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Theorem plypf1 24729
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 24712 . . . . 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 2818 . . . . . . . . 9 (ℂflds ℂ) = (ℂflds ℂ)
4 cnfldbas 20477 . . . . . . . . 9 ℂ = (Base‘ℂfld)
5 eqid 2818 . . . . . . . . 9 (0g‘(ℂflds ℂ)) = (0g‘(ℂflds ℂ))
6 cnex 10606 . . . . . . . . . 10 ℂ ∈ V
76a1i 11 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ℂ ∈ V)
8 fzfid 13329 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (0...𝑛) ∈ Fin)
9 cnring 20495 . . . . . . . . . 10 fld ∈ Ring
10 ringcmn 19260 . . . . . . . . . 10 (ℂfld ∈ Ring → ℂfld ∈ CMnd)
119, 10mp1i 13 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ℂfld ∈ CMnd)
124subrgss 19465 . . . . . . . . . . . . 13 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ)
1312ad2antrr 722 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ⊆ ℂ)
14 elmapi 8417 . . . . . . . . . . . . . . 15 (𝑎 ∈ ((𝑆 ∪ {0}) ↑m0) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
1514ad2antll 725 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
16 subrgsubg 19470 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld))
17 cnfld0 20497 . . . . . . . . . . . . . . . . . . . 20 0 = (0g‘ℂfld)
1817subg0cl 18225 . . . . . . . . . . . . . . . . . . 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 4734 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → {0} ⊆ 𝑆)
22 ssequn2 4156 . . . . . . . . . . . . . . . 16 ({0} ⊆ 𝑆 ↔ (𝑆 ∪ {0}) = 𝑆)
2321, 22sylib 219 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑆 ∪ {0}) = 𝑆)
2423feq3d 6494 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑎:ℕ0⟶(𝑆 ∪ {0}) ↔ 𝑎:ℕ0𝑆))
2515, 24mpbid 233 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 𝑎:ℕ0𝑆)
26 elfznn0 12988 . . . . . . . . . . . . 13 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
27 ffvelrn 6841 . . . . . . . . . . . . 13 ((𝑎:ℕ0𝑆𝑘 ∈ ℕ0) → (𝑎𝑘) ∈ 𝑆)
2825, 26, 27syl2an 595 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) ∈ 𝑆)
2913, 28sseldd 3965 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) ∈ ℂ)
3029adantrl 712 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → (𝑎𝑘) ∈ ℂ)
31 simprl 767 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑧 ∈ ℂ)
3226ad2antll 725 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑘 ∈ ℕ0)
33 expcl 13435 . . . . . . . . . . 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 10649 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → ((𝑎𝑘) · (𝑧𝑘)) ∈ ℂ)
36 eqid 2818 . . . . . . . . . 10 (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))
376mptex 6977 . . . . . . . . . . 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 6676 . . . . . . . . . . 11 (0g‘(ℂflds ℂ)) ∈ V
4039a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (0g‘(ℂflds ℂ)) ∈ V)
4136, 8, 38, 40fsuppmptdm 8832 . . . . . . . . 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 19031 . . . . . . . 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 13329 . . . . . . . . . 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 20540 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑎𝑘) · (𝑧𝑘)))) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
4645mpteq2dva 5152 . . . . . . . 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 2853 . . . . . . 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 19294 . . . . . . . . . 10 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → (ℂflds ℂ) ∈ Ring)
499, 6, 48mp2an 688 . . . . . . . . 9 (ℂflds ℂ) ∈ Ring
50 ringcmn 19260 . . . . . . . . 9 ((ℂflds ℂ) ∈ Ring → (ℂflds ℂ) ∈ CMnd)
5149, 50mp1i 13 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (ℂflds ℂ) ∈ CMnd)
52 cncrng 20494 . . . . . . . . . . 11 fld ∈ CRing
53 plypf1.e . . . . . . . . . . . 12 𝐸 = (eval1‘ℂfld)
54 eqid 2818 . . . . . . . . . . . 12 (Poly1‘ℂfld) = (Poly1‘ℂfld)
5553, 54, 3, 4evl1rhm 20423 . . . . . . . . . . 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 20329 . . . . . . . . . . 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 19495 . . . . . . . . . 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 19470 . . . . . . . . 9 ((𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)) → (𝐸𝐴) ∈ (SubGrp‘(ℂflds ℂ)))
