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Theorem plypf1 24416
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 24399 . . . . 5 (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
21simprbi 492 . . . 4 (𝑓 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
3 eqid 2778 . . . . . . . . 9 (ℂflds ℂ) = (ℂflds ℂ)
4 cnfldbas 20157 . . . . . . . . 9 ℂ = (Base‘ℂfld)
5 eqid 2778 . . . . . . . . 9 (0g‘(ℂflds ℂ)) = (0g‘(ℂflds ℂ))
6 cnex 10355 . . . . . . . . . 10 ℂ ∈ V
76a1i 11 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → ℂ ∈ V)
8 fzfid 13096 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (0...𝑛) ∈ Fin)
9 cnring 20175 . . . . . . . . . 10 fld ∈ Ring
10 ringcmn 18979 . . . . . . . . . 10 (ℂfld ∈ Ring → ℂfld ∈ CMnd)
119, 10mp1i 13 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → ℂfld ∈ CMnd)
124subrgss 19184 . . . . . . . . . . . . 13 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ)
1312ad2antrr 716 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ⊆ ℂ)
14 elmapi 8164 . . . . . . . . . . . . . . 15 (𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
1514ad2antll 719 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
16 subrgsubg 19189 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld))
17 cnfld0 20177 . . . . . . . . . . . . . . . . . . . 20 0 = (0g‘ℂfld)
1817subg0cl 17997 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆)
1916, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝑆 ∈ (SubRing‘ℂfld) → 0 ∈ 𝑆)
2019adantr 474 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → 0 ∈ 𝑆)
2120snssd 4573 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → {0} ⊆ 𝑆)
22 ssequn2 4009 . . . . . . . . . . . . . . . 16 ({0} ⊆ 𝑆 ↔ (𝑆 ∪ {0}) = 𝑆)
2321, 22sylib 210 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (𝑆 ∪ {0}) = 𝑆)
2423feq3d 6280 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (𝑎:ℕ0⟶(𝑆 ∪ {0}) ↔ 𝑎:ℕ0𝑆))
2515, 24mpbid 224 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → 𝑎:ℕ0𝑆)
26 elfznn0 12756 . . . . . . . . . . . . 13 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
27 ffvelrn 6623 . . . . . . . . . . . . 13 ((𝑎:ℕ0𝑆𝑘 ∈ ℕ0) → (𝑎𝑘) ∈ 𝑆)
2825, 26, 27syl2an 589 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) ∈ 𝑆)
2913, 28sseldd 3822 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) ∈ ℂ)
3029adantrl 706 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → (𝑎𝑘) ∈ ℂ)
31 simprl 761 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑧 ∈ ℂ)
3226ad2antll 719 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑘 ∈ ℕ0)
33 expcl 13201 . . . . . . . . . . 11 ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑧𝑘) ∈ ℂ)
3431, 32, 33syl2anc 579 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → (𝑧𝑘) ∈ ℂ)
3530, 34mulcld 10399 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → ((𝑎𝑘) · (𝑧𝑘)) ∈ ℂ)
36 eqid 2778 . . . . . . . . . 10 (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))
376mptex 6760 . . . . . . . . . . 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 6461 . . . . . . . . . . 11 (0g‘(ℂflds ℂ)) ∈ V
4039a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (0g‘(ℂflds ℂ)) ∈ V)
4136, 8, 38, 40fsuppmptdm 8576 . . . . . . . . 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 18775 . . . . . . . 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 13096 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑧 ∈ ℂ) → (0...𝑛) ∈ Fin)
4435anassrs 461 . . . . . . . . . 10 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎𝑘) · (𝑧𝑘)) ∈ ℂ)
4543, 44gsumfsum 20220 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑎𝑘) · (𝑧𝑘)))) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
4645mpteq2dva 4981 . . . . . . . 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 2814 . . . . . . 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 19013 . . . . . . . . . 10 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → (ℂflds ℂ) ∈ Ring)
499, 6, 48mp2an 682 . . . . . . . . 9 (ℂflds ℂ) ∈ Ring
50 ringcmn 18979 . . . . . . . . 9 ((ℂflds ℂ) ∈ Ring → (ℂflds ℂ) ∈ CMnd)
5149, 50mp1i 13 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (ℂflds ℂ) ∈ CMnd)
52 cncrng 20174 . . . . . . . . . . 11 fld ∈ CRing
53 plypf1.e . . . . . . . . . . . 12 𝐸 = (eval1‘ℂfld)
