MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plypf1 Structured version   Visualization version   GIF version

Theorem plypf1 26265
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 26248 . . . . 5 (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
21simprbi 496 . . . 4 (𝑓 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
3 eqid 2734 . . . . . . . . 9 (ℂflds ℂ) = (ℂflds ℂ)
4 cnfldbas 21385 . . . . . . . . 9 ℂ = (Base‘ℂfld)
5 eqid 2734 . . . . . . . . 9 (0g‘(ℂflds ℂ)) = (0g‘(ℂflds ℂ))
6 cnex 11233 . . . . . . . . . 10 ℂ ∈ V
76a1i 11 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ℂ ∈ V)
8 fzfid 14010 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (0...𝑛) ∈ Fin)
9 cnring 21420 . . . . . . . . . 10 fld ∈ Ring
10 ringcmn 20295 . . . . . . . . . 10 (ℂfld ∈ Ring → ℂfld ∈ CMnd)
119, 10mp1i 13 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ℂfld ∈ CMnd)
124subrgss 20588 . . . . . . . . . . . . 13 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ)
1312ad2antrr 726 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ⊆ ℂ)
14 elmapi 8887 . . . . . . . . . . . . . . 15 (𝑎 ∈ ((𝑆 ∪ {0}) ↑m0) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
1514ad2antll 729 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
16 subrgsubg 20593 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld))
17 cnfld0 21422 . . . . . . . . . . . . . . . . . . . 20 0 = (0g‘ℂfld)
1817subg0cl 19164 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆)
1916, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝑆 ∈ (SubRing‘ℂfld) → 0 ∈ 𝑆)
2019adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 0 ∈ 𝑆)
2120snssd 4813 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → {0} ⊆ 𝑆)
22 ssequn2 4198 . . . . . . . . . . . . . . . 16 ({0} ⊆ 𝑆 ↔ (𝑆 ∪ {0}) = 𝑆)
2321, 22sylib 218 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑆 ∪ {0}) = 𝑆)
2423feq3d 6723 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑎:ℕ0⟶(𝑆 ∪ {0}) ↔ 𝑎:ℕ0𝑆))
2515, 24mpbid 232 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 𝑎:ℕ0𝑆)
26 elfznn0 13656 . . . . . . . . . . . . 13 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
27 ffvelcdm 7100 . . . . . . . . . . . . 13 ((𝑎:ℕ0𝑆𝑘 ∈ ℕ0) → (𝑎𝑘) ∈ 𝑆)
2825, 26, 27syl2an 596 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) ∈ 𝑆)
2913, 28sseldd 3995 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) ∈ ℂ)
3029adantrl 716 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → (𝑎𝑘) ∈ ℂ)
31 simprl 771 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑧 ∈ ℂ)
3226ad2antll 729 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → 𝑘 ∈ ℕ0)
33 expcl 14116 . . . . . . . . . . 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 11278 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ (0...𝑛))) → ((𝑎𝑘) · (𝑧𝑘)) ∈ ℂ)
36 eqid 2734 . . . . . . . . . 10 (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))
376mptex 7242 . . . . . . . . . . 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 6919 . . . . . . . . . . 11 (0g‘(ℂflds ℂ)) ∈ V
4039a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (0g‘(ℂflds ℂ)) ∈ V)
4136, 8, 38, 40fsuppmptdm 9413 . . . . . . . . 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 20014 . . . . . . . 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 14010 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑧 ∈ ℂ) → (0...𝑛) ∈ Fin)
4435anassrs 467 . . . . . . . . . 10 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎𝑘) · (𝑧𝑘)) ∈ ℂ)
4543, 44gsumfsum 21469 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑧 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑎𝑘) · (𝑧𝑘)))) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
4645mpteq2dva 5247 . . . . . . . 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 2774 . . . . . . 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 20337 . . . . . . . . . 10 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → (ℂflds ℂ) ∈ Ring)
499, 6, 48mp2an 692 . . . . . . . . 9 (ℂflds ℂ) ∈ Ring
50 ringcmn 20295 . . . . . . . . 9 ((ℂflds ℂ) ∈ Ring → (ℂflds ℂ) ∈ CMnd)
5149, 50mp1i 13 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (ℂflds ℂ) ∈ CMnd)
52 cncrng 21418 . . . . . . . . . . 11 fld ∈ CRing
53 plypf1.e . . . . . . . . . . . 12 𝐸 = (eval1‘ℂfld)
