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Theorem plypf1 25444
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 = (ℂflds 𝑆)
plypf1.p 𝑃 = (Poly1𝑅)
plypf1.a 𝐴 = (Base‘𝑃)
plypf1.e 𝐸 = (eval1‘ℂfld)
Assertion
Ref Expression
plypf1 (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸𝐴))

Proof of Theorem plypf1
Dummy variables 𝑓 𝑎 𝑘 𝑛 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elply 25427 . . . . 5 (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
21simprbi 497 . . . 4 (𝑓 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
3 eqid 2737 . . . . . . . . 9 (ℂflds ℂ) = (ℂflds ℂ)
4 cnfldbas 20672 . . . . . . . . 9 ℂ = (Base‘ℂfld)
5 eqid 2737 . . . . . . . . 9 (0g‘(ℂflds ℂ)) = (0g‘(ℂflds ℂ))
6 cnex 11022 . . . . . . . . . 10 ℂ ∈ V
76a1i 11 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ℂ ∈ V)
8 fzfid 13763 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (0...𝑛) ∈ Fin)
9 cnring 20691 . . . . . . . . . 10 fld ∈ Ring
10 ringcmn 19887 . . . . . . . . . 10 (ℂfld ∈ Ring → ℂfld ∈ CMnd)
119, 10mp1i 13 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ℂfld ∈ CMnd)
124subrgss 20096 . . . . . . . . . . . . 13 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ)
1312ad2antrr 723 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ⊆ ℂ)
14 elmapi 8683 . . . . . . . . . . . . . . 15 (𝑎 ∈ ((𝑆 ∪ {0}) ↑m0) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
1514ad2antll 726 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
16 subrgsubg 20101 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld))
17 cnfld0 20693 . . . . . . . . . . . . . . . . . . . 20 0 = (0g‘ℂfld)
1817subg0cl 18830 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆)
1916, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝑆 ∈ (SubRing‘ℂfld) → 0 ∈ 𝑆)
2019adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 0 ∈ 𝑆)
2120snssd 4752 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → {0} ⊆ 𝑆)
22 ssequn2 4127 . . . . . . . . . . . . . . . 16 ({0} ⊆ 𝑆 ↔ (𝑆 ∪ {0}) = 𝑆)
2321, 22sylib 217 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑆 ∪ {0}) = 𝑆)
2423feq3d 6622 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑎:ℕ0⟶(𝑆 ∪ {0}) ↔ 𝑎:ℕ0𝑆))
2515, 24mpbid 231 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 𝑎:ℕ0𝑆)
26 elfznn0 13419 . . . . . . . . . . . . 13 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
27 ffvelcdm 6996 . . . . . . . . . . . . 13 ((𝑎:ℕ0𝑆𝑘 ∈ ℕ0) → (𝑎𝑘) ∈ 𝑆)
2825, 26, 27syl2an 596 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) ∈ 𝑆)
2913, 28sseldd 3931 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) ∈ ℂ)
3029adantrl 713 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → (𝑎𝑘) ∈ ℂ)
31 simprl 768 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑧 ∈ ℂ)
3226ad2antll 726 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑘 ∈ ℕ0)
33 expcl 13870 . . . . . . . . . . 11 ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑧𝑘) ∈ ℂ)
3431, 32, 33syl2anc 584 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → (𝑧𝑘) ∈ ℂ)
3530, 34mulcld 11065 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → ((𝑎𝑘) · (𝑧𝑘)) ∈ ℂ)
36 eqid 2737 . . . . . . . . . 10 (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))
376mptex 7136 . . . . . . . . . . 11 (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))) ∈ V
3837a1i 11 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))) ∈ V)
39 fvex 6822 . . . . . . . . . . 11 (0g‘(ℂflds ℂ)) ∈ V
4039a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (0g‘(ℂflds ℂ)) ∈ V)
4136, 8, 38, 40fsuppmptdm 9207 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))) finSupp (0g‘(ℂflds ℂ)))
423, 4, 5, 7, 8, 11, 35, 41pwsgsum 19650 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ((ℂflds ℂ) Σg (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))) = (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑎𝑘) · (𝑧𝑘))))))
43 fzfid 13763 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑧 ∈ ℂ) → (0...𝑛) ∈ Fin)
4435anassrs 468 . . . . . . . . . 10 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎𝑘) · (𝑧𝑘)) ∈ ℂ)
4543, 44gsumfsum 20736 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑎𝑘) · (𝑧𝑘)))) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
4645mpteq2dva 5185 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑎𝑘) · (𝑧𝑘))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
4742, 46eqtrd 2777 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ((ℂflds ℂ) Σg (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
483pwsring 19921 . . . . . . . . . 10 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → (ℂflds ℂ) ∈ Ring)
499, 6, 48mp2an 689 . . . . . . . . 9 (ℂflds ℂ) ∈ Ring
50 ringcmn 19887 . . . . . . . . 9 ((ℂflds ℂ) ∈ Ring → (ℂflds ℂ) ∈ CMnd)
5149, 50mp1i 13 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (ℂflds ℂ) ∈ CMnd)
52 cncrng 20690 . . . . . . . . . . 11 fld ∈ CRing
53 plypf1.e . . . . . . . . . . . 12 𝐸 = (eval1‘ℂfld)
54 eqid 2737 . . . . . . . . . . . 12 (Poly1‘ℂfld) = (Poly1‘ℂfld)
5553, 54, 3, 4evl1rhm 21569 . . . . . . . . . . 11 (ℂfld ∈ CRing → 𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂflds ℂ)))
5652, 55ax-mp 5 . . . . . . . . . 10 𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂflds ℂ))
57 plypf1.r . . . . . . . . . . . 12 𝑅 = (ℂflds 𝑆)
58 plypf1.p . . . . . . . . . . . 12 𝑃 = (Poly1𝑅)
59 plypf1.a . . . . . . . . . . . 12 𝐴 = (Base‘𝑃)
6054, 57, 58, 59subrgply1 21475 . . . . . . . . . . 11 (𝑆 ∈ (SubRing‘ℂfld) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
6160adantr 481 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
62 rhmima 20126 . . . . . . . . . 10 ((𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂflds ℂ)) ∧ 𝐴 ∈ (SubRing‘(Poly1‘ℂfld))) → (𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)))
6356, 61, 62sylancr 587 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)))
64 subrgsubg 20101 . . . . . . . . 9 ((𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)) → (𝐸𝐴) ∈ (SubGrp‘(ℂflds ℂ)))
65 subgsubm 18844 . . . . . . . . 9 ((𝐸𝐴) ∈ (SubGrp‘(ℂflds ℂ)) → (𝐸𝐴) ∈ (SubMnd‘(ℂflds ℂ)))
6663, 64, 653syl 18 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝐸𝐴) ∈ (SubMnd‘(ℂflds ℂ)))
67 eqid 2737 . . . . . . . . . . . 12 (Base‘(ℂflds ℂ)) = (Base‘(ℂflds ℂ))
689a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ℂfld ∈ Ring)
696a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ℂ ∈ V)
70 fconst6g 6698 . . . . . . . . . . . . . 14 ((𝑎𝑘) ∈ ℂ → (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ)
7129, 70syl 17 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ)
723, 4, 67pwselbasb 17266 . . . . . . . . . . . . . 14 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)) ↔ (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ))
739, 6, 72mp2an 689 . . . . . . . . . . . . 13 ((ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)) ↔ (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ)
7471, 73sylibr 233 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)))
7534anass1rs 652 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑧𝑘) ∈ ℂ)
7675fmpttd 7026 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ)
773, 4, 67pwselbasb 17266 . . . . . . . . . . . . . 14 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ))
789, 6, 77mp2an 689 . . . . . . . . . . . . 13 ((𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ)
7976, 78sylibr 233 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)))
80 cnfldmul 20674 . . . . . . . . . . . 12 · = (.r‘ℂfld)
81 eqid 2737 . . . . . . . . . . . 12 (.r‘(ℂflds ℂ)) = (.r‘(ℂflds ℂ))
823, 67, 68, 69, 74, 79, 80, 81pwsmulrval 17269 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)})(.r‘(ℂflds ℂ))(𝑧 ∈ ℂ ↦ (𝑧𝑘))) = ((ℂ × {(𝑎𝑘)}) ∘f · (𝑧 ∈ ℂ ↦ (𝑧𝑘))))
8329adantr 481 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑎𝑘) ∈ ℂ)
84 fconstmpt 5665 . . . . . . . . . . . . 13 (ℂ × {(𝑎𝑘)}) = (𝑧 ∈ ℂ ↦ (𝑎𝑘))
8584a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}) = (𝑧 ∈ ℂ ↦ (𝑎𝑘)))
86 eqidd 2738 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) = (𝑧 ∈ ℂ ↦ (𝑧𝑘)))
8769, 83, 75, 85, 86offval2 7591 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)}) ∘f · (𝑧 ∈ ℂ ↦ (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))
8882, 87eqtrd 2777 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)})(.r‘(ℂflds ℂ))(𝑧 ∈ ℂ ↦ (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))
8963adantr 481 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)))
90 eqid 2737 . . . . . . . . . . . . . 14 (algSc‘(Poly1‘ℂfld)) = (algSc‘(Poly1‘ℂfld))
9153, 54, 4, 90evl1sca 21571 . . . . . . . . . . . . 13 ((ℂfld ∈ CRing ∧ (𝑎𝑘) ∈ ℂ) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) = (ℂ × {(𝑎𝑘)}))
9252, 29, 91sylancr 587 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) = (ℂ × {(𝑎𝑘)}))
93 eqid 2737 . . . . . . . . . . . . . . . 16 (Base‘(Poly1‘ℂfld)) = (Base‘(Poly1‘ℂfld))
9493, 67rhmf 20037 . . . . . . . . . . . . . . 15 (𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂflds ℂ)) → 𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ)))
9556, 94ax-mp 5 . . . . . . . . . . . . . 14 𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ))
96 ffn 6635 . . . . . . . . . . . . . 14 (𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ)) → 𝐸 Fn (Base‘(Poly1‘ℂfld)))
9795, 96mp1i 13 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐸 Fn (Base‘(Poly1‘ℂfld)))
9893subrgss 20096 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubRing‘(Poly1‘ℂfld)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
9960, 98syl 17 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubRing‘ℂfld) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
10099ad2antrr 723 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
101 simpll 764 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ∈ (SubRing‘ℂfld))
10254, 90, 57, 58, 101, 59, 4, 29subrg1asclcl 21502 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴 ↔ (𝑎𝑘) ∈ 𝑆))
10328, 102mpbird 256 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴)
104 fnfvima 7146 . . . . . . . . . . . . 13 ((𝐸 Fn (Base‘(Poly1‘ℂfld)) ∧ 𝐴 ⊆ (Base‘(Poly1‘ℂfld)) ∧ ((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) ∈ (𝐸𝐴))
10597, 100, 103, 104syl3anc 1370 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) ∈ (𝐸𝐴))