54 eqid 2778 . . . . . . . . . . . 12 (Poly1‘ℂfld) = (Poly1‘ℂfld)
5553, 54, 3, 4evl1rhm 20103 . . . . . . . . . . 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 20010 . . . . . . . . . . 11 (𝑆 ∈ (SubRing‘ℂfld) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
6160adantr 474 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
62 rhmima 19214 . . . . . . . . . 10 ((𝐸 ∈ ((Poly1‘ℂfld) RingHom (ℂflds ℂ)) ∧ 𝐴 ∈ (SubRing‘(Poly1‘ℂfld))) → (𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)))
6356, 61, 62sylancr 581 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)))
64 subrgsubg 19189 . . . . . . . . 9 ((𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)) → (𝐸𝐴) ∈ (SubGrp‘(ℂflds ℂ)))
65 subgsubm 18011 . . . . . . . . 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 2778 . . . . . . . . . . . 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 6346 . . . . . . . . . . . . . 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 16545 . . . . . . . . . . . . . 14 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)) ↔ (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ))
739, 6, 72mp2an 682 . . . . . . . . . . . . 13 ((ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)) ↔ (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ)
7471, 73sylibr 226 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)))
7534anass1rs 645 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑧𝑘) ∈ ℂ)
7675fmpttd 6651 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ)
773, 4, 67pwselbasb 16545 . . . . . . . . . . . . . 14 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ))
789, 6, 77mp2an 682 . . . . . . . . . . . . 13 ((𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ)
7976, 78sylibr 226 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)))
80 cnfldmul 20159 . . . . . . . . . . . 12 · = (.r‘ℂfld)
81 eqid 2778 . . . . . . . . . . . 12 (.r‘(ℂflds ℂ)) = (.r‘(ℂflds ℂ))
823, 67, 68, 69, 74, 79, 80, 81pwsmulrval 16548 . . . . . . . . . . 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 474 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑎𝑘) ∈ ℂ)
84 fconstmpt 5413 . . . . . . . . . . . . 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 2779 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) = (𝑧 ∈ ℂ ↦ (𝑧𝑘)))
8769, 83, 75, 85, 86offval2 7193 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))
8882, 87eqtrd 2814 . . . . . . . . . 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 474 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)))
90 eqid 2778 . . . . . . . . . . . . . 14 (algSc‘(Poly1‘ℂfld)) = (algSc‘(Poly1‘ℂfld))
9153, 54, 4, 90evl1sca 20105 . . . . . . . . . . . . 13 ((ℂfld ∈ CRing ∧ (𝑎𝑘) ∈ ℂ) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) = (ℂ × {(𝑎𝑘)}))
9252, 29, 91sylancr 581 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) = (ℂ × {(𝑎𝑘)}))
93 eqid 2778 . . . . . . . . . . . . . . . 16 (Base‘(Poly1‘ℂfld)) = (Base‘(Poly1‘ℂfld))
9493, 67rhmf 19126 . . . . . . . . . . . . . . 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 6293 . . . . . . . . . . . . . 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 19184 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubRing‘(Poly1‘ℂfld)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
9960, 98syl 17 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubRing‘ℂfld) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
10099ad2antrr 716 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
101 simpll 757 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ∈ (SubRing‘ℂfld))
10254, 90, 57, 58, 101, 59, 4, 29subrg1asclcl 20037 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴 ↔ (𝑎𝑘) ∈ 𝑆))
10328, 102mpbird 249 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → ((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴)
104 fnfvima 6771 . . . . . . . . . . . . 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 1439 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) ∈ (𝐸𝐴))
10692, 105eqeltrrd 2860 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}) ∈ (𝐸𝐴))
10767subrgss 19184 . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . . . 19 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