54 eqid 2734 . . . . . . . . . . . 12 (Poly1‘ℂfld) = (Poly1‘ℂfld)
5553, 54, 3, 4evl1rhm 22351 . . . . . . . . . . 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 22249 . . . . . . . . . . 11 (𝑆 ∈ (SubRing‘ℂfld) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
6160adantr 480 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
62 rhmima 20620 . . . . . . . . . 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 20593 . . . . . . . . 9 ((𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)) → (𝐸𝐴) ∈ (SubGrp‘(ℂflds ℂ)))
65 subgsubm 19178 . . . . . . . . 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 2734 . . . . . . . . . . . 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 6797 . . . . . . . . . . . . . 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 17534 . . . . . . . . . . . . . 14 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)) ↔ (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ))
739, 6, 72mp2an 692 . . . . . . . . . . . . 13 ((ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)) ↔ (ℂ × {(𝑎𝑘)}):ℂ⟶ℂ)
7471, 73sylibr 234 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}) ∈ (Base‘(ℂflds ℂ)))
7534anass1rs 655 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑧𝑘) ∈ ℂ)
7675fmpttd 7134 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ)
773, 4, 67pwselbasb 17534 . . . . . . . . . . . . . 14 ((ℂfld ∈ Ring ∧ ℂ ∈ V) → ((𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ))
789, 6, 77mp2an 692 . . . . . . . . . . . . 13 ((𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)) ↔ (𝑧 ∈ ℂ ↦ (𝑧𝑘)):ℂ⟶ℂ)
7976, 78sylibr 234 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (Base‘(ℂflds ℂ)))
80 cnfldmul 21389 . . . . . . . . . . . 12 · = (.r‘ℂfld)
81 eqid 2734 . . . . . . . . . . . 12 (.r‘(ℂflds ℂ)) = (.r‘(ℂflds ℂ))
823, 67, 68, 69, 74, 79, 80, 81pwsmulrval 17537 . . . . . . . . . . 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 480 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → (𝑎𝑘) ∈ ℂ)
84 fconstmpt 5750 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) = (𝑧 ∈ ℂ ↦ (𝑧𝑘)))
8769, 83, 75, 85, 86offval2 7716 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)}) ∘f · (𝑧 ∈ ℂ ↦ (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))
8882, 87eqtrd 2774 . . . . . . . . . 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 480 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)))
90 eqid 2734 . . . . . . . . . . . . . 14 (algSc‘(Poly1‘ℂfld)) = (algSc‘(Poly1‘ℂfld))
9153, 54, 4, 90evl1sca 22353 . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . 16 (Base‘(Poly1‘ℂfld)) = (Base‘(Poly1‘ℂfld))
9493, 67rhmf 20501 . . . . . . . . . . . . . . 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 6736 . . . . . . . . . . . . . 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 20588 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubRing‘(Poly1‘ℂfld)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
9960, 98syl 17 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubRing‘ℂfld) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
10099ad2antrr 726 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ⊆ (Base‘(Poly1‘ℂfld)))
101 simpll 767 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑆 ∈ (SubRing‘ℂfld))
10254, 90, 57, 58, 101, 59, 4, 29subrg1asclcl 22278 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴 ↔ (𝑎𝑘) ∈ 𝑆))
10328, 102mpbird 257 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴)
104 fnfvima 7252 . . . . . . . . . . . . 13 ((𝐸 Fn (Base‘(Poly1‘ℂfld)) ∧ 𝐴 ⊆ (Base‘(Poly1‘ℂfld)) ∧ ((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘)) ∈ 𝐴) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) ∈ (𝐸𝐴))
10597, 100, 103, 104syl3anc 1370 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘((algSc‘(Poly1‘ℂfld))‘(𝑎𝑘))) ∈ (𝐸𝐴))
10692, 105eqeltrrd 2839 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (ℂ × {(𝑎𝑘)}) ∈ (𝐸𝐴))
10767subrgss 20588 . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . 19 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ (SubRing‘(Poly1‘ℂfld)))
110 eqid 2734 . . . . . . . . . . . . . . . . . . . 20 (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘(Poly1‘ℂfld))