65 subgsubm 18239 . . . . . . . . 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 2818 . . . . . . . . . . . 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 6561 . . . . . . . . . . . . . 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 16749 . . . . . . . . . . . . . 14 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)) ↔ (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ))
739, 6, 72mp2an 688 . . . . . . . . . . . . 13 ((ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)) ↔ (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ)
7471, 73sylibr 235 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)))
7534anass1rs 651 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑧𝑘) ∈ ℂ)
7675fmpttd 6871 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ)
773, 4, 67pwselbasb 16749 . . . . . . . . . . . . . 14 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ))
789, 6, 77mp2an 688 . . . . . . . . . . . . 13 ((𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ)
7976, 78sylibr 235 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)))
80 cnfldmul 20479 . . . . . . . . . . . 12 · = (.r‘ℂfld)
81 eqid 2818 . . . . . . . . . . . 12 (.r‘(ℂflds ℂ)) = (.r‘(ℂflds ℂ))
823, 67, 68, 69, 74, 79, 80, 81pwsmulrval 16752 . . . . . . . . . . 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 5607 . . . . . . . . . . . . 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 2819 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) = (𝑧 ∈ ℂ ↦ (𝑧𝑘)))
8769, 83, 75, 85, 86offval2 7415 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)}) ∘f · (𝑧 ∈ ℂ ↦ (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))
8882, 87eqtrd 2853 . . . . . . . . . 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 2818 . . . . . . . . . . . . . 14 (algSc‘(Poly1‘ℂfld)) = (algSc‘(Poly1‘ℂfld))
9153, 54, 4, 90evl1sca 20425 . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . 16 (Base‘(Poly1‘ℂfld)) = (Base‘(Poly1‘ℂfld))
9493, 67rhmf 19407 . . . . . . . . . . . . . . 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 6507 . . . . . . . . . . . . . 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 19465 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubRing‘(Poly1‘ℂfld)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
9960, 98syl 17 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubRing‘ℂfld) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
10099ad2antrr 722 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
101 simpll 763 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ∈ (SubRing‘ℂfld))
10254, 90, 57, 58, 101, 59, 4, 29subrg1asclcl 20356 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴 ↔ (𝑎𝑘) ∈ 𝑆))
10328, 102mpbird 258 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴)
104 fnfvima 6986 . . . . . . . . . . . . 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 1363 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) ∈ (𝐸𝐴))
10692, 105eqeltrrd 2911 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}) ∈ (𝐸𝐴))
10767subrgss 19465 . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . 19 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
110 eqid 2818 . . . . . . . . . . . . . . . . . . . 20 (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘(Poly1‘ℂfld))
111110subrgsubm 19477 . . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . . . 19 (var1‘ℂfld) = (var1‘ℂfld)
115114, 101, 57, 58, 59subrgvr1cl 20358 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (var1‘ℂfld) ∈ 𝐴)
116 eqid 2818 . . . . . . . . . . . . . . . . . . 19 (.g‘(mulGrp‘(Poly1‘ℂfld))) = (.g‘(mulGrp‘(Poly1‘ℂfld)))
117116submmulgcl 18208 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (SubMnd‘(mulGrp‘(Poly1‘ℂfld))) ∧ 𝑘 ∈ ℕ0 ∧ (var1‘ℂfld) ∈ 𝐴) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
118112, 113, 115, 117syl3anc 1363 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
119 fnfvima 6986 . . . . . . . . . . . . . . . . 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 1363 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (𝐸𝐴))
121108, 120sseldd 3965 . . . . . . . . . . . . . . 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 16750 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))):ℂ⟶ℂ)
