGrp‘(Poly1‘ℂfld))))
112	109, 111	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ (SubMnd‘(mulGrp‘(Poly1‘ℂfld))))
113	26	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
114		eqid 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (var1‘ℂfld) = (var1‘ℂfld)
115	114, 101, 57, 58, 59	subrgvr1cl 21442	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (var1‘ℂfld) ∈ 𝐴)
116		eqid 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (.g‘(mulGrp‘(Poly1‘ℂfld))) = (.g‘(mulGrp‘(Poly1‘ℂfld)))
117	116	submmulgcl 18755	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ (SubMnd‘(mulGrp‘(Poly1‘ℂfld))) ∧ 𝑘 ∈ ℕ0 ∧ (var1‘ℂfld) ∈ 𝐴) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
118	112, 113, 115, 117	syl3anc 1370	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
119		fnfvima 7118	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 Fn (Base‘(Poly1‘ℂfld)) ∧ 𝐴 ⊆ (Base‘(Poly1‘ℂfld)) ∧ (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (𝐸 “ 𝐴))
120	97, 100, 118, 119	syl3anc 1370	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (𝐸 “ 𝐴))
121	108, 120	sseldd 3923	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(ℂfld ↑s ℂ)))
122	3, 4, 67, 68, 69, 121	pwselbas 17209	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))):ℂ⟶ℂ)
123	122	feqmptd 6846	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) = (𝑧 ∈ ℂ ↦ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧)))
124	52	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CRing)
125		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
126	53, 114, 4, 54, 93, 124, 125	evl1vard 21512	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → ((var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(var1‘ℂfld))‘𝑧) = 𝑧))
127		eqid 2739	. . . . . . . . . . . . . . . . 17 ⊢ (.g‘(mulGrp‘ℂfld)) = (.g‘(mulGrp‘ℂfld))
128	113	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ0)
129	53, 54, 4, 93, 124, 125, 126, 116, 127, 128	evl1expd 21520	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → ((𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑘(.g‘(mulGrp‘ℂfld))𝑧)))
130	129	simprd 496	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑘(.g‘(mulGrp‘ℂfld))𝑧))
131		cnfldexp 20640	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧↑𝑘))
132	125, 128, 131	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧↑𝑘))
133	130, 132	eqtrd 2779	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑧↑𝑘))
134	133	mpteq2dva 5175	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧)) = (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)))
135	123, 134	eqtrd 2779	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) = (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)))
136	135, 120	eqeltrrd 2841	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)) ∈ (𝐸 “ 𝐴))
137	81	subrgmcl 20045	. . . . . . . . . . 11 ⊢ (((𝐸 “ 𝐴) ∈ (SubRing‘(ℂfld ↑s ℂ)) ∧ (ℂ × {(𝑎‘𝑘)}) ∈ (𝐸 “ 𝐴) ∧ (𝑧 ∈ ℂ ↦ (𝑧↑𝑘)) ∈ (𝐸 “ 𝐴)) → ((ℂ × {(𝑎‘𝑘)})(.r‘(ℂfld ↑s ℂ))(𝑧 ∈ ℂ ↦ (𝑧↑𝑘))) ∈ (𝐸 “ 𝐴))
138	89, 106, 136, 137	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎‘𝑘)})(.r‘(ℂfld ↑s ℂ))(𝑧 ∈ ℂ ↦ (𝑧↑𝑘))) ∈ (𝐸 “ 𝐴))
139	88, 138	eqeltrrd 2841	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))) ∈ (𝐸 “ 𝐴))
140	139	fmpttd 6998	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘)))):(0...𝑛)⟶(𝐸 “ 𝐴))
141	36, 8, 139, 40	fsuppmptdm 9148	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘)))) finSupp (0g‘(ℂfld ↑s ℂ)))
142	5, 51, 8, 66, 140, 141	gsumsubmcl 19529	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → ((ℂfld ↑s ℂ) Σg (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎‘𝑘) · (𝑧↑𝑘))))) ∈ (𝐸 “ 𝐴))
143	47, 142	eqeltrrd 2841	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ∈ (𝐸 “ 𝐴))
144		eleq1 2827	. . . . . 6 ⊢ (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) → (𝑓 ∈ (𝐸 “ 𝐴) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ∈ (𝐸 “ 𝐴)))