111110subrgsubm 20601 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
114 eqid 2734 . . . . . . . . . . . . . . . . . . 19 (var1‘ℂfld) = (var1‘ℂfld)
115114, 101, 57, 58, 59subrgvr1cl 22280 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (var1‘ℂfld) ∈ 𝐴)
116 eqid 2734 . . . . . . . . . . . . . . . . . . 19 (.g‘(mulGrp‘(Poly1‘ℂfld))) = (.g‘(mulGrp‘(Poly1‘ℂfld)))
117116submmulgcl 19147 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (SubMnd‘(mulGrp‘(Poly1‘ℂfld))) ∧ 𝑘 ∈ ℕ0 ∧ (var1‘ℂfld) ∈ 𝐴) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
118112, 113, 115, 117syl3anc 1370 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴)
119 fnfvima 7252 . . . . . . . . . . . . . . . . 17 ((𝐸 Fn (Base‘(Poly1‘ℂfld)) ∧ 𝐴 ⊆ (Base‘(Poly1‘ℂfld)) ∧ (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ 𝐴) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (𝐸𝐴))
12097, 100, 118, 119syl3anc 1370 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (𝐸𝐴))
121108, 120sseldd 3995 . . . . . . . . . . . . . . 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 17535 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))):ℂ⟶ℂ)
123122feqmptd 6976 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
12653, 114, 4, 54, 93, 124, 125evl1vard 22356 . . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . . 17 (.g‘(mulGrp‘ℂfld)) = (.g‘(mulGrp‘ℂfld))
128113adantr 480 . . . . . . . . . . . . . . . . 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 22364 . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 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 21434 . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . 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 5247 . . . . . . . . . . . . 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 2774 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐸‘(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) = (𝑧 ∈ ℂ ↦ (𝑧𝑘)))
136135, 120eqeltrrd 2839 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (𝐸𝐴))
13781subrgmcl 20600 . . . . . . . . . . 11 (((𝐸𝐴) ∈ (SubRing‘(ℂflds ℂ)) ∧ (ℂ × {(𝑎𝑘)}) ∈ (𝐸𝐴) ∧ (𝑧 ∈ ℂ ↦ (𝑧𝑘)) ∈ (𝐸𝐴)) → ((ℂ × {(𝑎𝑘)})(.r‘(ℂflds ℂ))(𝑧 ∈ ℂ ↦ (𝑧𝑘))) ∈ (𝐸𝐴))
13889, 106, 136, 137syl3anc 1370 . . . . . . . . . 10 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → ((ℂ × {(𝑎𝑘)})(.r‘(ℂflds ℂ))(𝑧 ∈ ℂ ↦ (𝑧𝑘))) ∈ (𝐸𝐴))
13988, 138eqeltrrd 2839 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴))
140139fmpttd 7134 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘)))):(0...𝑛)⟶(𝐸𝐴))
14136, 8, 139, 40fsuppmptdm 9413 . . . . . . . 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 19951 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → ((ℂflds ℂ) Σg (𝑘 ∈ (0...𝑛) ↦ (𝑧 ∈ ℂ ↦ ((𝑎𝑘) · (𝑧𝑘))))) ∈ (𝐸𝐴))
14347, 142eqeltrrd 2839 . . . . . 6 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴))
144 eleq1 2826 . . . . . 6 (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) → (𝑓 ∈ (𝐸𝐴) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ∈ (𝐸𝐴)))
145143, 144syl5ibrcom 247 . . . . 5 ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) → 𝑓 ∈ (𝐸𝐴)))
146145rexlimdvva 3210 . . . 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 6739 . . . . . 6 (𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ)) → Fun 𝐸)
14995, 148ax-mp 5 . . . . 5 Fun 𝐸
150 fvelima 6973 . . . . 5 ((Fun 𝐸𝑓 ∈ (𝐸𝐴)) → ∃𝑎𝐴 (𝐸𝑎) = 𝑓)
151149, 150mpan 690 . . . 4 (𝑓 ∈ (𝐸𝐴) → ∃𝑎𝐴 (𝐸𝑎) = 𝑓)
15299sselda 3994 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝑎 ∈ (Base‘(Poly1‘ℂfld)))
153 eqid 2734 . . . . . . . . . . . 12 ( ·𝑠 ‘(Poly1‘ℂfld)) = ( ·𝑠 ‘(Poly1‘ℂfld))
154 eqid 2734 . . . . . . . . . . . 12 (coe1𝑎) = (coe1𝑎)
15554, 114, 93, 153, 110, 116, 154ply1coe 22317 . . . . . . . . . . 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 6910 . . . . . . . . 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 2734 . . . . . . . . . 10 (0g‘(Poly1‘ℂfld)) = (0g‘(Poly1‘ℂfld))
15954ply1ring 22264 . . . . . . . . . . . 12 (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ Ring)
1609, 159ax-mp 5 . . . . . . . . . . 11 (Poly1‘ℂfld) ∈ Ring
161 ringcmn 20295 . . . . . . . . . . 11 ((Poly1‘ℂfld) ∈ Ring → (Poly1‘ℂfld) ∈ CMnd)
162160, 161mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (Poly1‘ℂfld) ∈ CMnd)