123122feqmptd 6726 . . . . . . . . . . . . 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 20428 . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . 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 20436 . . . . . . . . . . . . . . . 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 20506 . . . . . . . . . . . . . . . 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 2853 . . . . . . . . . . . . . 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 5152 . . . . . . . . . . . . 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 2853 . . . . . . . . . . . 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 2911 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (𝐸𝐴))
13781subrgmcl 19476 . . . . . . . . . . 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 1363 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)})(.r‘(ℂflds ℂ))(𝑧 ∈ ℂ ↦ (𝑧𝑘))) ∈ (𝐸𝐴))
13988, 138eqeltrrd 2911 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴))
140139fmpttd 6871 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))):(0...𝑛)⟶(𝐸𝐴))
14136, 8, 139, 40fsuppmptdm 8832 . . . . . . . 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 18968 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ((ℂflds ℂ) Σg (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))) ∈ (𝐸𝐴))
14347, 142eqeltrrd 2911 . . . . . 6 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴))
144 eleq1 2897 . . . . . 6 (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) → (𝑓 ∈ (𝐸𝐴) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴)))
145143, 144syl5ibrcom 248 . . . . 5 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) → 𝑓 ∈ (𝐸𝐴)))
146145rexlimdvva 3291 . . . 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 6510 . . . . . 6 (𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ)) → Fun 𝐸)
14995, 148ax-mp 5 . . . . 5 Fun 𝐸
150 fvelima 6724 . . . . 5 ((Fun 𝐸𝑓 ∈ (𝐸𝐴)) → ∃𝑎𝐴 (𝐸𝑎) = 𝑓)
151149, 150mpan 686 . . . 4 (𝑓 ∈ (𝐸𝐴) → ∃𝑎𝐴 (𝐸𝑎) = 𝑓)
15299sselda 3964 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝑎 ∈ (Base‘(Poly1‘ℂfld)))
153 eqid 2818 . . . . . . . . . . . 12 ( ·𝑠 ‘(Poly1‘ℂfld)) = ( ·𝑠 ‘(Poly1‘ℂfld))
154 eqid 2818 . . . . . . . . . . . 12 (coe1𝑎) = (coe1𝑎)
15554, 114, 93, 153, 110, 116, 154ply1coe 20392 . . . . . . . . . . 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 6667 . . . . . . . . 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 2818 . . . . . . . . . 10 (0g‘(Poly1‘ℂfld)) = (0g‘(Poly1‘ℂfld))
15954ply1ring 20344 . . . . . . . . . . . 12 (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ Ring)
1609, 159ax-mp 5 . . . . . . . . . . 11 (Poly1‘ℂfld) ∈ Ring
161 ringcmn 19260 . . . . . . . . . . 11 ((Poly1‘ℂfld) ∈ Ring → (Poly1‘ℂfld) ∈ CMnd)
162160, 161mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (Poly1‘ℂfld) ∈ CMnd)
163 ringmnd 19235 . . . . . . . . . . 11 ((ℂflds ℂ) ∈ Ring → (ℂflds ℂ) ∈ Mnd)
16449, 163mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (ℂflds ℂ) ∈ Mnd)
165 nn0ex 11891 . . . . . . . . . . 11 0 ∈ V
166165a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℕ0 ∈ V)
167 rhmghm 19406 . . . . . . . . . . . 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 18306 . . . . . . . . . . 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 20348 . . . . . . . . . . . . 13 (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ LMod)
1729, 171mp1i 13 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (Poly1‘ℂfld) ∈ LMod)
17312ad2antrr 722 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → 𝑆 ⊆ ℂ)
174 eqid 2818 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
175154, 59, 58, 174coe1f 20307 . . . . . . . . . . . . . . . 16 (𝑎𝐴 → (coe1𝑎):ℕ0⟶(Base‘𝑅))
17657subrgbas 19473 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 = (Base‘𝑅))
177176feq3d 6494 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (SubRing‘ℂfld) → ((coe1𝑎):ℕ0𝑆 ↔ (coe1𝑎):ℕ0⟶(Base‘𝑅)))
178175, 177syl5ibr 247 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubRing‘ℂfld) → (𝑎𝐴 → (coe1𝑎):ℕ0𝑆))
179178imp 407 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎):ℕ0𝑆)
180179ffvelrnda 6843 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ 𝑆)
181173, 180sseldd 3965 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ ℂ)
182110ringmgp 19232 . . . . . . . . . . . . . 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 20313 . . . . . . . . . . . . . 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 19174 . . . . . . . . . . . . . 14 (Base‘(Poly1‘ℂfld)) = (Base‘(mulGrp‘(Poly1‘ℂfld)))