10692, 105eqeltrrd 2839 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}) ∈ (𝐸𝐴))
10767subrgss 20096 . . . . . . . . . . . . . . . . 17 ((𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)) → (𝐸𝐴) ⊆ (Base‘(ℂflds ℂ)))
10889, 107syl 17 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸𝐴) ⊆ (Base‘(ℂflds ℂ)))
10960ad2antrr 723 . . . . . . . . . . . . . . . . . . 19 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
110 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘(Poly1‘ℂfld))
111110subrgsubm 20108 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ (SubRing‘(Poly1‘ℂfld)) → 𝐴 ∈ (SubMnd‘(mulGrp‘(Poly1‘ℂfld))))
112109, 111syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ (SubMnd‘(mulGrp‘(Poly1‘ℂfld))))
11326adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
114 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (var1‘ℂfld) = (var1‘ℂfld)
115114, 101, 57, 58, 59subrgvr1cl 21504 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (var1‘ℂfld) ∈ 𝐴)
116 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (.g‘(mulGrp‘(Poly1‘ℂfld))) = (.g‘(mulGrp‘(Poly1‘ℂfld)))
117116submmulgcl 18813 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (SubMnd‘(mulGrp‘(Poly1‘ℂfld))) ∧ 𝑘 ∈ ℕ0 ∧ (var1‘ℂfld) ∈ 𝐴) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
118112, 113, 115, 117syl3anc 1370 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
119 fnfvima 7146 . . . . . . . . . . . . . . . . 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))) ∈ (𝐸𝐴))
12097, 100, 118, 119syl3anc 1370 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (𝐸𝐴))
121108, 120sseldd 3931 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(ℂflds ℂ)))
1223, 4, 67, 68, 69, 121pwselbas 17267 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))):ℂ⟶ℂ)
123122feqmptd 6874 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) = (𝑧 ∈ ℂ ↦ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧)))
12452a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CRing)
125 simpr 485 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
12653, 114, 4, 54, 93, 124, 125evl1vard 21574 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → ((var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(var1‘ℂfld))‘𝑧) = 𝑧))
127 eqid 2737 . . . . . . . . . . . . . . . . 17 (.g‘(mulGrp‘ℂfld)) = (.g‘(mulGrp‘ℂfld))
128113adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ0)
12953, 54, 4, 93, 124, 125, 126, 116, 127, 128evl1expd 21582 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 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)))
130129simprd 496 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑘(.g‘(mulGrp‘ℂfld))𝑧))
131 cnfldexp 20702 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧𝑘))
132125, 128, 131syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧𝑘))
133130, 132eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑧𝑘))
134133mpteq2dva 5185 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧)) = (𝑧 ∈ ℂ ↦ (𝑧𝑘)))
135123, 134eqtrd 2777 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) = (𝑧 ∈ ℂ ↦ (𝑧𝑘)))
136135, 120eqeltrrd 2839 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (𝐸𝐴))
13781subrgmcl 20107 . . . . . . . . . . 11 (((𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)) ∧ (ℂ × {(𝑎𝑘)}) ∈ (𝐸𝐴) ∧ (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (𝐸𝐴)) → ((ℂ × {(𝑎𝑘)})(.r‘(ℂflds ℂ))(𝑧 ∈ ℂ ↦ (𝑧𝑘))) ∈ (𝐸𝐴))
13889, 106, 136, 137syl3anc 1370 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)})(.r‘(ℂflds ℂ))(𝑧 ∈ ℂ ↦ (𝑧𝑘))) ∈ (𝐸𝐴))
13988, 138eqeltrrd 2839 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴))
140139fmpttd 7026 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))):(0...𝑛)⟶(𝐸𝐴))
14136, 8, 139, 40fsuppmptdm 9207 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))) finSupp (0g‘(ℂflds ℂ)))
1425, 51, 8, 66, 140, 141gsumsubmcl 19587 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ((ℂflds ℂ) Σg (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))) ∈ (𝐸𝐴))
14347, 142eqeltrrd 2839 . . . . . 6 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴))
144 eleq1 2825 . . . . . 6 (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) → (𝑓 ∈ (𝐸𝐴) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴)))
145143, 144syl5ibrcom 246 . . . . 5 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) → 𝑓 ∈ (𝐸𝐴)))
146145rexlimdvva 3202 . . . 4 (𝑆 ∈ (SubRing‘ℂfld) → (∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) → 𝑓 ∈ (𝐸𝐴)))
1472, 146syl5 34 . . 3 (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (Poly‘𝑆) → 𝑓 ∈ (𝐸𝐴)))
148 ffun 6638 . . . . . 6 (𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ)) → Fun 𝐸)
14995, 148ax-mp 5 . . . . 5 Fun 𝐸
150 fvelima 6872 . . . . 5 ((Fun 𝐸𝑓 ∈ (𝐸𝐴)) → ∃𝑎𝐴 (𝐸𝑎) = 𝑓)
151149, 150mpan 687 . . . 4 (𝑓 ∈ (𝐸𝐴) → ∃𝑎𝐴 (𝐸𝑎) = 𝑓)
15299sselda 3930 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝑎 ∈ (Base‘(Poly1‘ℂfld)))
153 eqid 2737 . . . . . . . . . . . 12 ( ·𝑠 ‘(Poly1‘ℂfld)) = ( ·𝑠 ‘(Poly1‘ℂfld))
154 eqid 2737 . . . . . . . . . . . 12 (coe1𝑎) = (coe1𝑎)
15554, 114, 93, 153, 110, 116, 154ply1coe 21538 . . . . . . . . . . 11 ((ℂfld ∈ Ring ∧ 𝑎 ∈ (Base‘(Poly1‘ℂfld))) → 𝑎 = ((Poly1‘ℂfld) Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))))
1569, 152, 155sylancr 587 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝑎 = ((Poly1‘ℂfld) Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))))
157156fveq2d 6813 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) = (𝐸‘((Poly1‘ℂfld) Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))))
158 eqid 2737 . . . . . . . . . 10 (0g‘(Poly1‘ℂfld)) = (0g‘(Poly1‘ℂfld))
15954ply1ring 21490 . . . . . . . . . . . 12 (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ Ring)
1609, 159ax-mp 5 . . . . . . . . . . 11 (Poly1‘ℂfld) ∈ Ring
161 ringcmn 19887 . . . . . . . . . . 11 ((Poly1‘ℂfld) ∈ Ring → (Poly1‘ℂfld) ∈ CMnd)
162160, 161mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (Poly1‘ℂfld) ∈ CMnd)
163 ringmnd 19860 . . . . . . . . . . 11 ((ℂflds ℂ) ∈ Ring → (ℂflds ℂ) ∈ Mnd)
16449, 163mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (ℂflds ℂ) ∈ Mnd)
165 nn0ex 12309 . . . . . . . . . . 11 0 ∈ V
166165a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℕ0 ∈ V)
167 rhmghm 20036 . . . . . . . . . . . 12 (𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂflds ℂ)) → 𝐸 ∈ ((Poly1‘ℂfld) GrpHom (ℂflds ℂ)))
16856, 167ax-mp 5 . . . . . . . . . . 11 𝐸 ∈ ((Poly1‘ℂfld) GrpHom (ℂflds ℂ))
169 ghmmhm 18911 . . . . . . . . . . 11 (𝐸 ∈ ((Poly1‘ℂfld) GrpHom (ℂflds ℂ)) → 𝐸 ∈ ((Poly1‘ℂfld) MndHom (ℂflds ℂ)))
170168, 169mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝐸 ∈ ((Poly1‘ℂfld) MndHom (ℂflds ℂ)))
17154ply1lmod 21494 . . . . . . . . . . . . 13 (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ LMod)
1729, 171mp1i 13 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (Poly1‘ℂfld) ∈ LMod)
17312ad2antrr 723 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → 𝑆 ⊆ ℂ)
174 eqid 2737 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
175154, 59, 58, 174coe1f 21453 . . . . . . . . . . . . . . . 16 (𝑎𝐴 → (coe1𝑎):ℕ0⟶(Base‘𝑅))
17657subrgbas 20104 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 = (Base‘𝑅))
177176feq3d 6622 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (SubRing‘ℂfld) → ((coe1𝑎):ℕ0𝑆 ↔ (coe1𝑎):ℕ0⟶(Base‘𝑅)))
178175, 177syl5ibr 245 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubRing‘ℂfld) → (𝑎𝐴 → (coe1𝑎):ℕ0𝑆))
179178imp 407 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎):ℕ0𝑆)
180179ffvelcdmda 6998 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ 𝑆)
181173, 180sseldd 3931 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ ℂ)
182110ringmgp 19856 . . . . . . . . . . . . . 14 ((Poly1‘ℂfld) ∈ Ring → (mulGrp‘(Poly1‘ℂfld)) ∈ Mnd)
183160, 182mp1i 13 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘(Poly1‘ℂfld)) ∈ Mnd)
184 simpr 485 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
185114, 54, 93vr1cl 21459 . . . . . . . . . . . . . 14 (ℂfld ∈ Ring → (var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)))
1869, 185mp1i 13 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)))
187110, 93mgpbas 19793 . . . . . . . . . . . . . 14 (Base‘(Poly1‘ℂfld)) = (Base‘(mulGrp‘(Poly1‘ℂfld)))
188187, 116mulgnn0cl 18787 . . . . . . . . . . . . 13 (((mulGrp‘(Poly1‘ℂfld)) ∈ Mnd ∧ 𝑘 ∈ ℕ0 ∧ (var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld))) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)))
189183, 184, 186, 188syl3anc 1370 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)))
19054ply1sca 21495 . . . . . . . . . . . . . 14 (ℂfld ∈ Ring → ℂfld = (Scalar‘(Poly1‘ℂfld)))
1919, 190ax-mp 5 . . . . . . . . . . . . 13 fld = (Scalar‘(Poly1‘ℂfld))
19293, 191, 153, 4lmodvscl 20211 . . . . . . . . . . . 12 (((Poly1‘ℂfld) ∈ LMod ∧ ((coe1𝑎)‘𝑘) ∈ ℂ ∧ (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld))) → (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)))
193172, 181, 189, 192syl3anc 1370 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)))
194193fmpttd 7026 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℕ0⟶(Base‘(Poly1‘ℂfld)))
195165mptex 7136 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V
196 funmpt 6506 . . . . . . . . . . . . 13 Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))
197 fvex 6822 . . . . . . . . . . . . 13 (0g‘(Poly1‘ℂfld)) ∈ V
198195, 196, 1973pm3.2i 1338 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∧ (0g‘(Poly1‘ℂfld)) ∈ V)
199198a1i 11 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∧ (0g‘(Poly1‘ℂfld)) ∈ V))
200154, 93, 54, 17coe1sfi 21455 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (coe1𝑎) finSupp 0)
201152, 200syl 17 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎) finSupp 0)
202201fsuppimpd 9203 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((coe1𝑎) supp 0) ∈ Fin)
203179feqmptd 6874 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎) = (𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)))
204203oveq1d 7328 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((coe1𝑎) supp 0) = ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0))
205 eqimss2 3987 . . . . . . . . . . . . 13 (((coe1𝑎) supp 0) = ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0) → ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0) ⊆ ((coe1𝑎) supp 0))
206204, 205syl 17 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0) ⊆ ((coe1𝑎) supp 0))
2079, 171mp1i 13 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (Poly1‘ℂfld) ∈ LMod)
20893, 191, 153, 17, 158lmod0vs 20227 . . . . . . . . . . . . 13 (((Poly1‘ℂfld) ∈ LMod ∧ 𝑥 ∈ (Base‘(Poly1‘ℂfld))) → (0( ·𝑠 ‘(Poly1‘ℂfld))𝑥) = (0g‘(Poly1‘ℂfld)))
209207, 208sylan 580 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑥 ∈ (Base‘(Poly1‘ℂfld))) → (0( ·𝑠 ‘(Poly1‘ℂfld))𝑥) = (0g‘(Poly1‘ℂfld)))
210 c0ex 11039 . . . . . . . . . . . . 13 0 ∈ V
211210a1i 11 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 0 ∈ V)
212206, 209, 180, 189, 211suppssov1 8059 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) supp (0g‘(Poly1‘ℂfld))) ⊆ ((coe1𝑎) supp 0))
213 suppssfifsupp 9211 . . . . . . . . . . 11 ((((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∧ (0g‘(Poly1‘ℂfld)) ∈ V) ∧ (((coe1𝑎) supp 0) ∈ Fin ∧ ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) supp (0g‘(Poly1‘ℂfld))) ⊆ ((coe1𝑎) supp 0))) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) finSupp (0g‘(Poly1‘ℂfld)))
214199, 202, 212, 213syl12anc 834 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) finSupp (0g‘(Poly1‘ℂfld)))
21593, 158, 162, 164, 166, 170, 194, 214gsummhm 19606 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((ℂflds ℂ) Σg (𝐸 ∘ (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))) = (𝐸‘((Poly1‘ℂfld) Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))))
21695a1i 11 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ)))
217216, 193cofmpt 7041 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸 ∘ (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))) = (𝑘 ∈ ℕ0 ↦ (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))))
2189a1i 11 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ℂfld ∈ Ring)
2196a1i 11 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ℂ ∈ V)
22095ffvelcdmi 6997 . . . . . . . . . . . . . . . 16 ((((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ (Base‘(ℂflds ℂ)))
221193, 220syl 17 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ (Base‘(ℂflds ℂ)))
2223, 4, 67, 218, 219, 221pwselbas 17267 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℂ⟶ℂ)
223222feqmptd 6874 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) = (𝑧 ∈ ℂ ↦ ((𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))‘𝑧)))
22452a1i 11 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CRing)
225 simpr 485 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
22653, 114, 4, 54, 93, 224, 225evl1vard 21574 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(var1‘ℂfld))‘𝑧) = 𝑧))
227184adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ0)
22853, 54, 4, 93, 224, 225, 226, 116, 127, 227evl1expd 21582 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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)))
229225, 227, 131syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧𝑘))
230229eqeq2d 2748 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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) = (𝑧𝑘)))
231230anbi2d 629 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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) = (𝑧𝑘))))
232228, 231mpbid 231 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))‘𝑧) = (𝑧𝑘)))
233181adantr 481 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((coe1𝑎)‘𝑘) ∈ ℂ)
23453, 54, 4, 93, 224, 225, 232, 233, 153, 80evl1vsd 21581 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))‘𝑧) = (((coe1𝑎)‘𝑘) · (𝑧𝑘))))
235234simprd 496 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))‘𝑧) = (((coe1𝑎)‘𝑘) · (𝑧𝑘)))
236235mpteq2dva 5185 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))‘𝑧)) = (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))
237223, 236eqtrd 2777 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) = (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))
238237mpteq2dva 5185 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))) = (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))))
239217, 238eqtrd 2777 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸 ∘ (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))) = (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))))
240239oveq2d 7329 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((ℂflds ℂ) Σg (𝐸 ∘ (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))) = ((ℂflds ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))))
241157, 215, 2403eqtr2d 2783 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) = ((ℂflds ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))))
2426a1i 11 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℂ ∈ V)
2439, 10mp1i 13 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℂfld ∈ CMnd)
244181adantlr 712 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ ℂ)
24533adantll 711 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧𝑘) ∈ ℂ)
246244, 245mulcld 11065 . . . . . . . . . 10 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
247246anasss 467 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0)) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
248165mptex 7136 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∈ V
249 funmpt 6506 . . . . . . . . . . . 12 Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))
250248, 249, 393pm3.2i 1338 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∧ (0g‘(ℂflds ℂ)) ∈ V)
251250a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∧ (0g‘(ℂflds ℂ)) ∈ V))
252 fzfid 13763 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin)
253 eldifn 4072 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → ¬ 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
254253adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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)))
255152ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑎 ∈ (Base‘(Poly1‘ℂfld)))
256 eldifi 4071 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → 𝑘 ∈ ℕ0)
257256adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℕ0)
258 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 ( deg1 ‘ℂfld) = ( deg1 ‘ℂfld)
259258, 54, 93, 17, 154deg1ge 25334 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 ∈ (Base‘(Poly1‘ℂfld)) ∧ 𝑘 ∈ ℕ0 ∧ ((coe1𝑎)‘𝑘) ≠ 0) → 𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎))
2602593expia 1120 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ (Base‘(Poly1‘ℂfld)) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘) ≠ 0 → 𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎)))
261255, 257, 260syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) ≠ 0 → 𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎)))
262 0xr 11092 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ*
263258, 54, 93deg1xrcl 25318 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
264152, 263syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
265264ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
266 xrmax2 12980 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℝ* ∧ (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*) → (( deg1 ‘ℂfld)‘𝑎) ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))
267262, 265, 266sylancr 587 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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))
268257nn0red 12364 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ)
269268rexrd 11095 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ*)
270 ifcl 4514 . . . . . . . . . . . . . . . . . . . . . . . 24 (((( deg1 ‘ℂfld)‘𝑎) ∈ ℝ* ∧ 0 ∈ ℝ*) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℝ*)
271265, 262, 270sylancl 586 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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 12962 . . . . . . . . . . . . . . . . . . . . . . 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)))
273269, 265, 271, 272syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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)))
274267, 273mpan2d 691 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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)))
275261, 274syld 47 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) ≠ 0 → 𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
276275, 257jctild 526 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) ≠ 0 → (𝑘 ∈ ℕ0𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))))
277258, 54, 93deg1cl 25319 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
278152, 277syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
279 elun 4093 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}))
280278, 279sylib 217 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}))
281 nn0ge0 12328 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → 0 ≤ (( deg1 ‘ℂfld)‘𝑎))
282281iftrued 4477 . . . . . . . . . . . . . . . . . . . . . . . 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)
284282, 283eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
285 mnflt0 12931 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -∞ < 0
286 mnfxr 11102 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 -∞ ∈ ℝ*
287 xrltnle 11112 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞))
288286, 262, 287mp2an 689 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (-∞ < 0 ↔ ¬ 0 ≤ -∞)
289285, 288mpbi 229 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ¬ 0 ≤ -∞
290 elsni 4586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → (( deg1 ‘ℂfld)‘𝑎) = -∞)
291290breq2d 5097 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → (0 ≤ (( deg1 ‘ℂfld)‘𝑎) ↔ 0 ≤ -∞))
292289, 291mtbiri 326 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → ¬ 0 ≤ (( deg1 ‘ℂfld)‘𝑎))
293292iffalsed 4480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) = 0)
294 0nn0 12318 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℕ0
295293, 294eqeltrdi 2846 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
296284, 295jaoi 854 . . . . . . . . . . . . . . . . . . . . . 22 (((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
297280, 296syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
298297ad2antrr 723 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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 13418 . . . . . . . . . . . . . . . . . . . 20 (if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0 → (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↔ (𝑘 ∈ ℕ0𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))))
300298, 299syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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 ∈ ℕ0𝑘 ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))))
301276, 300sylibrd 258 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) ≠ 0 → 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))))
302301necon1bd 2959 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 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)) → ((coe1𝑎)‘𝑘) = 0))
303254, 302mpd 15 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → ((coe1𝑎)‘𝑘) = 0)
304303oveq1d 7328 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = (0 · (𝑧𝑘)))
305256, 245sylan2 593 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑧𝑘) ∈ ℂ)
306305mul02d 11243 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (0 · (𝑧𝑘)) = 0)
307304, 306eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = 0)
308307an32s 649 . . . . . . . . . . . . 13 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) ∧ 𝑧 ∈ ℂ) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = 0)
309308mpteq2dva 5185 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ 0))
310 fconstmpt 5665 . . . . . . . . . . . . 13 (ℂ × {0}) = (𝑧 ∈ ℂ ↦ 0)
311 ringmnd 19860 . . . . . . . . . . . . . . 15 (ℂfld ∈ Ring → ℂfld ∈ Mnd)
3129, 311ax-mp 5 . . . . . . . . . . . . . 14 fld ∈ Mnd
3133, 17pws0g 18488 . . . . . . . . . . . . . 14 ((ℂfld ∈ Mnd ∧ ℂ ∈ V) → (ℂ × {0}) = (0g‘(ℂflds ℂ)))
314312, 6, 313mp2an 689 . . . . . . . . . . . . 13 (ℂ × {0}) = (0g‘(ℂflds ℂ))
315310, 314eqtr3i 2767 . . . . . . . . . . . 12 (𝑧 ∈ ℂ ↦ 0) = (0g‘(ℂflds ℂ))
316309, 315eqtrdi 2793 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) = (0g‘(ℂflds ℂ)))
317316, 166suppss2 8061 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) supp (0g‘(ℂflds ℂ))) ⊆ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
318 suppssfifsupp 9211 . . . . . . . . . 10 ((((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∧ (0g‘(ℂflds ℂ)) ∈ V) ∧ ((0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin ∧ ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) supp (0g‘(ℂflds ℂ))) ⊆ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) finSupp (0g‘(ℂflds ℂ)))
319251, 252, 317, 318syl12anc 834 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) finSupp (0g‘(ℂflds ℂ)))
3203, 4, 5, 242, 166, 243, 247, 319pwsgsum 19650 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((ℂflds ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))) = (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))))
321 fz0ssnn0 13421 . . . . . . . . . . . 12 (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ⊆ ℕ0
322 resmpt 5962 . . . . . . . . . . . 12 ((0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ⊆ ℕ0 → ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ↾ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) = (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))
323321, 322ax-mp 5 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ↾ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) = (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))
324323oveq2i 7324 . . . . . . . . . 10 (ℂfld Σg ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ↾ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) = (ℂfld Σg (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))
3259, 10mp1i 13 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CMnd)
326165a1i 11 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ℕ0 ∈ V)
327246fmpttd 7026 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))):ℕ0⟶ℂ)
328307, 326suppss2 8061 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) supp 0) ⊆ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
329165mptex 7136 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V
330 funmpt 6506 . . . . . . . . . . . . . 14 Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))
331329, 330, 2103pm3.2i 1338 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∧ 0 ∈ V)
332331a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∧ 0 ∈ V))
333 fzfid 13763 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin)
334 suppssfifsupp 9211 . . . . . . . . . . . 12 ((((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∧ 0 ∈ V) ∧ ((0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin ∧ ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) supp 0) ⊆ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) finSupp 0)
335332, 333, 328, 334syl12anc 834 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) finSupp 0)
3364, 17, 325, 326, 327, 328, 335gsumres 19581 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ↾ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) = (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))))
337 elfznn0 13419 . . . . . . . . . . . 12 (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) → 𝑘 ∈ ℕ0)
338337, 246sylan2 593 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