163 ringmnd 20260 . . . . . . . . . . 11 ((ℂflds ℂ) ∈ Ring → (ℂflds ℂ) ∈ Mnd)
16449, 163mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (ℂflds ℂ) ∈ Mnd)
165 nn0ex 12529 . . . . . . . . . . 11 0 ∈ V
166165a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℕ0 ∈ V)
167 rhmghm 20500 . . . . . . . . . . . 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 19256 . . . . . . . . . . 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 22268 . . . . . . . . . . . . 13 (ℂfld ∈ Ring → (Poly1‘ℂfld) ∈ LMod)
1729, 171mp1i 13 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (Poly1‘ℂfld) ∈ LMod)
17312ad2antrr 726 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → 𝑆 ⊆ ℂ)
174 eqid 2734 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
175154, 59, 58, 174coe1f 22228 . . . . . . . . . . . . . . . 16 (𝑎𝐴 → (coe1𝑎):ℕ0⟶(Base‘𝑅))
17657subrgbas 20597 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 = (Base‘𝑅))
177176feq3d 6723 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (SubRing‘ℂfld) → ((coe1𝑎):ℕ0𝑆 ↔ (coe1𝑎):ℕ0⟶(Base‘𝑅)))
178175, 177imbitrrid 246 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubRing‘ℂfld) → (𝑎𝐴 → (coe1𝑎):ℕ0𝑆))
179178imp 406 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎):ℕ0𝑆)
180179ffvelcdmda 7103 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ 𝑆)
181173, 180sseldd 3995 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ ℂ)
182110, 93mgpbas 20157 . . . . . . . . . . . . 13 (Base‘(Poly1‘ℂfld)) = (Base‘(mulGrp‘(Poly1‘ℂfld)))
183110ringmgp 20256 . . . . . . . . . . . . . 14 ((Poly1‘ℂfld) ∈ Ring → (mulGrp‘(Poly1‘ℂfld)) ∈ Mnd)
184160, 183mp1i 13 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘(Poly1‘ℂfld)) ∈ Mnd)
185 simpr 484 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
186114, 54, 93vr1cl 22234 . . . . . . . . . . . . . 14 (ℂfld ∈ Ring → (var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)))
1879, 186mp1i 13 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)))
188182, 116, 184, 185, 187mulgnn0cld 19125 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)) ∈ (Base‘(Poly1‘ℂfld)))
18954ply1sca 22269 . . . . . . . . . . . . . 14 (ℂfld ∈ Ring → ℂfld = (Scalar‘(Poly1‘ℂfld)))
1909, 189ax-mp 5 . . . . . . . . . . . . 13 fld = (Scalar‘(Poly1‘ℂfld))
19193, 190, 153, 4lmodvscl 20892 . . . . . . . . . . . 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)))
192172, 181, 188, 191syl3anc 1370 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))) ∈ (Base‘(Poly1‘ℂfld)))
193192fmpttd 7134 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℕ0⟶(Base‘(Poly1‘ℂfld)))
194165mptex 7242 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V
195 funmpt 6605 . . . . . . . . . . . . 13 Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld))))
196 fvex 6919 . . . . . . . . . . . . 13 (0g‘(Poly1‘ℂfld)) ∈ V
197194, 195, 1963pm3.2i 1338 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∧ (0g‘(Poly1‘ℂfld)) ∈ V)
198197a1i 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))
199154, 93, 54, 17coe1sfi 22230 . . . . . . . . . . . . 13 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → (coe1𝑎) finSupp 0)
200152, 199syl 17 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎) finSupp 0)
201200fsuppimpd 9406 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((coe1𝑎) supp 0) ∈ Fin)
202179feqmptd 6976 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (coe1𝑎) = (𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)))
203202oveq1d 7445 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((coe1𝑎) supp 0) = ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0))
204 eqimss2 4054 . . . . . . . . . . . . 13 (((coe1𝑎) supp 0) = ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0) → ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0) ⊆ ((coe1𝑎) supp 0))
205203, 204syl 17 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝑘 ∈ ℕ0 ↦ ((coe1𝑎)‘𝑘)) supp 0) ⊆ ((coe1𝑎) supp 0))
2069, 171mp1i 13 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (Poly1‘ℂfld) ∈ LMod)
20793, 190, 153, 17, 158lmod0vs 20909 . . . . . . . . . . . . 13 (((Poly1‘ℂfld) ∈ LMod ∧ 𝑥 ∈ (Base‘(Poly1‘ℂfld))) → (0( ·𝑠 ‘(Poly1‘ℂfld))𝑥) = (0g‘(Poly1‘ℂfld)))