110 eqid 2778 . . . . . . . . . . . . . . . . . . . 20 (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘(Poly1‘ℂfld))
111110subrgsubm 19196 . . . . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
114 eqid 2778 . . . . . . . . . . . . . . . . . . 19 (var1‘ℂfld) = (var1‘ℂfld)
115114, 101, 57, 58, 59subrgvr1cl 20039 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (var1‘ℂfld) ∈ 𝐴)
116 eqid 2778 . . . . . . . . . . . . . . . . . . 19 (.g‘(mulGrp‘(Poly1‘ℂfld))) = (.g‘(mulGrp‘(Poly1‘ℂfld)))
117116submmulgcl 17980 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (SubMnd‘(mulGrp‘(Poly1‘ℂfld))) ∧ 𝑘 ∈ ℕ0 ∧ (var1‘ℂfld) ∈ 𝐴) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
118112, 113, 115, 117syl3anc 1439 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
119 fnfvima 6771 . . . . . . . . . . . . . . . . 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 1439 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (𝐸𝐴))
121108, 120sseldd 3822 . . . . . . . . . . . . . . 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 16546 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))):ℂ⟶ℂ)
123122feqmptd 6511 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
12653, 114, 4, 54, 93, 124, 125evl1vard 20108 . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . . 17 (.g‘(mulGrp‘ℂfld)) = (.g‘(mulGrp‘ℂfld))
128113adantr 474 . . . . . . . . . . . . . . . . 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 20116 . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . 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 20186 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧𝑘))
132125, 128, 131syl2anc 579 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧𝑘))
133130, 132eqtrd 2814 . . . . . . . . . . . . . 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 4981 . . . . . . . . . . . . 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 2814 . . . . . . . . . . . 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 2860 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (𝐸𝐴))
13781subrgmcl 19195 . . . . . . . . . . 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 1439 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)})(.r‘(ℂflds ℂ))(𝑧 ∈ ℂ ↦ (𝑧𝑘))) ∈ (𝐸𝐴))
13988, 138eqeltrrd 2860 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴))
140139fmpttd 6651 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))):(0...𝑛)⟶(𝐸𝐴))
14136, 8, 139, 40fsuppmptdm 8576 . . . . . . . 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 18716 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → ((ℂflds ℂ) Σg (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))) ∈ (𝐸𝐴))
14347, 142eqeltrrd 2860 . . . . . 6 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴))
144 eleq1 2847 . . . . . 6 (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) → (𝑓 ∈ (𝐸𝐴) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴)))
145143, 144syl5ibrcom 239 . . . . 5 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) → 𝑓 ∈ (𝐸𝐴)))
146145rexlimdvva 3221 . . . 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 6296 . . . . . 6 (𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ)) → Fun 𝐸)
14995, 148ax-mp 5 . . . . 5 Fun 𝐸
150 fvelima 6510 . . . . 5 ((Fun 𝐸𝑓 ∈ (𝐸𝐴)) → ∃𝑎𝐴 (𝐸𝑎) = 𝑓)
151149, 150mpan 680 . . . 4 (𝑓 ∈ (𝐸𝐴) → ∃𝑎𝐴 (𝐸𝑎) = 𝑓)
15299sselda 3821 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝑎 ∈ (Base‘(Poly1‘ℂfld)))
153 eqid 2778 . . . . . . . . . . . 12 ( ·𝑠 ‘(Poly1‘ℂfld)) = ( ·𝑠 ‘(Poly1‘ℂfld))
154 eqid 2778 . . . . . . . . . . . 12 (coe1𝑎) = (coe1𝑎)
15554, 114, 93, 153, 110, 116, 154ply1coe 20073 . . . . . . . . . . 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 581 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝑎 = ((Poly1‘ℂfld) Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))))
157156fveq2d 6452 . . . . . . . . 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 2778 . . . . . . . . . 10 (0g‘(Poly1‘ℂfld)) = (0g‘(Poly1‘ℂfld))
15954ply1ring 20025 . . . . . . . . . . . 12 (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ Ring)
1609, 159ax-mp 5 . . . . . . . . . . 11 (Poly1‘ℂfld) ∈ Ring
161 ringcmn 18979 . . . . . . . . . . 11 ((Poly1‘ℂfld) ∈ Ring → (Poly1‘ℂfld) ∈ CMnd)