339333, 338gsumfsum 20736 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) = Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘)))
340324, 336, 3393eqtr3a 2801 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) = Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘)))
341340mpteq2dva 5185 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘))))
342241, 320, 3413eqtrd 2781 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘))))
34312adantr 481 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝑆 ⊆ ℂ)
344 elplyr 25433 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0 ∧ (coe1𝑎):ℕ0𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ (Poly‘𝑆))
345343, 297, 179, 344syl3anc 1370 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ (Poly‘𝑆))
346342, 345eqeltrd 2838 . . . . . 6 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) ∈ (Poly‘𝑆))
347 eleq1 2825 . . . . . 6 ((𝐸𝑎) = 𝑓 → ((𝐸𝑎) ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (Poly‘𝑆)))
348346, 347syl5ibcom 244 . . . . 5 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝐸𝑎) = 𝑓𝑓 ∈ (Poly‘𝑆)))
349348rexlimdva 3149 . . . 4 (𝑆 ∈ (SubRing‘ℂfld) → (∃𝑎𝐴 (𝐸𝑎) = 𝑓𝑓 ∈ (Poly‘𝑆)))
350151, 349syl5 34 . . 3 (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (𝐸𝐴) → 𝑓 ∈ (Poly‘𝑆)))
351147, 350impbid 211 . 2 (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (𝐸𝐴)))
352351eqrdv 2735 1 (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844  w3a 1086   = wceq 1540  wcel 2105  wne 2941  wrex 3071  Vcvv 3441  cdif 3893  cun 3894  wss 3896  ifcif 4469  {csn 4569   class class class wbr 5085  cmpt 5168   × cxp 5603  cres 5607  cima 5608  ccom 5609  Fun wfun 6457   Fn wfn 6458  wf 6459  cfv 6463  (class class class)co 7313  f cof 7569   supp csupp 8022  m cmap 8661  Fincfn 8779   finSupp cfsupp 9196  cc 10939  0cc0 10941   · cmul 10946  -∞cmnf 11077  *cxr 11078   < clt 11079  cle 11080  0cn0 12303  ...cfz 13309  cexp 13852  Σcsu 15466  Basecbs 16979  s cress 17008  .rcmulr 17030  Scalarcsca 17032   ·𝑠 cvsca 17033  0gc0g 17217   Σg cgsu 17218  s cpws 17224  Mndcmnd 18452   MndHom cmhm 18495  SubMndcsubmnd 18496  .gcmg 18767  SubGrpcsubg 18816   GrpHom cghm 18898  CMndccmn 19453  mulGrpcmgp 19787  Ringcrg 19850  CRingccrg 19851   RingHom crh 20023  SubRingcsubrg 20091  LModclmod 20194  fldccnfld 20668  algSccascl 21130  var1cv1 21418  Poly1cpl1 21419  coe1cco1 21420  eval1ce1 21551   deg1 cdg1 25287  Polycply 25416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626  ax-inf2 9467  ax-cnex 10997  ax-resscn 10998  ax-1cn 10999  ax-icn 11000  ax-addcl 11001  ax-addrcl 11002  ax-mulcl 11003  ax-mulrcl 11004  ax-mulcom 11005  ax-addass 11006  ax-mulass 11007  ax-distr 11008  ax-i2m1 11009  ax-1ne0 11010  ax-1rid 11011  ax-rnegex 11012  ax-rrecex 11013  ax-cnre 11014  ax-pre-lttri 11015  ax-pre-lttrn 11016  ax-pre-ltadd 11017  ax-pre-mulgt0 11018  ax-pre-sup 11019  ax-addf 11020  ax-mulf 11021
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4849  df-int 4891  df-iun 4937  df-iin 4938  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-se 5561  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-pred 6222  df-ord 6289  df-on 6290  df-lim 6291  df-suc 6292  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-isom 6472  df-riota 7270  df-ov 7316  df-oprab 7317  df-mpo 7318  df-of 7571  df-ofr 7572  df-om 7756  df-1st 7874  df-2nd 7875  df-supp 8023  df-frecs 8142  df-wrecs 8173  df-recs 8247  df-rdg 8286  df-1o 8342  df-er 8544  df-map 8663  df-pm 8664  df-ixp 8732  df-en 8780  df-dom 8781  df-sdom 8782  df-fin 8783  df-fsupp 9197  df-sup 9269  df-oi 9337  df-card 9765  df-pnf 11081  df-mnf 11082  df-xr 11083  df-ltxr 11084  df-le 11085  df-sub 11277  df-neg 11278  df-div 11703  df-nn 12044  df-2 12106  df-3 12107  df-4 12108  df-5 12109  df-6 12110  df-7 12111  df-8 12112  df-9 12113  df-n0 12304  df-z 12390  df-dec 12508  df-uz 12653  df-rp 12801  df-fz 13310  df-fzo 13453  df-seq 13792  df-exp 13853  df-hash 14115  df-cj 14879  df-re 14880  df-im 14881  df-sqrt 15015  df-abs 15016  df-clim 15266  df-sum 15467  df-struct 16915  df-sets 16932  df-slot 16950  df-ndx 16962  df-base 16980  df-ress 17009  df-plusg 17042  df-mulr 17043  df-starv 17044  df-sca 17045  df-vsca 17046  df-ip 17047  df-tset 17048  df-ple 17049  df-ds 17051  df-unif 17052  df-hom 17053  df-cco 17054  df-0g 17219  df-gsum 17220  df-prds 17225  df-pws 17227  df-mre 17362  df-mrc 17363  df-acs 17365  df-mgm 18393  df-sgrp 18442  df-mnd 18453  df-mhm 18497  df-submnd 18498  df-grp 18647  df-minusg 18648  df-sbg 18649  df-mulg 18768  df-subg 18819  df-ghm 18899  df-cntz 18990  df-cmn 19455  df-abl 19456  df-mgp 19788  df-ur 19805  df-srg 19809  df-ring 19852  df-cring 19853  df-rnghom 20026  df-subrg 20093  df-lmod 20196  df-lss 20265  df-lsp 20305  df-cnfld 20669  df-assa 21131  df-asp 21132  df-ascl 21133  df-psr 21183  df-mvr 21184  df-mpl 21185  df-opsr 21187  df-evls 21353  df-evl 21354  df-psr1 21422  df-vr1 21423  df-ply1 21424  df-coe1 21425  df-evl1 21553  df-mdeg 25288  df-deg1 25289  df-ply 25420
This theorem is referenced by: (None)
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