145	143, 144	syl5ibrcom 246	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) → 𝑓 ∈ (𝐸 “ 𝐴)))
146	145	rexlimdvva 3224	. . . 4 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) → 𝑓 ∈ (𝐸 “ 𝐴)))
147	2, 146	syl5 34	. . 3 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (Poly‘𝑆) → 𝑓 ∈ (𝐸 “ 𝐴)))
148		ffun 6612	. . . . . 6 ⊢ (𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂfld ↑s ℂ)) → Fun 𝐸)
149	95, 148	ax-mp 5	. . . . 5 ⊢ Fun 𝐸
150		fvelima 6844	. . . . 5 ⊢ ((Fun 𝐸 ∧ 𝑓 ∈ (𝐸 “ 𝐴)) → ∃𝑎 ∈ 𝐴 (𝐸‘𝑎) = 𝑓)
151	149, 150	mpan 687	. . . 4 ⊢ (𝑓 ∈ (𝐸 “ 𝐴) → ∃𝑎 ∈ 𝐴 (𝐸‘𝑎) = 𝑓)
152	99	sselda 3922	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ (Base‘(Poly1‘ℂfld)))
153		eqid 2739	. . . . . . . . . . . 12 ⊢ ( ·𝑠 ‘(Poly1‘ℂfld)) = ( ·𝑠 ‘(Poly1‘ℂfld))
154		eqid 2739	. . . . . . . . . . . 12 ⊢ (coe1‘𝑎) = (coe1‘𝑎)
155	54, 114, 93, 153, 110, 116, 154	ply1coe 21476	. . . . . . . . . . 11 ⊢ ((ℂfld ∈ Ring ∧ 𝑎 ∈ (Base‘(Poly1‘ℂfld))) → 𝑎 = ((Poly1‘ℂfld) Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))))
156	9, 152, 155	sylancr 587	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → 𝑎 = ((Poly1‘ℂfld) Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))))
157	156	fveq2d 6787	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝐸‘𝑎) = (𝐸‘((Poly1‘ℂfld) Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))))
158		eqid 2739	. . . . . . . . . 10 ⊢ (0g‘(Poly1‘ℂfld)) = (0g‘(Poly1‘ℂfld))
159	54	ply1ring 21428	. . . . . . . . . . . 12 ⊢ (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ Ring)
160	9, 159	ax-mp 5	. . . . . . . . . . 11 ⊢ (Poly1‘ℂfld) ∈ Ring
161		ringcmn 19829	. . . . . . . . . . 11 ⊢ ((Poly1‘ℂfld) ∈ Ring → (Poly1‘ℂfld) ∈ CMnd)
162	160, 161	mp1i 13	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (Poly1‘ℂfld) ∈ CMnd)
163		ringmnd 19802	. . . . . . . . . . 11 ⊢ ((ℂfld ↑s ℂ) ∈ Ring → (ℂfld ↑s ℂ) ∈ Mnd)
164	49, 163	mp1i 13	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (ℂfld ↑s ℂ) ∈ Mnd)
165		nn0ex 12248	. . . . . . . . . . 11 ⊢ ℕ0 ∈ V
166	165	a1i 11	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ℕ0 ∈ V)
167		rhmghm 19978	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂfld ↑s ℂ)) → 𝐸 ∈ ((Poly1‘ℂfld) GrpHom (ℂfld ↑s ℂ)))
168	56, 167	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐸 ∈ ((Poly1‘ℂfld) GrpHom (ℂfld ↑s ℂ))
169		ghmmhm 18853	. . . . . . . . . . 11 ⊢ (𝐸 ∈ ((Poly1‘ℂfld) GrpHom (ℂfld ↑s ℂ)) → 𝐸 ∈ ((Poly1‘ℂfld) MndHom (ℂfld ↑s ℂ)))
170	168, 169	mp1i 13	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → 𝐸 ∈ ((Poly1‘ℂfld) MndHom (ℂfld ↑s ℂ)))
171	54	ply1lmod 21432	. . . . . . . . . . . . 13 ⊢ (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ LMod)
172	9, 171	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (Poly1‘ℂfld) ∈ LMod)
173	12	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → 𝑆 ⊆ ℂ)
174		eqid 2739	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑅) = (Base‘𝑅)
175	154, 59, 58, 174	coe1f 21391	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝐴 → (coe1‘𝑎):ℕ0⟶(Base‘𝑅))
176	57	subrgbas 20042	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 = (Base‘𝑅))
177	176	feq3d 6596	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → ((coe1‘𝑎):ℕ0⟶𝑆 ↔ (coe1‘𝑎):ℕ0⟶(Base‘𝑅)))
178	175, 177	syl5ibr 245	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (𝑎 ∈ 𝐴 → (coe1‘𝑎):ℕ0⟶𝑆))
179	178	imp 407	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (coe1‘𝑎):ℕ0⟶𝑆)
180	179	ffvelrnda 6970	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝑎)‘𝑘) ∈ 𝑆)
181	173, 180	sseldd 3923	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝑎)‘𝑘) ∈ ℂ)
182	110	ringmgp 19798	. . . . . . . . . . . . . 14 ⊢ ((Poly1‘ℂfld) ∈ Ring → (mulGrp‘(Poly1‘ℂfld)) ∈ Mnd)
183	160, 182	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘(Poly1‘ℂfld)) ∈ Mnd)
184		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
185	114, 54, 93	vr1cl 21397	. . . . . . . . . . . . . 14 ⊢ (ℂfld ∈ Ring → (var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)))