188187, 116mulgnn0cl 18182 . . . . . . . . . . . . 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 1363 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)))
19054ply1sca 20349 . . . . . . . . . . . . . 14 (ℂfld ∈ Ring → ℂfld = (Scalar‘(Poly1‘ℂfld)))
1919, 190ax-mp 5 . . . . . . . . . . . . 13 fld = (Scalar‘(Poly1‘ℂfld))
19293, 191, 153, 4lmodvscl 19580 . . . . . . . . . . . 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 1363 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)))
194193fmpttd 6871 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℕ0⟶(Base‘(Poly1‘ℂfld)))
195165mptex 6977 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V
196 funmpt 6386 . . . . . . . . . . . . 13 Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))
197 fvex 6676 . . . . . . . . . . . . 13 (0g‘(Poly1‘ℂfld)) ∈ V
198195, 196, 1973pm3.2i 1331 . . . . . . . . . . . 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 20309 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (coe1𝑎) finSupp 0)
201152, 200syl 17 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎) finSupp 0)
202201fsuppimpd 8828 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((coe1𝑎) supp 0) ∈ Fin)
203179feqmptd 6726 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎) = (𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)))
204203oveq1d 7160 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((coe1𝑎) supp 0) = ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0))
205 eqimss2 4021 . . . . . . . . . . . . 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 19596 . . . . . . . . . . . . 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 10623 . . . . . . . . . . . . 13 0 ∈ V
211210a1i 11 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 0 ∈ V)
212206, 209, 180, 189, 211suppssov1 7851 . . . . . . . . . . 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 8836 . . . . . . . . . . 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 832 . . . . . . . . . 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 18987 . . . . . . . . 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 6886 . . . . . . . . . . 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)
22095ffvelrni 6842 . . . . . . . . . . . . . . . 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 16750 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℂ⟶ℂ)
223222feqmptd 6726 . . . . . . . . . . . . 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 20428 . . . . . . . . . . . . . . . . . 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 20436 . . . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . 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 20435 . . . . . . . . . . . . . . 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 5152 . . . . . . . . . . . . 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 2853 . . . . . . . . . . . 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 5152 . . . . . . . . . . 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 2853 . . . . . . . . . 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 7161 . . . . . . . . 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 2859 . . . . . . . 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 711 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ ℂ)
24533adantll 710 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧𝑘) ∈ ℂ)
246244, 245mulcld 10649 . . . . . . . . . 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 6977 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∈ V
249 funmpt 6386 . . . . . . . . . . . 12 Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))
250248, 249, 393pm3.2i 1331 . . . . . . . . . . 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 13329 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin)
253 eldifn 4101 . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . . . 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 4100 . . . . . . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . . . . . . . . 24 ( deg1 ‘ℂfld) = ( deg1 ‘ℂfld)
259258, 54, 93, 17, 154deg1ge 24619 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 ∈ (Base‘(Poly1‘ℂfld)) ∧ 𝑘 ∈ ℕ0 ∧ ((coe1𝑎)‘𝑘) ≠ 0) → 𝑘 ≤ (( deg1 ‘ℂfld)‘𝑎))
2602593expia 1113 . . . . . . . . . . . . . . . . . . . . . 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 10676 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ*
263258, 54, 93deg1xrcl 24603 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
264152, 263syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (( deg1 ‘ℂfld)‘𝑎) ∈ ℝ*)
265264ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 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 12557 . . . . . . . . . . . . . . . . . . . . . . 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 11944 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ)
269268rexrd 10679 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ*)
270 ifcl 4507 . . . . . . . . . . . . . . . . . . . . . . . 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 12539 . . . . . . . . . . . . . . . . . . . . . . 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 1363 . . . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . . . 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 24604 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
278152, 277syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
279 elun 4122 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 ‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}))
280278, 279sylib 219 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘ℂfld)‘𝑎) ∈ {-∞}))
281 nn0ge0 11910 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → 0 ≤ (( deg1 ‘ℂfld)‘𝑎))
282281iftrued 4471 . . . . . . . . . . . . . . . . . . . . . . . 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 2910 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 ‘ℂfld)‘𝑎) ∈ ℕ0 → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
285 mnflt0 12508 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -∞ < 0
286 mnfxr 10686 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 -∞ ∈ ℝ*
287 xrltnle 10696 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞))
288286, 262, 287mp2an 688 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (-∞ < 0 ↔ ¬ 0 ≤ -∞)
289285, 288mpbi 231 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ¬ 0 ≤ -∞
290 elsni 4574 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → (( deg1 ‘ℂfld)‘𝑎) = -∞)
291290breq2d 5069 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → (0 ≤ (( deg1 ‘ℂfld)‘𝑎) ↔ 0 ≤ -∞))
292289, 291mtbiri 328 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → ¬ 0 ≤ (( deg1 ‘ℂfld)‘𝑎))
293292iffalsed 4474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) = 0)
294 0nn0 11900 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℕ0
295293, 294syl6eqel 2918 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 ‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0) ∈ ℕ0)
296284, 295jaoi 851 . . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . 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 12987 . . . . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . . . 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 3031 . . . . . . . . . . . . . . . . 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 7160 . . . . . . . . . . . . . . 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 592 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (𝑧𝑘) ∈ ℂ)
306305mul02d 10826 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (0 · (𝑧𝑘)) = 0)
307304, 306eqtrd 2853 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = 0)
308307an32s 648 . . . . . . . . . . . . 13 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)))) ∧ 𝑧 ∈ ℂ) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = 0)
309308mpteq2dva 5152 . . . . . . . . . . . 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 5607 . . . . . . . . . . . . 13 (ℂ × {0}) = (𝑧 ∈ ℂ ↦ 0)
311 ringmnd 19235 . . . . . . . . . . . . . . 15 (ℂfld ∈ Ring → ℂfld ∈ Mnd)
3129, 311ax-mp 5 . . . . . . . . . . . . . 14 fld ∈ Mnd
3133, 17pws0g 17935 . . . . . . . . . . . . . 14 ((ℂfld ∈ Mnd ∧ ℂ ∈ V) → (ℂ × {0}) = (0g‘(ℂflds ℂ)))
314312, 6, 313mp2an 688 . . . . . . . . . . . . 13 (ℂ × {0}) = (0g‘(ℂflds ℂ))
315310, 314eqtr3i 2843 . . . . . . . . . . . 12 (𝑧 ∈ ℂ ↦ 0) = (0g‘(ℂflds ℂ))
316309, 315syl6eq 2869 . . . . . . . . . . 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 7853 . . . . . . . . . 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 8836 . . . . . . . . . 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 832 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) finSupp (0g‘(ℂflds ℂ)))
3203, 4, 5, 242, 166, 243, 247, 319pwsgsum 19031 . . . . . . . 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 12990 . . . . . . . . . . . 12 (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ⊆ ℕ0
322 resmpt 5898 . . . . . . . . . . . 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 7156 . . . . . . . . . 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 6871 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))):ℕ0⟶ℂ)
328307, 326suppss2 7853 . . . . . . . . . . 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 6977 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V
330 funmpt 6386 . . . . . . . . . . . . . 14 Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))
331329, 330, 2103pm3.2i 1331 . . . . . . . . . . . . 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 13329 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) ∈ Fin)
334 suppssfifsupp 8836 . . . . . . . . . . . 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 832 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) finSupp 0)
3364, 17, 325, 326, 327, 328, 335gsumres 18962 . . . . . . . . . 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 12988 . . . . . . . . . . . 12 (𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0)) → 𝑘 ∈ ℕ0)
338337, 246sylan2 592 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...if(0 ≤ (( deg1 ‘ℂfld)‘𝑎), (( deg1 ‘ℂfld)‘𝑎), 0))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
339333, 338gsumfsum 20540 . . . . . . . . . 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 2877 . . . . . . . . 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 5152 . . . . . . . 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 2857 . . . . . . 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 24718 . . . . . . . 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 1363 . . . . . . 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 2910 . . . . . 6 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) ∈ (Poly‘𝑆))
347 eleq1 2897 . . . . . 6 ((𝐸𝑎) = 𝑓 → ((𝐸𝑎) ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (Poly‘𝑆)))
348346, 347syl5ibcom 246 . . . . 5 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝐸𝑎) = 𝑓𝑓 ∈ (Poly‘𝑆)))
349348rexlimdva 3281 . . . 4 (𝑆 ∈ (SubRing‘ℂfld) → (∃𝑎𝐴 (𝐸𝑎) = 𝑓𝑓 ∈ (Poly‘𝑆)))
350151, 349syl5 34 . . 3 (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (𝐸𝐴) → 𝑓 ∈ (Poly‘𝑆)))
351147, 350impbid 213 . 2 (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (𝐸𝐴)))
352351eqrdv 2816 1 (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wrex 3136  Vcvv 3492  cdif 3930  cun 3931  wss 3933  ifcif 4463  {csn 4557   class class class wbr 5057  cmpt 5137   × cxp 5546  cres 5550  cima 5551  ccom 5552  Fun wfun 6342   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396   supp csupp 7819  m cmap 8395  Fincfn 8497   finSupp cfsupp 8821  cc 10523  0cc0 10525   · cmul 10530  -∞cmnf 10661  *cxr 10662   < clt 10663  cle 10664  0cn0 11885  ...cfz 12880  cexp 13417  Σcsu 15030  Basecbs 16471  s cress 16472  .rcmulr 16554  Scalarcsca 16556   ·𝑠 cvsca 16557  0gc0g 16701   Σg cgsu 16702  s cpws 16708  Mndcmnd 17899   MndHom cmhm 17942  SubMndcsubmnd 17943  .gcmg 18162  SubGrpcsubg 18211   GrpHom cghm 18293  CMndccmn 18835  mulGrpcmgp 19168  Ringcrg 19226  CRingccrg 19227   RingHom crh 19393  SubRingcsubrg 19460  LModclmod 19563  algSccascl 20012  var1cv1 20272  Poly1cpl1 20273  coe1cco1 20274  eval1ce1 20405  fldccnfld 20473   deg1 cdg1 24575  Polycply 24701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-0g 16703  df-gsum 16704  df-prds 16709  df-pws 16711  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-subg 18214  df-ghm 18294  df-cntz 18385  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-srg 19185  df-ring 19228  df-cring 19229  df-rnghom 19396  df-subrg 19462  df-lmod 19565  df-lss 19633  df-lsp 19673  df-assa 20013  df-asp 20014  df-ascl 20015  df-psr 20064  df-mvr 20065  df-mpl 20066  df-opsr 20068  df-evls 20214  df-evl 20215  df-psr1 20276  df-vr1 20277  df-ply1 20278  df-coe1 20279  df-evl1 20407  df-cnfld 20474  df-mdeg 24576  df-deg1 24577  df-ply 24705
This theorem is referenced by: (None)
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