208206, 207sylan 580 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑥 ∈ (Base‘(Poly1‘ℂfld))) → (0( ·𝑠 ‘(Poly1‘ℂfld))𝑥) = (0g‘(Poly1‘ℂfld)))
209 c0ex 11252 . . . . . . . . . . . . 13 0 ∈ V
210209a1i 11 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 0 ∈ V)
211205, 208, 180, 188, 210suppssov1 8220 . . . . . . . . . . 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))
212 suppssfifsupp 9417 . . . . . . . . . . 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)))
213198, 201, 211, 212syl12anc 837 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) finSupp (0g‘(Poly1‘ℂfld)))
21493, 158, 162, 164, 166, 170, 193, 213gsummhm 19970 . . . . . . . . 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)))))))
21595a1i 11 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝐸:(Base‘(Poly1‘ℂfld))⟶(Base‘(ℂflds ℂ)))
216215, 192cofmpt 7151 . . . . . . . . . . 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))))))
2179a1i 11 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ℂfld ∈ Ring)
2186a1i 11 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → ℂ ∈ V)
21995ffvelcdmi 7102 . . . . . . . . . . . . . . . 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)))
220192, 219syl 17 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))) ∈ (Base‘(ℂflds ℂ)))
2213, 4, 67, 217, 218, 220pwselbas 17535 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) → (𝐸‘(((coe1𝑎)‘𝑘)( ·𝑠 ‘(Poly1‘ℂfld))(𝑘(.g‘(mulGrp‘(Poly1‘ℂfld)))(var1‘ℂfld)))):ℂ⟶ℂ)
222221feqmptd 6976 . . . . . . . . . . . . 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)))
22352a1i 11 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CRing)
224 simpr 484 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
22553, 114, 4, 54, 93, 223, 224evl1vard 22356 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((var1‘ℂfld) ∈ (Base‘(Poly1‘ℂfld)) ∧ ((𝐸‘(var1‘ℂfld))‘𝑧) = 𝑧))
226185adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ0)
22753, 54, 4, 93, 223, 224, 225, 116, 127, 226evl1expd 22364 . . . . . . . . . . . . . . . . 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)))
228224, 226, 131syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝑘(.g‘(mulGrp‘ℂfld))𝑧) = (𝑧𝑘))
229228eqeq2d 2745 . . . . . . . . . . . . . . . . . 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)))
230229anbi2d 630 . . . . . . . . . . . . . . . . 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))))
231227, 230mpbid 232 . . . . . . . . . . . . . . . 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)))
232181adantr 480 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((coe1𝑎)‘𝑘) ∈ ℂ)
23353, 54, 4, 93, 223, 224, 231, 232, 153, 80evl1vsd 22363 . . . . . . . . . . . . . . 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))))
234233simprd 495 . . . . . . . . . . . . . 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)))
235234mpteq2dva 5247 . . . . . . . . . . . . 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))))
236222, 235eqtrd 2774 . . . . . . . . . . . 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))))
237236mpteq2dva 5247 . . . . . . . . . . 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)))))
238216, 237eqtrd 2774 . . . . . . . . . 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)))))
239238oveq2d 7446 . . . . . . . . 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))))))
240157, 214, 2393eqtr2d 2780 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) = ((ℂflds ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))))
2416a1i 11 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℂ ∈ V)
2429, 10mp1i 13 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ℂfld ∈ CMnd)
243181adantlr 715 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → ((coe1𝑎)‘𝑘) ∈ ℂ)
24433adantll 714 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (𝑧𝑘) ∈ ℂ)
245243, 244mulcld 11278 . . . . . . . . . 10 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
246245anasss 466 . . . . . . . . 9 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ (𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0)) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
247165mptex 7242 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∈ V
248 funmpt 6605 . . . . . . . . . . . 12 Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))