162160, 161mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (Poly1‘ℂfld) ∈ CMnd)
163 ringmnd 18954 . . . . . . . . . . 11 ((ℂflds ℂ) ∈ Ring → (ℂflds ℂ) ∈ Mnd)
16449, 163mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (ℂflds ℂ) ∈ Mnd)
165 nn0ex 11654 . . . . . . . . . . 11 0 ∈ V
166165a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℕ0 ∈ V)
167 rhmghm 19125 . . . . . . . . . . . 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 18065 . . . . . . . . . . 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 20029 . . . . . . . . . . . . 13 (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ LMod)
1729, 171mp1i 13 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (Poly1‘ℂfld) ∈ LMod)
17312ad2antrr 716 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → 𝑆 ⊆ ℂ)
174 eqid 2778 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
175154, 59, 58, 174coe1f 19988 . . . . . . . . . . . . . . . 16 (𝑎𝐴 → (coe1𝑎):ℕ0⟶(Base‘𝑅))
17657subrgbas 19192 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 = (Base‘𝑅))
177176feq3d 6280 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (SubRing‘ℂfld) → ((coe1𝑎):ℕ0𝑆 ↔ (coe1𝑎):ℕ0⟶(Base‘𝑅)))
178175, 177syl5ibr 238 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubRing‘ℂfld) → (𝑎𝐴 → (coe1𝑎):ℕ0𝑆))
179178imp 397 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎):ℕ0𝑆)
180179ffvelrnda 6625 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ 𝑆)
181173, 180sseldd 3822 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ ℂ)
182110ringmgp 18951 . . . . . . . . . . . . . 14 ((Poly1‘ℂfld) ∈ Ring → (mulGrp‘(Poly1‘ℂfld)) ∈ Mnd)
183160, 182mp1i 13 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘(Poly1‘ℂfld)) ∈ Mnd)
184 simpr 479 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
185114, 54, 93vr1cl 19994 . . . . . . . . . . . . . 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 18893 . . . . . . . . . . . . . 14 (Base‘(Poly1‘ℂfld)) = (Base‘(mulGrp‘(Poly1‘ℂfld)))
188187, 116mulgnn0cl 17955 . . . . . . . . . . . . 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 1439 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)))
19054ply1sca 20030 . . . . . . . . . . . . . 14 (ℂfld ∈ Ring → ℂfld = (Scalar‘(Poly1‘ℂfld)))
1919, 190ax-mp 5 . . . . . . . . . . . . 13 fld = (Scalar‘(Poly1‘ℂfld))
19293, 191, 153, 4lmodvscl 19283 . . . . . . . . . . . 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 1439 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)))
194193fmpttd 6651 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℕ0⟶(Base‘(Poly1‘ℂfld)))
195165mptex 6760 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V
196 funmpt 6175 . . . . . . . . . . . . 13 Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))
197 fvex 6461 . . . . . . . . . . . . 13 (0g‘(Poly1‘ℂfld)) ∈ V
198195, 196, 1973pm3.2i 1395 . . . . . . . . . . . 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 19990 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (coe1𝑎) finSupp 0)
201152, 200syl 17 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎) finSupp 0)
202201fsuppimpd 8572 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((coe1𝑎) supp 0) ∈ Fin)
203179feqmptd 6511 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎) = (𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)))
204203oveq1d 6939 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((coe1𝑎) supp 0) = ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0))
205 eqimss2 3877 . . . . . . . . . . . . 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 19299 . . . . . . . . . . . . 13 (((Poly1‘ℂfld) ∈ LMod ∧ 𝑥 ∈ (Base‘(Poly1‘ℂfld))) → (0( ·𝑠 ‘(Poly1‘ℂfld))𝑥) = (0g‘(Poly1‘ℂfld)))
209207, 208sylan 575 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑥 ∈ (Base‘(Poly1‘ℂfld))) → (0( ·𝑠 ‘(Poly1‘ℂfld))𝑥) = (0g‘(Poly1‘ℂfld)))
210 c0ex 10372 . . . . . . . . . . . . 13 0 ∈ V
211210a1i 11 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 0 ∈ V)
212206, 209, 180, 189, 211suppssov1 7611 . . . . . . . . . . 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 8580 . . . . . . . . . . 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 827 . . . . . . . . . 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 18735 . . . . . . . . 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)))))))