186	9, 185	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)))
187	110, 93	mgpbas 19735	. . . . . . . . . . . . . 14 ⊢ (Base‘(Poly1‘ℂfld)) = (Base‘(mulGrp‘(Poly1‘ℂfld)))
188	187, 116	mulgnn0cl 18729	. . . . . . . . . . . . 13 ⊢ (((mulGrp‘(Poly1‘ℂfld)) ∈ Mnd ∧ 𝑘 ∈ ℕ0 ∧ (var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld))) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)))
189	183, 184, 186, 188	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)))
190	54	ply1sca 21433	. . . . . . . . . . . . . 14 ⊢ (ℂfld ∈ Ring → ℂfld = (Scalar‘(Poly1‘ℂfld)))
191	9, 190	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ℂfld = (Scalar‘(Poly1‘ℂfld))
192	93, 191, 153, 4	lmodvscl 20149	. . . . . . . . . . . 12 ⊢ (((Poly1‘ℂfld) ∈ LMod ∧ ((coe1‘𝑎)‘𝑘) ∈ ℂ ∧ (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld))) → (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)))
193	172, 181, 189, 192	syl3anc 1370	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)))
194	193	fmpttd 6998	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℕ0⟶(Base‘(Poly1‘ℂfld)))
195	165	mptex 7108	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V
196		funmpt 6479	. . . . . . . . . . . . 13 ⊢ Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))
197		fvex 6796	. . . . . . . . . . . . 13 ⊢ (0g‘(Poly1‘ℂfld)) ∈ V
198	195, 196, 197	3pm3.2i 1338	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∧ (0g‘(Poly1‘ℂfld)) ∈ V)
199	198	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∧ (0g‘(Poly1‘ℂfld)) ∈ V))
200	154, 93, 54, 17	coe1sfi 21393	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (coe1‘𝑎) finSupp 0)
201	152, 200	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (coe1‘𝑎) finSupp 0)
202	201	fsuppimpd 9144	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((coe1‘𝑎) supp 0) ∈ Fin)
203	179	feqmptd 6846	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (coe1‘𝑎) = (𝑘 ∈ ℕ0 ↦ ((coe1‘𝑎)‘𝑘)))
204	203	oveq1d 7299	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((coe1‘𝑎) supp 0) = ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝑎)‘𝑘)) supp 0))
205		eqimss2 3979	. . . . . . . . . . . . 13 ⊢ (((coe1‘𝑎) supp 0) = ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝑎)‘𝑘)) supp 0) → ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝑎)‘𝑘)) supp 0) ⊆ ((coe1‘𝑎) supp 0))
206	204, 205	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((𝑘 ∈ ℕ0 ↦ ((coe1‘𝑎)‘𝑘)) supp 0) ⊆ ((coe1‘𝑎) supp 0))
207	9, 171	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (Poly1‘ℂfld) ∈ LMod)
208	93, 191, 153, 17, 158	lmod0vs 20165	. . . . . . . . . . . . 13 ⊢ (((Poly1‘ℂfld) ∈ LMod ∧ 𝑥 ∈ (Base‘(Poly1‘ℂfld))) → (0( ·𝑠 ‘(Poly1‘ℂfld))𝑥) = (0g‘(Poly1‘ℂfld)))
209	207, 208	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (Base‘(Poly1‘ℂfld))) → (0( ·𝑠 ‘(Poly1‘ℂfld))𝑥) = (0g‘(Poly1‘ℂfld)))
210		c0ex 10978	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
211	210	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → 0 ∈ V)
212	206, 209, 180, 189, 211	suppssov1 8023	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) supp (0g‘(Poly1‘ℂfld))) ⊆ ((coe1‘𝑎) supp 0))
213		suppssfifsupp 9152	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∧ (0g‘(Poly1‘ℂfld)) ∈ V) ∧ (((coe1‘𝑎) supp 0) ∈ Fin ∧ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) supp (0g‘(Poly1‘ℂfld))) ⊆ ((coe1‘𝑎) supp 0))) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) finSupp (0g‘(Poly1‘ℂfld)))
214	199, 202, 212, 213	syl12anc 834	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) finSupp (0g‘(Poly1‘ℂfld)))
215	93, 158, 162, 164, 166, 170, 194, 214	gsummhm 19548	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((ℂfld ↑s ℂ) Σg (𝐸 ∘ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))) = (𝐸‘((Poly1‘ℂfld) Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))))
216	95	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → 𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂfld ↑s ℂ)))