249247, 248, 393pm3.2i 1338 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) ∧ (0g‘(ℂflds ℂ)) ∈ V)
250249a1i 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))
251 fzfid 14010 . . . . . . . . . 10 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)) ∈ Fin)
252 eldifn 4141 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0))) → ¬ 𝑘 ∈ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))
253252adantl 481 . . . . . . . . . . . . . . . . 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)))
254152ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → 𝑎 ∈ (Base‘(Poly1‘ℂfld)))
255 eldifi 4140 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0))) → 𝑘 ∈ ℕ0)
256255adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℕ0)
257 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . . 24 (deg1‘ℂfld) = (deg1‘ℂfld)
258257, 54, 93, 17, 154deg1ge 26151 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎 ∈ (Base‘(Poly1‘ℂfld)) ∧ 𝑘 ∈ ℕ0 ∧ ((coe1𝑎)‘𝑘) ≠ 0) → 𝑘 ≤ ((deg1‘ℂfld)‘𝑎))
2592583expia 1120 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ (Base‘(Poly1‘ℂfld)) ∧ 𝑘 ∈ ℕ0) → (((coe1𝑎)‘𝑘) ≠ 0 → 𝑘 ≤ ((deg1‘ℂfld)‘𝑎)))
260254, 256, 259syl2anc 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)))
261 0xr 11305 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ*
262257, 54, 93deg1xrcl 26135 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → ((deg1‘ℂfld)‘𝑎) ∈ ℝ*)
263152, 262syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((deg1‘ℂfld)‘𝑎) ∈ ℝ*)
264263ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → ((deg1‘ℂfld)‘𝑎) ∈ ℝ*)
265 xrmax2 13214 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℝ* ∧ ((deg1‘ℂfld)‘𝑎) ∈ ℝ*) → ((deg1‘ℂfld)‘𝑎) ≤ if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0))
266261, 264, 265sylancr 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))
267256nn0red 12585 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ)
268267rexrd 11308 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → 𝑘 ∈ ℝ*)
269 ifcl 4575 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((deg1‘ℂfld)‘𝑎) ∈ ℝ* ∧ 0 ∈ ℝ*) → if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0) ∈ ℝ*)
270264, 261, 269sylancl 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*)
271 xrletr 13196 . . . . . . . . . . . . . . . . . . . . . . 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)))
272268, 264, 270, 271syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → ((𝑘 ≤ ((deg1‘ℂfld)‘𝑎) ∧ ((deg1‘ℂfld)‘𝑎) ≤ if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)) → 𝑘 ≤ if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))
273266, 272mpan2d 694 . . . . . . . . . . . . . . . . . . . . 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)))
274260, 273syld 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)))
275274, 256jctild 525 . . . . . . . . . . . . . . . . . . 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))))
276257, 54, 93deg1cl 26136 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 ∈ (Base‘(Poly1‘ℂfld)) → ((deg1‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
277152, 276syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((deg1‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
278 elun 4162 . . . . . . . . . . . . . . . . . . . . . . 23 (((deg1‘ℂfld)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ (((deg1‘ℂfld)‘𝑎) ∈ ℕ0 ∨ ((deg1‘ℂfld)‘𝑎) ∈ {-∞}))
279277, 278sylib 218 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (((deg1‘ℂfld)‘𝑎) ∈ ℕ0 ∨ ((deg1‘ℂfld)‘𝑎) ∈ {-∞}))
280 nn0ge0 12548 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((deg1‘ℂfld)‘𝑎) ∈ ℕ0 → 0 ≤ ((deg1‘ℂfld)‘𝑎))
281280iftrued 4538 . . . . . . . . . . . . . . . . . . . . . . . 24 (((deg1‘ℂfld)‘𝑎) ∈ ℕ0 → if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0) = ((deg1‘ℂfld)‘𝑎))
282 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24 (((deg1‘ℂfld)‘𝑎) ∈ ℕ0 → ((deg1‘ℂfld)‘𝑎) ∈ ℕ0)
283281, 282eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . 23 (((deg1‘ℂfld)‘𝑎) ∈ ℕ0 → if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0) ∈ ℕ0)
284 mnflt0 13164 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -∞ < 0
285 mnfxr 11315 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 -∞ ∈ ℝ*