216 eqidd 2779 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) = (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))
21795a1i 11 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ)))
218217feqmptd 6511 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝐸 = (𝑥 ∈ (Base‘(Poly1‘ℂfld)) ↦ (𝐸𝑥)))
219 fveq2 6448 . . . . . . . . . . . 12 (𝑥 = (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) → (𝐸𝑥) = (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))))
220193, 216, 218, 219fmptco 6663 . . . . . . . . . . 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))))))
2219a1i 11 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ℂfld ∈ Ring)
2226a1i 11 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ℂ ∈ V)
22395ffvelrni 6624 . . . . . . . . . . . . . . . 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)))
224193, 223syl 17 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ (Base‘(ℂflds ℂ)))
2253, 4, 67, 221, 222, 224pwselbas 16546 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℂ⟶ℂ)
226225feqmptd 6511 . . . . . . . . . . . . 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)))
22752a1i 11 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CRing)
228 simpr 479 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
22953, 114, 4, 54, 93, 227, 228evl1vard 20108 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(var1‘ℂfld))‘𝑧) = 𝑧))
230184adantr 474 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ0)
23153, 54, 4, 93, 227, 228, 229, 116, 127, 230evl1expd 20116 . . . . . . . . . . . . . . . . 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)))
232228, 230, 131syl2anc 579 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧𝑘))
233232eqeq2d 2788 . . . . . . . . . . . . . . . . . 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)))
234233anbi2d 622 . . . . . . . . . . . . . . . . 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))))
235231, 234mpbid 224 . . . . . . . . . . . . . . . 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)))
236181adantr 474 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((coe1𝑎)‘𝑘) ∈ ℂ)
23753, 54, 4, 93, 227, 228, 235, 236, 153, 80evl1vsd 20115 . . . . . . . . . . . . . . 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))))
238237simprd 491 . . . . . . . . . . . . . 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)))
239238mpteq2dva 4981 . . . . . . . . . . . . 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))))
240226, 239eqtrd 2814 . . . . . . . . . . . 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))))
241240mpteq2dva 4981 . . . . . . . . . . 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)))))
242220, 241eqtrd 2814 . . . . . . . . . 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)))))
243242oveq2d 6940 . . . . . . . . 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))))))
244157, 215, 2433eqtr2d 2820 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) = ((ℂflds ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))))
2456a1i 11 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℂ ∈ V)
2469, 10mp1i 13 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℂfld ∈ CMnd)
247181adantlr 705 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ ℂ)
24833adantll 704 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧𝑘) ∈ ℂ)
249247, 248mulcld 10399 . . . . . . . . . 10 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
250249anasss 460 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0)) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
251165mptex 6760 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∈ V
252 funmpt 6175 . . . . . . . . . . . 12 Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))
253251, 252, 393pm3.2i 1395 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∧ (0g‘(ℂflds ℂ)) ∈ V)
254253a1i 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))
255 fzfid 13096 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin)
256 eldifn 3956 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → ¬ 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