217	216, 193	cofmpt 7013	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝐸 ∘ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))) = (𝑘 ∈ ℕ0 ↦ (𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))))
218	9	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → ℂfld ∈ Ring)
219	6	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → ℂ ∈ V)
220	95	ffvelrni 6969	. . . . . . . . . . . . . . . 16 ⊢ ((((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)) → (𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ (Base‘(ℂfld ↑s ℂ)))
221	193, 220	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ (Base‘(ℂfld ↑s ℂ)))
222	3, 4, 67, 218, 219, 221	pwselbas 17209	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℂ⟶ℂ)
223	222	feqmptd 6846	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) = (𝑧 ∈ ℂ ↦ ((𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))‘𝑧)))
224	52	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CRing)
225		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
226	53, 114, 4, 54, 93, 224, 225	evl1vard 21512	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(var1‘ℂfld))‘𝑧) = 𝑧))
227	184	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ0)
228	53, 54, 4, 93, 224, 225, 226, 116, 127, 227	evl1expd 21520	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑘(.g‘(mulGrp‘ℂfld))𝑧)))
229	225, 227, 131	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧↑𝑘))
230	229	eqeq2d 2750	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑘(.g‘(mulGrp‘ℂfld))𝑧) ↔ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑧↑𝑘)))
231	230	anbi2d 629	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (((𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑘(.g‘(mulGrp‘ℂfld))𝑧)) ↔ ((𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑧↑𝑘))))
232	228, 231	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑧↑𝑘)))
233	181	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((coe1‘𝑎)‘𝑘) ∈ ℂ)
234	53, 54, 4, 93, 224, 225, 232, 233, 153, 80	evl1vsd 21519	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))‘𝑧) = (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))
235	234	simprd 496	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))‘𝑧) = (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))
236	235	mpteq2dva 5175	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))‘𝑧)) = (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))
237	223, 236	eqtrd 2779	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) = (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))
238	237	mpteq2dva 5175	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝑘 ∈ ℕ0 ↦ (𝐸‘(((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))) = (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))))
239	217, 238	eqtrd 2779	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝐸 ∘ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))) = (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))))
240	239	oveq2d 7300	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((ℂfld ↑s ℂ) Σg (𝐸 ∘ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))) = ((ℂfld ↑s ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))))
241	157, 215, 240	3eqtr2d 2785	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝐸‘𝑎) = ((ℂfld ↑s ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))))
242	6	a1i 11	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ℂ ∈ V)
243	9, 10	mp1i 13	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ℂfld ∈ CMnd)
244	181	adantlr 712	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝑎)‘𝑘) ∈ ℂ)
245	33	adantll 711	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧↑𝑘) ∈ ℂ)
246	244, 245	mulcld 11004	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
247	246	anasss 467	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0)) → (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
248	165	mptex 7108	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) ∈ V
249		funmpt 6479	. . . . . . . . . . . 12 ⊢ Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))
250	248, 249, 39	3pm3.2i 1338	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) ∧ (0g‘(ℂfld ↑s ℂ)) ∈ V)
251	250	a1i 11	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) ∧ (0g‘(ℂfld ↑s ℂ)) ∈ V))
252		fzfid 13702	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin)
253		eldifn 4063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → ¬ 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
254	253	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → ¬ 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
255	152	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑎 ∈ (Base‘(Poly1‘ℂfld)))
256		eldifi 4062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → 𝑘 ∈ ℕ0)
257	256	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℕ0)
258		eqid 2739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ( deg1 ‘ℂfld) = ( deg1 ‘ℂfld)
259	258, 54, 93, 17, 154	deg1ge 25272	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ∈ (Base‘(Poly1‘ℂfld)) ∧ 𝑘 ∈ ℕ0 ∧ ((coe1‘𝑎)‘𝑘) ≠ 0) → 𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎))
260	259	3expia 1120	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ (Base‘(Poly1‘ℂfld)) ∧ 𝑘 ∈ ℕ0) → (((coe1‘𝑎)‘𝑘) ≠ 0 → 𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎)))
261	255, 257, 260	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1‘𝑎)‘𝑘) ≠ 0 → 𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎)))
262		0xr 11031	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℝ*
263	258, 54, 93	deg1xrcl 25256	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
264	152, 263	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
265	264	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
266		xrmax2 12919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℝ* ∧ (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*) → (( deg1 ‘ℂfld)‘𝑎) ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))
267	262, 265, 266	sylancr 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (( deg1 ‘ℂfld)‘𝑎) ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))
268	257	nn0red 12303	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ)
269	268	rexrd 11034	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ*)
270		ifcl 4505	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((( deg1 ‘ℂfld)‘𝑎) ∈ ℝ* ∧ 0 ∈ ℝ*) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℝ*)
271	265, 262, 270	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℝ*)
272		xrletr 12901	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ* ∧ (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ* ∧ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℝ*) → ((𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎) ∧ (( deg1 ‘ℂfld)‘𝑎) ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) → 𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
273	269, 265, 271, 272	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → ((𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎) ∧ (( deg1 ‘ℂfld)‘𝑎) ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) → 𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
274	267, 273	mpan2d 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎) → 𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
275	261, 274	syld 47	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1‘𝑎)‘𝑘) ≠ 0 → 𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
276	275, 257	jctild 526	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1‘𝑎)‘𝑘) ≠ 0 → (𝑘 ∈ ℕ0 ∧ 𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))))
277	258, 54, 93	deg1cl 25257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
278	152, 277	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
279		elun 4084	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}))