286 xrltnle 11325 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞))
287285, 261, 286mp2an 692 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (-∞ < 0 ↔ ¬ 0 ≤ -∞)
288284, 287mpbi 230 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ¬ 0 ≤ -∞
289 elsni 4647 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((deg1‘ℂfld)‘𝑎) ∈ {-∞} → ((deg1‘ℂfld)‘𝑎) = -∞)
290289breq2d 5159 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((deg1‘ℂfld)‘𝑎) ∈ {-∞} → (0 ≤ ((deg1‘ℂfld)‘𝑎) ↔ 0 ≤ -∞))
291288, 290mtbiri 327 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((deg1‘ℂfld)‘𝑎) ∈ {-∞} → ¬ 0 ≤ ((deg1‘ℂfld)‘𝑎))
292291iffalsed 4541 . . . . . . . . . . . . . . . . . . . . . . . 24 (((deg1‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0) = 0)
293 0nn0 12538 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℕ0
294292, 293eqeltrdi 2846 . . . . . . . . . . . . . . . . . . . . . . 23 (((deg1‘ℂfld)‘𝑎) ∈ {-∞} → if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0) ∈ ℕ0)
295283, 294jaoi 857 . . . . . . . . . . . . . . . . . . . . . 22 ((((deg1‘ℂfld)‘𝑎) ∈ ℕ0 ∨ ((deg1‘ℂfld)‘𝑎) ∈ {-∞}) → if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0) ∈ ℕ0)
296279, 295syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0) ∈ ℕ0)
297296ad2antrr 726 . . . . . . . . . . . . . . . . . . . 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)
298 fznn0 13655 . . . . . . . . . . . . . . . . . . . 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))))
299297, 298syl 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))))
300275, 299sylibrd 259 . . . . . . . . . . . . . . . . . 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))))
301300necon1bd 2955 . . . . . . . . . . . . . . . . 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))
302253, 301mpd 15 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → ((coe1𝑎)‘𝑘) = 0)
303302oveq1d 7445 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = (0 · (𝑧𝑘)))
304255, 244sylan2 593 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → (𝑧𝑘) ∈ ℂ)
305304mul02d 11456 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → (0 · (𝑧𝑘)) = 0)
306303, 305eqtrd 2774 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = 0)
307306an32s 652 . . . . . . . . . . . . 13 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑘 ∈ (ℕ0 ∖ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))) ∧ 𝑧 ∈ ℂ) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) = 0)
308307mpteq2dva 5247 . . . . . . . . . . . 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))
309 fconstmpt 5750 . . . . . . . . . . . . 13 (ℂ × {0}) = (𝑧 ∈ ℂ ↦ 0)
310 ringmnd 20260 . . . . . . . . . . . . . . 15 (ℂfld ∈ Ring → ℂfld ∈ Mnd)
3119, 310ax-mp 5 . . . . . . . . . . . . . 14 fld ∈ Mnd
3123, 17pws0g 18798 . . . . . . . . . . . . . 14 ((ℂfld ∈ Mnd ∧ ℂ ∈ V) → (ℂ × {0}) = (0g‘(ℂflds ℂ)))
313311, 6, 312mp2an 692 . . . . . . . . . . . . 13 (ℂ × {0}) = (0g‘(ℂflds ℂ))
314309, 313eqtr3i 2764 . . . . . . . . . . . 12 (𝑧 ∈ ℂ ↦ 0) = (0g‘(ℂflds ℂ))
315308, 314eqtrdi 2790 . . . . . . . . . . 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)))
316315, 166suppss2 8223 . . . . . . . . . 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)))
317 suppssfifsupp 9417 . . . . . . . . . 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)))
318250, 251, 316, 317syl12anc 837 . . . . . . . . 9 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))) finSupp (0g‘(ℂflds ℂ)))
3193, 4, 5, 241, 166, 242, 246, 318pwsgsum 20014 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((ℂflds ℂ) Σg (𝑘 ∈ ℕ0 ↦ (𝑧 ∈ ℂ ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))) = (𝑧 ∈ ℂ ↦ (ℂfld Σg (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))))))
320 fz0ssnn0 13658 . . . . . . . . . . . 12 (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)) ⊆ ℕ0
321 resmpt 6056 . . . . . . . . . . . 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))))
322320, 321ax-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)))
323322oveq2i 7441 . . . . . . . . . 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))))
3249, 10mp1i 13 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ℂfld ∈ CMnd)
325165a1i 11 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ℕ0 ∈ V)