257256adantl 475 . . . . . . . . . . . . . . . . 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)))
258152ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑎 ∈ (Base‘(Poly1‘ℂfld)))
259 eldifi 3955 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → 𝑘 ∈ ℕ0)
260259adantl 475 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℕ0)
261 eqid 2778 . . . . . . . . . . . . . . . . . . . . . . . 24 ( deg1 ‘ℂfld) = ( deg1 ‘ℂfld)
262261, 54, 93, 17, 154deg1ge 24306 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 ∈ (Base‘(Poly1‘ℂfld)) ∧ 𝑘 ∈ ℕ0 ∧ ((coe1𝑎)‘𝑘) ≠ 0) → 𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎))
2632623expia 1111 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ (Base‘(Poly1‘ℂfld)) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘) ≠ 0 → 𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎)))
264258, 260, 263syl2anc 579 . . . . . . . . . . . . . . . . . . . . 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)))
265 0xr 10425 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ*
266261, 54, 93deg1xrcl 24290 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
267152, 266syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
268267ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
269 xrmax2 12324 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℝ* ∧ (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*) → (( deg1 ‘ℂfld)‘𝑎) ≤ if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))
270265, 268, 269sylancr 581 . . . . . . . . . . . . . . . . . . . . . 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))
271260nn0red 11708 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ)
272271rexrd 10428 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ*)
273 ifcl 4351 . . . . . . . . . . . . . . . . . . . . . . . 24 (((( deg1 ‘ℂfld)‘𝑎) ∈ ℝ* ∧ 0 ∈ ℝ*) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℝ*)
274268, 265, 273sylancl 580 . . . . . . . . . . . . . . . . . . . . . . 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*)
275 xrletr 12306 . . . . . . . . . . . . . . . . . . . . . . 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)))
276272, 268, 274, 275syl3anc 1439 . . . . . . . . . . . . . . . . . . . . . 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)))
277270, 276mpan2d 684 . . . . . . . . . . . . . . . . . . . . 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)))
278264, 277syld 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)))
279278, 260jctild 521 . . . . . . . . . . . . . . . . . . 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))))
280261, 54, 93deg1cl 24291 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
281152, 280syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
282 elun 3976 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}))
283281, 282sylib 210 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}))
284 nn0ge0 11674 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → 0 ≤ (( deg1 ‘ℂfld)‘𝑎))
285284iftrued 4315 . . . . . . . . . . . . . . . . . . . . . . . 24 ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) = (( deg1 ‘ℂfld)‘𝑎))
286 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24 ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → (( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0)
287285, 286eqeltrd 2859 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
288 mnflt0 12275 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -∞ < 0
289 mnfxr 10436 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 -∞ ∈ ℝ*
290 xrltnle 10446 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞))
291289, 265, 290mp2an 682 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (-∞ < 0 ↔ ¬ 0 ≤ -∞)
292288, 291mpbi 222 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ¬ 0 ≤ -∞
293 elsni 4415 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → (( deg1 ‘ℂfld)‘𝑎) = -∞)
294293breq2d 4900 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → (0 ≤ (( deg1 ‘ℂfld)‘𝑎) ↔ 0 ≤ -∞))
295292, 294mtbiri 319 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → ¬ 0 ≤ (( deg1 ‘ℂfld)‘𝑎))
296295iffalsed 4318 . . . . . . . . . . . . . . . . . . . . . . . 24 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) = 0)
297 0nn0 11664 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℕ0
298296, 297syl6eqel 2867 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
299287, 298jaoi 846 . . . . . . . . . . . . . . . . . . . . . 22 (((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
300283, 299syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
301300ad2antrr 716 . . . . . . . . . . . . . . . . . . . 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)
302 fznn0 12755 . . . . . . . . . . . . . . . . . . . 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))))
303301, 302syl 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))))
304279, 303sylibrd 251 . . . . . . . . . . . . . . . . . 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))))
305304necon1bd 2987 . . . . . . . . . . . . . . . . 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))
306257, 305mpd 15 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → ((coe1𝑎)‘𝑘) = 0)
307306oveq1d 6939 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = (0 · (𝑧𝑘)))
308259, 248sylan2 586 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑧𝑘) ∈ ℂ)
309308mul02d 10576 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (0 · (𝑧𝑘)) = 0)
310307, 309eqtrd 2814 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = 0)
311310an32s 642 . . . . . . . . . . . . 13 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) ∧ 𝑧 ∈ ℂ) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = 0)
312311mpteq2dva 4981 . . . . . . . . . . . 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))
313 fconstmpt 5413 . . . . . . . . . . . . 13 (ℂ × {0}) = (𝑧 ∈ ℂ ↦ 0)
314 ringmnd 18954 . . . . . . . . . . . . . . 15 (ℂfld ∈ Ring → ℂfld ∈ Mnd)
3159, 314ax-mp 5 . . . . . . . . . . . . . 14 fld ∈ Mnd
3163, 17pws0g 17723 . . . . . . . . . . . . . 14 ((ℂfld ∈ Mnd ∧ ℂ ∈ V) → (ℂ × {0}) = (0g‘(ℂflds ℂ)))
317315, 6, 316mp2an 682 . . . . . . . . . . . . 13 (ℂ × {0}) = (0g‘(ℂflds ℂ))
318313, 317eqtr3i 2804 . . . . . . . . . . . 12 (𝑧 ∈ ℂ ↦ 0) = (0g‘(ℂflds ℂ))
319312, 318syl6eq 2830 . . . . . . . . . . 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)))
320319, 166suppss2 7613 . . . . . . . . . 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)))
321 suppssfifsupp 8580 . . . . . . . . . 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)))
322254, 255, 320, 321syl12anc 827 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) finSupp (0g‘(ℂflds ℂ)))
3233, 4, 5, 245, 166, 246, 250, 322pwsgsum 18775 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((ℂflds ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))) = (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))))
324 fz0ssnn0 12758 . . . . . . . . . . . 12 (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ⊆ ℕ0
325 resmpt 5701 . . . . . . . . . . . 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))))
326324, 325ax-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)))
327326oveq2i 6935 . . . . . . . . . 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))))
3289, 10mp1i 13 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CMnd)
329165a1i 11 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ℕ0 ∈ V)
330249fmpttd 6651 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))):ℕ0⟶ℂ)
331310, 329suppss2 7613 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) supp 0) ⊆ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))
332165mptex 6760 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V
333 funmpt 6175 . . . . . . . . . . . . . 14 Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))
334332, 333, 2103pm3.2i 1395 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∧ 0 ∈ V)
335334a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∧ 0 ∈ V))
336 fzfid 13096 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin)
337 suppssfifsupp 8580 . . . . . . . . . . . 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)
338335, 336, 331, 337syl12anc 827 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) finSupp 0)
3394, 17, 328, 329, 330, 331, 338gsumres 18711 . . . . . . . . . 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)))))
340 elfznn0 12756 . . . . . . . . . . . 12 (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) → 𝑘 ∈ ℕ0)
341340, 249sylan2 586 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
342336, 341gsumfsum 20220 . . . . . . . . . 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)))
343327, 339, 3423eqtr3a 2838 . . . . . . . . 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)))
344343mpteq2dva 4981 . . . . . . . 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))))
345244, 323, 3443eqtrd 2818 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘))))
34612adantr 474 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝑆 ⊆ ℂ)
347 elplyr 24405 . . . . . . . 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))
348346, 300, 179, 347syl3anc 1439 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ (Poly‘𝑆))
349345, 348eqeltrd 2859 . . . . . 6 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) ∈ (Poly‘𝑆))
350 eleq1 2847 . . . . . 6 ((𝐸𝑎) = 𝑓 → ((𝐸𝑎) ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (Poly‘𝑆)))
351349, 350syl5ibcom 237 . . . . 5 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝐸𝑎) = 𝑓𝑓 ∈ (Poly‘𝑆)))
352351rexlimdva 3213 . . . 4 (𝑆 ∈ (SubRing‘ℂfld) → (∃𝑎𝐴 (𝐸𝑎) = 𝑓𝑓 ∈ (Poly‘𝑆)))
353151, 352syl5 34 . . 3 (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (𝐸𝐴) → 𝑓 ∈ (Poly‘𝑆)))
354147, 353impbid 204 . 2 (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (𝐸𝐴)))
355354eqrdv 2776 1 (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 836  w3a 1071   = wceq 1601  wcel 2107  wne 2969  wrex 3091  Vcvv 3398  cdif 3789  cun 3790  wss 3792  ifcif 4307  {csn 4398   class class class wbr 4888  cmpt 4967   × cxp 5355  cres 5359  cima 5360  ccom 5361  Fun wfun 6131   Fn wfn 6132  wf 6133  cfv 6137  (class class class)co 6924  𝑓 cof 7174   supp csupp 7578  𝑚 cmap 8142  Fincfn 8243   finSupp cfsupp 8565  cc 10272  0cc0 10274   · cmul 10279  -∞cmnf 10411  *cxr 10412   < clt 10413  cle 10414  0cn0 11647  ...cfz 12648  cexp 13183  Σcsu 14833  Basecbs 16266  s cress 16267  .rcmulr 16350  Scalarcsca 16352   ·𝑠 cvsca 16353  0gc0g 16497   Σg cgsu 16498  s cpws 16504  Mndcmnd 17691   MndHom cmhm 17730  SubMndcsubmnd 17731  .gcmg 17938  SubGrpcsubg 17983   GrpHom cghm 18052  CMndccmn 18590  mulGrpcmgp 18887  Ringcrg 18945  CRingccrg 18946   RingHom crh 19112  SubRingcsubrg 19179  LModclmod 19266  algSccascl 19719  var1cv1 19953  Poly1cpl1 19954  coe1cco1 19955  eval1ce1 20086  fldccnfld 20153   deg1 cdg1 24262  Polycply 24388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352  ax-addf 10353  ax-mulf 10354
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-ofr 7177  df-om 7346  df-1st 7447  df-2nd 7448  df-supp 7579  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-ixp 8197  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-fsupp 8566  df-sup 8638  df-oi 8706  df-card 9100  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-4 11445  df-5 11446  df-6 11447  df-7 11448  df-8 11449  df-9 11450  df-n0 11648  df-z 11734  df-dec 11851  df-uz 11998  df-rp 12143  df-fz 12649  df-fzo 12790  df-seq 13125  df-exp 13184  df-hash 13442  df-cj 14252  df-re 14253  df-im 14254  df-sqrt 14388  df-abs 14389  df-clim 14636  df-sum 14834  df-struct 16268  df-ndx 16269  df-slot 16270  df-base 16272  df-sets 16273  df-ress 16274  df-plusg 16362  df-mulr 16363  df-starv 16364  df-sca 16365  df-vsca 16366  df-ip 16367  df-tset 16368  df-ple 16369  df-ds 16371  df-unif 16372  df-hom 16373  df-cco 16374  df-0g 16499  df-gsum 16500  df-prds 16505  df-pws 16507  df-mre 16643  df-mrc 16644  df-acs 16646  df-mgm 17639  df-sgrp 17681  df-mnd 17692  df-mhm 17732  df-submnd 17733  df-grp 17823  df-minusg 17824  df-sbg 17825  df-mulg 17939  df-subg 17986  df-ghm 18053  df-cntz 18144  df-cmn 18592  df-abl 18593  df-mgp 18888  df-ur 18900  df-srg 18904  df-ring 18947  df-cring 18948  df-rnghom 19115  df-subrg 19181  df-lmod 19268  df-lss 19336  df-lsp 19378  df-assa 19720  df-asp 19721  df-ascl 19722  df-psr 19764  df-mvr 19765  df-mpl 19766  df-opsr 19768  df-evls 19913  df-evl 19914  df-psr1 19957  df-vr1 19958  df-ply1 19959  df-coe1 19960  df-evl1 20088  df-cnfld 20154  df-mdeg 24263  df-deg1 24264  df-ply 24392
This theorem is referenced by: (None)
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