280	278, 279	sylib 217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}))
281		nn0ge0 12267	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → 0 ≤ (( deg1 ‘ℂfld)‘𝑎))
282	281	iftrued 4468	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) = (( deg1 ‘ℂfld)‘𝑎))
283		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → (( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0)
284	282, 283	eqeltrd 2840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
285		mnflt0 12870	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ -∞ < 0
286		mnfxr 11041	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ -∞ ∈ ℝ*
287		xrltnle 11051	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞))
288	286, 262, 287	mp2an 689	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (-∞ < 0 ↔ ¬ 0 ≤ -∞)
289	285, 288	mpbi 229	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ¬ 0 ≤ -∞
290		elsni 4579	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → (( deg1 ‘ℂfld)‘𝑎) = -∞)
291	290	breq2d 5087	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → (0 ≤ (( deg1 ‘ℂfld)‘𝑎) ↔ 0 ≤ -∞))
292	289, 291	mtbiri 327	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → ¬ 0 ≤ (( deg1 ‘ℂfld)‘𝑎))
293	292	iffalsed 4471	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) = 0)
294		0nn0 12257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ ℕ0
295	293, 294	eqeltrdi 2848	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
296	284, 295	jaoi 854	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
297	280, 296	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
298	297	ad2antrr 723	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
299		fznn0 13357	. . . . . . . . . . . . . . . . . . . 20 ⊢ (if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0 → (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↔ (𝑘 ∈ ℕ0 ∧ 𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))))
300	298, 299	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↔ (𝑘 ∈ ℕ0 ∧ 𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))))
301	276, 300	sylibrd 258	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1‘𝑎)‘𝑘) ≠ 0 → 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))))
302	301	necon1bd 2962	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (¬ 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) → ((coe1‘𝑎)‘𝑘) = 0))
303	254, 302	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → ((coe1‘𝑎)‘𝑘) = 0)
304	303	oveq1d 7299	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)) = (0 · (𝑧↑𝑘)))
305	256, 245	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑧↑𝑘) ∈ ℂ)
306	305	mul02d 11182	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (0 · (𝑧↑𝑘)) = 0)
307	304, 306	eqtrd 2779	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)) = 0)
308	307	an32s 649	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) ∧ 𝑧 ∈ ℂ) → (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)) = 0)
309	308	mpteq2dva 5175	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ 0))
310		fconstmpt 5650	. . . . . . . . . . . . 13 ⊢ (ℂ × {0}) = (𝑧 ∈ ℂ ↦ 0)
311		ringmnd 19802	. . . . . . . . . . . . . . 15 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd)
312	9, 311	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ℂfld ∈ Mnd
313	3, 17	pws0g 18430	. . . . . . . . . . . . . 14 ⊢ ((ℂfld ∈ Mnd ∧ ℂ ∈ V) → (ℂ × {0}) = (0g‘(ℂfld ↑s ℂ)))
314	312, 6, 313	mp2an 689	. . . . . . . . . . . . 13 ⊢ (ℂ × {0}) = (0g‘(ℂfld ↑s ℂ))
315	310, 314	eqtr3i 2769	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ ↦ 0) = (0g‘(ℂfld ↑s ℂ))
316	309, 315	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) = (0g‘(ℂfld ↑s ℂ)))
317	316, 166	suppss2 8025	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) supp (0g‘(ℂfld ↑s ℂ))) ⊆ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
318		suppssfifsupp 9152	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) ∧ (0g‘(ℂfld ↑s ℂ)) ∈ V) ∧ ((0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin ∧ ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) supp (0g‘(ℂfld ↑s ℂ))) ⊆ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) finSupp (0g‘(ℂfld ↑s ℂ)))
319	251, 252, 317, 318	syl12anc 834	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) finSupp (0g‘(ℂfld ↑s ℂ)))
320	3, 4, 5, 242, 166, 243, 247, 319	pwsgsum 19592	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((ℂfld ↑s ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))) = (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))))
321		fz0ssnn0 13360	. . . . . . . . . . . 12 ⊢ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ⊆ ℕ0
322		resmpt 5948	. . . . . . . . . . . 12 ⊢ ((0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ⊆ ℕ0 → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ↾ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) = (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))
323	321, 322	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ↾ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) = (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))
324	323	oveq2i 7295	. . . . . . . . . 10 ⊢ (ℂfld Σg ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ↾ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) = (ℂfld Σg (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))
325	9, 10	mp1i 13	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CMnd)
326	165	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → ℕ0 ∈ V)
327	246	fmpttd 6998	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))):ℕ0⟶ℂ)
328	307, 326	suppss2 8025	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) supp 0) ⊆ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
329	165	mptex 7108	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ∈ V
330		funmpt 6479	. . . . . . . . . . . . . 14 ⊢ Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))
331	329, 330, 210	3pm3.2i 1338	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ∧ 0 ∈ V)
332	331	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ∧ 0 ∈ V))
333		fzfid 13702	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin)
334		suppssfifsupp 9152	. . . . . . . . . . . 12 ⊢ ((((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ∧ 0 ∈ V) ∧ ((0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin ∧ ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) supp 0) ⊆ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) finSupp 0)
335	332, 333, 328, 334	syl12anc 834	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) finSupp 0)
336	4, 17, 325, 326, 327, 328, 335	gsumres 19523	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg ((𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ↾ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) = (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))))
337		elfznn0 13358	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) → 𝑘 ∈ ℕ0)
338	337, 246	sylan2 593	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)) ∈ ℂ)
339	333, 338	gsumfsum 20674	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) = Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))
340	324, 336, 339	3eqtr3a 2803	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))) = Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘)))
341	340	mpteq2dva 5175	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))
342	241, 320, 341	3eqtrd 2783	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝐸‘𝑎) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))))
343	12	adantr 481	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → 𝑆 ⊆ ℂ)
344		elplyr 25371	. . . . . . . 8 ⊢ ((𝑆 ⊆ ℂ ∧ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0 ∧ (coe1‘𝑎):ℕ0⟶𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘𝑆))
345	343, 297, 179, 344	syl3anc 1370	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1‘𝑎)‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘𝑆))
346	342, 345	eqeltrd 2840	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → (𝐸‘𝑎) ∈ (Poly‘𝑆))
347		eleq1 2827	. . . . . 6 ⊢ ((𝐸‘𝑎) = 𝑓 → ((𝐸‘𝑎) ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (Poly‘𝑆)))
348	346, 347	syl5ibcom 244	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎 ∈ 𝐴) → ((𝐸‘𝑎) = 𝑓 → 𝑓 ∈ (Poly‘𝑆)))
349	348	rexlimdva 3214	. . . 4 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (∃𝑎 ∈ 𝐴 (𝐸‘𝑎) = 𝑓 → 𝑓 ∈ (Poly‘𝑆)))
350	151, 349	syl5 34	. . 3 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (𝐸 “ 𝐴) → 𝑓 ∈ (Poly‘𝑆)))
351	147, 350	impbid 211	. 2 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (𝐸 “ 𝐴)))
352	351	eqrdv 2737	1 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸 “ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2944  ∃wrex 3066  Vcvv 3433   ∖ cdif 3885   ∪ cun 3886   ⊆ wss 3888  ifcif 4460  {csn 4562   class class class wbr 5075   ↦ cmpt 5158   × cxp 5588   ↾ cres 5592   “ cima 5593   ∘ ccom 5594  Fun wfun 6431   Fn wfn 6432  ⟶wf 6433  ‘cfv 6437  (class class class)co 7284   ∘_f cof 7540   supp csupp 7986   ↑_m cmap 8624  Fincfn 8742   finSupp cfsupp 9137  ℂcc 10878  0cc0 10880   · cmul 10885  -∞cmnf 11016  ℝ^*cxr 11017   < clt 11018   ≤ cle 11019  ℕ₀cn0 12242  ...cfz 13248  ↑cexp 13791  Σcsu 15406  Basecbs 16921   ↾_s cress 16950  ._rcmulr 16972  Scalarcsca 16974   ·_𝑠 cvsca 16975  0_gc0g 17159   Σ_g cgsu 17160   ↑_s cpws 17166  Mndcmnd 18394   MndHom cmhm 18437  SubMndcsubmnd 18438  ._gcmg 18709  SubGrpcsubg 18758   GrpHom cghm 18840  CMndccmn 19395  mulGrpcmgp 19729  Ringcrg 19792  CRingccrg 19793   RingHom crh 19965  SubRingcsubrg 20029  LModclmod 20132  ℂ_fldccnfld 20606  algSccascl 21068  var₁cv1 21356  Poly₁cpl1 21357  coe₁cco1 21358  eval₁ce1 21489   deg₁ cdg1 25225  Polycply 25354

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 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-inf2 9408  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957  ax-pre-sup 10958  ax-addf 10959  ax-mulf 10960

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-iin 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-isom 6446  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-of 7542  df-ofr 7543  df-om 7722  df-1st 7840  df-2nd 7841  df-supp 7987  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-er 8507  df-map 8626  df-pm 8627  df-ixp 8695  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-fsupp 9138  df-sup 9210  df-oi 9278  df-card 9706  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-div 11642  df-nn 11983  df-2 12045  df-3 12046  df-4 12047  df-5 12048  df-6 12049  df-7 12050  df-8 12051  df-9 12052  df-n0 12243  df-z 12329  df-dec 12447  df-uz 12592  df-rp 12740  df-fz 13249  df-fzo 13392  df-seq 13731  df-exp 13792  df-hash 14054  df-cj 14819  df-re 14820  df-im 14821  df-sqrt 14955  df-abs 14956  df-clim 15206  df-sum 15407  df-struct 16857  df-sets 16874  df-slot 16892  df-ndx 16904  df-base 16922  df-ress 16951  df-plusg 16984  df-mulr 16985  df-starv 16986  df-sca 16987  df-vsca 16988  df-ip 16989  df-tset 16990  df-ple 16991  df-ds 16993  df-unif 16994  df-hom 16995  df-cco 16996  df-0g 17161  df-gsum 17162  df-prds 17167  df-pws 17169  df-mre 17304  df-mrc 17305  df-acs 17307  df-mgm 18335  df-sgrp 18384  df-mnd 18395  df-mhm 18439  df-submnd 18440  df-grp 18589  df-minusg 18590  df-sbg 18591  df-mulg 18710  df-subg 18761  df-ghm 18841  df-cntz 18932  df-cmn 19397  df-abl 19398  df-mgp 19730  df-ur 19747  df-srg 19751  df-ring 19794  df-cring 19795  df-rnghom 19968  df-subrg 20031  df-lmod 20134  df-lss 20203  df-lsp 20243  df-cnfld 20607  df-assa 21069  df-asp 21070  df-ascl 21071  df-psr 21121  df-mvr 21122  df-mpl 21123  df-opsr 21125  df-evls 21291  df-evl 21292  df-psr1 21360  df-vr1 21361  df-ply1 21362  df-coe1 21363  df-evl1 21491  df-mdeg 25226  df-deg1 25227  df-ply 25358

This theorem is referenced by: (None)