326245fmpttd 7134 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))):ℕ0⟶ℂ)
327306, 325suppss2 8223 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) supp 0) ⊆ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)))
328165mptex 7242 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V
329 funmpt 6605 . . . . . . . . . . . . . 14 Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘)))
330328, 329, 2093pm3.2i 1338 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∧ 0 ∈ V)
331330a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → ((𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ V ∧ Fun (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∧ 0 ∈ V))
332 fzfid 14010 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)) ∈ Fin)
333 suppssfifsupp 9417 . . . . . . . . . . . 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)
334331, 332, 327, 333syl12anc 837 . . . . . . . . . . 11 (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) → (𝑘 ∈ ℕ0 ↦ (((coe1𝑎)‘𝑘) · (𝑧𝑘))) finSupp 0)
3354, 17, 324, 325, 326, 327, 334gsumres 19945 . . . . . . . . . 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)))))
336 elfznn0 13656 . . . . . . . . . . . 12 (𝑘 ∈ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0)) → 𝑘 ∈ ℕ0)
337336, 245sylan2 593 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0))) → (((coe1𝑎)‘𝑘) · (𝑧𝑘)) ∈ ℂ)
338332, 337gsumfsum 21469 . . . . . . . . . 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)))
339323, 335, 3383eqtr3a 2798 . . . . . . . . 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)))
340339mpteq2dva 5247 . . . . . . . 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))))
341240, 319, 3403eqtrd 2778 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘))))
34212adantr 480 . . . . . . . 8 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → 𝑆 ⊆ ℂ)
343 elplyr 26254 . . . . . . . 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))
344342, 296, 179, 343syl3anc 1370 . . . . . . 7 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(0 ≤ ((deg1‘ℂfld)‘𝑎), ((deg1‘ℂfld)‘𝑎), 0))(((coe1𝑎)‘𝑘) · (𝑧𝑘))) ∈ (Poly‘𝑆))
345341, 344eqeltrd 2838 . . . . . 6 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → (𝐸𝑎) ∈ (Poly‘𝑆))
346 eleq1 2826 . . . . . 6 ((𝐸𝑎) = 𝑓 → ((𝐸𝑎) ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (Poly‘𝑆)))
347345, 346syl5ibcom 245 . . . . 5 ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐴) → ((𝐸𝑎) = 𝑓𝑓 ∈ (Poly‘𝑆)))
348347rexlimdva 3152 . . . 4 (𝑆 ∈ (SubRing‘ℂfld) → (∃𝑎𝐴 (𝐸𝑎) = 𝑓𝑓 ∈ (Poly‘𝑆)))
349151, 348syl5 34 . . 3 (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (𝐸𝐴) → 𝑓 ∈ (Poly‘𝑆)))
350147, 349impbid 212 . 2 (𝑆 ∈ (SubRing‘ℂfld) → (𝑓 ∈ (Poly‘𝑆) ↔ 𝑓 ∈ (𝐸𝐴)))
351350eqrdv 2732 1 (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wrex 3067  Vcvv 3477  cdif 3959  cun 3960  wss 3962  ifcif 4530  {csn 4630   class class class wbr 5147  cmpt 5230   × cxp 5686  cres 5690  cima 5691  ccom 5692  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694   supp csupp 8183  m cmap 8864  Fincfn 8983   finSupp cfsupp 9398  cc 11150  0cc0 11152   · cmul 11157  -∞cmnf 11290  *cxr 11291   < clt 11292  cle 11293  0cn0 12523  ...cfz 13543  cexp 14098  Σcsu 15718  Basecbs 17244  s cress 17273  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485   Σg cgsu 17486  s cpws 17492  Mndcmnd 18759   MndHom cmhm 18806  SubMndcsubmnd 18807  .gcmg 19097  SubGrpcsubg 19150   GrpHom cghm 19242  CMndccmn 19812  mulGrpcmgp 20151  Ringcrg 20250  CRingccrg 20251   RingHom crh 20485  SubRingcsubrg 20585  LModclmod 20874  fldccnfld 21381  algSccascl 21889  var1cv1 22192  Poly1cpl1 22193  coe1cco1 22194  eval1ce1 22333  deg1cdg1 26107  Polycply 26237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231  ax-mulf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-srg 20204  df-ring 20252  df-cring 20253  df-rhm 20488  df-subrng 20562  df-subrg 20586  df-lmod 20876  df-lss 20947  df-lsp 20987  df-cnfld 21382  df-assa 21890  df-asp 21891  df-ascl 21892  df-psr 21946  df-mvr 21947  df-mpl 21948  df-opsr 21950  df-evls 22115  df-evl 22116  df-psr1 22196  df-vr1 22197  df-ply1 22198  df-coe1 22199  df-evl1 22335  df-mdeg 26108  df-deg1 26109  df-ply 26241
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator