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Theorem plypf1 25589
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 = (β„‚fld β†Ύs 𝑆)
plypf1.p 𝑃 = (Poly1β€˜π‘…)
plypf1.a 𝐴 = (Baseβ€˜π‘ƒ)
plypf1.e 𝐸 = (eval1β€˜β„‚fld)
Assertion
Ref Expression
plypf1 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ (Polyβ€˜π‘†) = (𝐸 β€œ 𝐴))

Proof of Theorem plypf1
Dummy variables 𝑓 π‘Ž π‘˜ 𝑛 π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elply 25572 . . . . 5 (𝑓 ∈ (Polyβ€˜π‘†) ↔ (𝑆 βŠ† β„‚ ∧ βˆƒπ‘› ∈ β„•0 βˆƒπ‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0)𝑓 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
21simprbi 498 . . . 4 (𝑓 ∈ (Polyβ€˜π‘†) β†’ βˆƒπ‘› ∈ β„•0 βˆƒπ‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0)𝑓 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))
3 eqid 2737 . . . . . . . . 9 (β„‚fld ↑s β„‚) = (β„‚fld ↑s β„‚)
4 cnfldbas 20816 . . . . . . . . 9 β„‚ = (Baseβ€˜β„‚fld)
5 eqid 2737 . . . . . . . . 9 (0gβ€˜(β„‚fld ↑s β„‚)) = (0gβ€˜(β„‚fld ↑s β„‚))
6 cnex 11139 . . . . . . . . . 10 β„‚ ∈ V
76a1i 11 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ β„‚ ∈ V)
8 fzfid 13885 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (0...𝑛) ∈ Fin)
9 cnring 20835 . . . . . . . . . 10 β„‚fld ∈ Ring
10 ringcmn 20010 . . . . . . . . . 10 (β„‚fld ∈ Ring β†’ β„‚fld ∈ CMnd)
119, 10mp1i 13 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ β„‚fld ∈ CMnd)
124subrgss 20239 . . . . . . . . . . . . 13 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ 𝑆 βŠ† β„‚)
1312ad2antrr 725 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ 𝑆 βŠ† β„‚)
14 elmapi 8794 . . . . . . . . . . . . . . 15 (π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0) β†’ π‘Ž:β„•0⟢(𝑆 βˆͺ {0}))
1514ad2antll 728 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ π‘Ž:β„•0⟢(𝑆 βˆͺ {0}))
16 subrgsubg 20244 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ 𝑆 ∈ (SubGrpβ€˜β„‚fld))
17 cnfld0 20837 . . . . . . . . . . . . . . . . . . . 20 0 = (0gβ€˜β„‚fld)
1817subg0cl 18943 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (SubGrpβ€˜β„‚fld) β†’ 0 ∈ 𝑆)
1916, 18syl 17 . . . . . . . . . . . . . . . . . 18 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ 0 ∈ 𝑆)
2019adantr 482 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ 0 ∈ 𝑆)
2120snssd 4774 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ {0} βŠ† 𝑆)
22 ssequn2 4148 . . . . . . . . . . . . . . . 16 ({0} βŠ† 𝑆 ↔ (𝑆 βˆͺ {0}) = 𝑆)
2321, 22sylib 217 . . . . . . . . . . . . . . 15 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (𝑆 βˆͺ {0}) = 𝑆)
2423feq3d 6660 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (π‘Ž:β„•0⟢(𝑆 βˆͺ {0}) ↔ π‘Ž:β„•0βŸΆπ‘†))
2515, 24mpbid 231 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ π‘Ž:β„•0βŸΆπ‘†)
26 elfznn0 13541 . . . . . . . . . . . . 13 (π‘˜ ∈ (0...𝑛) β†’ π‘˜ ∈ β„•0)
27 ffvelcdm 7037 . . . . . . . . . . . . 13 ((π‘Ž:β„•0βŸΆπ‘† ∧ π‘˜ ∈ β„•0) β†’ (π‘Žβ€˜π‘˜) ∈ 𝑆)
2825, 26, 27syl2an 597 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (π‘Žβ€˜π‘˜) ∈ 𝑆)
2913, 28sseldd 3950 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (π‘Žβ€˜π‘˜) ∈ β„‚)
3029adantrl 715 . . . . . . . . . 10 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ (𝑧 ∈ β„‚ ∧ π‘˜ ∈ (0...𝑛))) β†’ (π‘Žβ€˜π‘˜) ∈ β„‚)
31 simprl 770 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ (𝑧 ∈ β„‚ ∧ π‘˜ ∈ (0...𝑛))) β†’ 𝑧 ∈ β„‚)
3226ad2antll 728 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ (𝑧 ∈ β„‚ ∧ π‘˜ ∈ (0...𝑛))) β†’ π‘˜ ∈ β„•0)
33 expcl 13992 . . . . . . . . . . 11 ((𝑧 ∈ β„‚ ∧ π‘˜ ∈ β„•0) β†’ (π‘§β†‘π‘˜) ∈ β„‚)
3431, 32, 33syl2anc 585 . . . . . . . . . 10 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ (𝑧 ∈ β„‚ ∧ π‘˜ ∈ (0...𝑛))) β†’ (π‘§β†‘π‘˜) ∈ β„‚)
3530, 34mulcld 11182 . . . . . . . . 9 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ (𝑧 ∈ β„‚ ∧ π‘˜ ∈ (0...𝑛))) β†’ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)) ∈ β„‚)
36 eqid 2737 . . . . . . . . . 10 (π‘˜ ∈ (0...𝑛) ↦ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))) = (π‘˜ ∈ (0...𝑛) ↦ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))
376mptex 7178 . . . . . . . . . . 11 (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ V
3837a1i 11 . . . . . . . . . 10 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ V)
39 fvex 6860 . . . . . . . . . . 11 (0gβ€˜(β„‚fld ↑s β„‚)) ∈ V
4039a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (0gβ€˜(β„‚fld ↑s β„‚)) ∈ V)
4136, 8, 38, 40fsuppmptdm 9323 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (π‘˜ ∈ (0...𝑛) ↦ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))) finSupp (0gβ€˜(β„‚fld ↑s β„‚)))
423, 4, 5, 7, 8, 11, 35, 41pwsgsum 19766 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ ((β„‚fld ↑s β„‚) Ξ£g (π‘˜ ∈ (0...𝑛) ↦ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))) = (𝑧 ∈ β„‚ ↦ (β„‚fld Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))))
43 fzfid 13885 . . . . . . . . . 10 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ 𝑧 ∈ β„‚) β†’ (0...𝑛) ∈ Fin)
4435anassrs 469 . . . . . . . . . 10 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (0...𝑛)) β†’ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)) ∈ β„‚)
4543, 44gsumfsum 20880 . . . . . . . . 9 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ 𝑧 ∈ β„‚) β†’ (β„‚fld Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))) = Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))
4645mpteq2dva 5210 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (𝑧 ∈ β„‚ ↦ (β„‚fld Ξ£g (π‘˜ ∈ (0...𝑛) ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))) = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))
4742, 46eqtrd 2777 . . . . . . 7 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ ((β„‚fld ↑s β„‚) Ξ£g (π‘˜ ∈ (0...𝑛) ↦ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))) = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))
483pwsring 20046 . . . . . . . . . 10 ((β„‚fld ∈ Ring ∧ β„‚ ∈ V) β†’ (β„‚fld ↑s β„‚) ∈ Ring)
499, 6, 48mp2an 691 . . . . . . . . 9 (β„‚fld ↑s β„‚) ∈ Ring
50 ringcmn 20010 . . . . . . . . 9 ((β„‚fld ↑s β„‚) ∈ Ring β†’ (β„‚fld ↑s β„‚) ∈ CMnd)
5149, 50mp1i 13 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (β„‚fld ↑s β„‚) ∈ CMnd)
52 cncrng 20834 . . . . . . . . . . 11 β„‚fld ∈ CRing
53 plypf1.e . . . . . . . . . . . 12 𝐸 = (eval1β€˜β„‚fld)
54 eqid 2737 . . . . . . . . . . . 12 (Poly1β€˜β„‚fld) = (Poly1β€˜β„‚fld)
5553, 54, 3, 4evl1rhm 21714 . . . . . . . . . . 11 (β„‚fld ∈ CRing β†’ 𝐸 ∈ ((Poly1β€˜β„‚fld) RingHom (β„‚fld ↑s β„‚)))
5652, 55ax-mp 5 . . . . . . . . . 10 𝐸 ∈ ((Poly1β€˜β„‚fld) RingHom (β„‚fld ↑s β„‚))
57 plypf1.r . . . . . . . . . . . 12 𝑅 = (β„‚fld β†Ύs 𝑆)
58 plypf1.p . . . . . . . . . . . 12 𝑃 = (Poly1β€˜π‘…)
59 plypf1.a . . . . . . . . . . . 12 𝐴 = (Baseβ€˜π‘ƒ)
6054, 57, 58, 59subrgply1 21620 . . . . . . . . . . 11 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ 𝐴 ∈ (SubRingβ€˜(Poly1β€˜β„‚fld)))
6160adantr 482 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ 𝐴 ∈ (SubRingβ€˜(Poly1β€˜β„‚fld)))
62 rhmima 20269 . . . . . . . . . 10 ((𝐸 ∈ ((Poly1β€˜β„‚fld) RingHom (β„‚fld ↑s β„‚)) ∧ 𝐴 ∈ (SubRingβ€˜(Poly1β€˜β„‚fld))) β†’ (𝐸 β€œ 𝐴) ∈ (SubRingβ€˜(β„‚fld ↑s β„‚)))
6356, 61, 62sylancr 588 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (𝐸 β€œ 𝐴) ∈ (SubRingβ€˜(β„‚fld ↑s β„‚)))
64 subrgsubg 20244 . . . . . . . . 9 ((𝐸 β€œ 𝐴) ∈ (SubRingβ€˜(β„‚fld ↑s β„‚)) β†’ (𝐸 β€œ 𝐴) ∈ (SubGrpβ€˜(β„‚fld ↑s β„‚)))
65 subgsubm 18957 . . . . . . . . 9 ((𝐸 β€œ 𝐴) ∈ (SubGrpβ€˜(β„‚fld ↑s β„‚)) β†’ (𝐸 β€œ 𝐴) ∈ (SubMndβ€˜(β„‚fld ↑s β„‚)))
6663, 64, 653syl 18 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (𝐸 β€œ 𝐴) ∈ (SubMndβ€˜(β„‚fld ↑s β„‚)))
67 eqid 2737 . . . . . . . . . . . 12 (Baseβ€˜(β„‚fld ↑s β„‚)) = (Baseβ€˜(β„‚fld ↑s β„‚))
689a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ β„‚fld ∈ Ring)
696a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ β„‚ ∈ V)
70 fconst6g 6736 . . . . . . . . . . . . . 14 ((π‘Žβ€˜π‘˜) ∈ β„‚ β†’ (β„‚ Γ— {(π‘Žβ€˜π‘˜)}):β„‚βŸΆβ„‚)
7129, 70syl 17 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (β„‚ Γ— {(π‘Žβ€˜π‘˜)}):β„‚βŸΆβ„‚)
723, 4, 67pwselbasb 17377 . . . . . . . . . . . . . 14 ((β„‚fld ∈ Ring ∧ β„‚ ∈ V) β†’ ((β„‚ Γ— {(π‘Žβ€˜π‘˜)}) ∈ (Baseβ€˜(β„‚fld ↑s β„‚)) ↔ (β„‚ Γ— {(π‘Žβ€˜π‘˜)}):β„‚βŸΆβ„‚))
739, 6, 72mp2an 691 . . . . . . . . . . . . 13 ((β„‚ Γ— {(π‘Žβ€˜π‘˜)}) ∈ (Baseβ€˜(β„‚fld ↑s β„‚)) ↔ (β„‚ Γ— {(π‘Žβ€˜π‘˜)}):β„‚βŸΆβ„‚)
7471, 73sylibr 233 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (β„‚ Γ— {(π‘Žβ€˜π‘˜)}) ∈ (Baseβ€˜(β„‚fld ↑s β„‚)))
7534anass1rs 654 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ (π‘§β†‘π‘˜) ∈ β„‚)
7675fmpttd 7068 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)):β„‚βŸΆβ„‚)
773, 4, 67pwselbasb 17377 . . . . . . . . . . . . . 14 ((β„‚fld ∈ Ring ∧ β„‚ ∈ V) β†’ ((𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)) ∈ (Baseβ€˜(β„‚fld ↑s β„‚)) ↔ (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)):β„‚βŸΆβ„‚))
789, 6, 77mp2an 691 . . . . . . . . . . . . 13 ((𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)) ∈ (Baseβ€˜(β„‚fld ↑s β„‚)) ↔ (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)):β„‚βŸΆβ„‚)
7976, 78sylibr 233 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)) ∈ (Baseβ€˜(β„‚fld ↑s β„‚)))
80 cnfldmul 20818 . . . . . . . . . . . 12 Β· = (.rβ€˜β„‚fld)
81 eqid 2737 . . . . . . . . . . . 12 (.rβ€˜(β„‚fld ↑s β„‚)) = (.rβ€˜(β„‚fld ↑s β„‚))
823, 67, 68, 69, 74, 79, 80, 81pwsmulrval 17380 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ ((β„‚ Γ— {(π‘Žβ€˜π‘˜)})(.rβ€˜(β„‚fld ↑s β„‚))(𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜))) = ((β„‚ Γ— {(π‘Žβ€˜π‘˜)}) ∘f Β· (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜))))
8329adantr 482 . . . . . . . . . . . 12 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ (π‘Žβ€˜π‘˜) ∈ β„‚)
84 fconstmpt 5699 . . . . . . . . . . . . 13 (β„‚ Γ— {(π‘Žβ€˜π‘˜)}) = (𝑧 ∈ β„‚ ↦ (π‘Žβ€˜π‘˜))
8584a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (β„‚ Γ— {(π‘Žβ€˜π‘˜)}) = (𝑧 ∈ β„‚ ↦ (π‘Žβ€˜π‘˜)))
86 eqidd 2738 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)) = (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)))
8769, 83, 75, 85, 86offval2 7642 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ ((β„‚ Γ— {(π‘Žβ€˜π‘˜)}) ∘f Β· (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜))) = (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))
8882, 87eqtrd 2777 . . . . . . . . . 10 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ ((β„‚ Γ— {(π‘Žβ€˜π‘˜)})(.rβ€˜(β„‚fld ↑s β„‚))(𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜))) = (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))
8963adantr 482 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝐸 β€œ 𝐴) ∈ (SubRingβ€˜(β„‚fld ↑s β„‚)))
90 eqid 2737 . . . . . . . . . . . . . 14 (algScβ€˜(Poly1β€˜β„‚fld)) = (algScβ€˜(Poly1β€˜β„‚fld))
9153, 54, 4, 90evl1sca 21716 . . . . . . . . . . . . 13 ((β„‚fld ∈ CRing ∧ (π‘Žβ€˜π‘˜) ∈ β„‚) β†’ (πΈβ€˜((algScβ€˜(Poly1β€˜β„‚fld))β€˜(π‘Žβ€˜π‘˜))) = (β„‚ Γ— {(π‘Žβ€˜π‘˜)}))
9252, 29, 91sylancr 588 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (πΈβ€˜((algScβ€˜(Poly1β€˜β„‚fld))β€˜(π‘Žβ€˜π‘˜))) = (β„‚ Γ— {(π‘Žβ€˜π‘˜)}))
93 eqid 2737 . . . . . . . . . . . . . . . 16 (Baseβ€˜(Poly1β€˜β„‚fld)) = (Baseβ€˜(Poly1β€˜β„‚fld))
9493, 67rhmf 20167 . . . . . . . . . . . . . . 15 (𝐸 ∈ ((Poly1β€˜β„‚fld) RingHom (β„‚fld ↑s β„‚)) β†’ 𝐸:(Baseβ€˜(Poly1β€˜β„‚fld))⟢(Baseβ€˜(β„‚fld ↑s β„‚)))
9556, 94ax-mp 5 . . . . . . . . . . . . . 14 𝐸:(Baseβ€˜(Poly1β€˜β„‚fld))⟢(Baseβ€˜(β„‚fld ↑s β„‚))
96 ffn 6673 . . . . . . . . . . . . . 14 (𝐸:(Baseβ€˜(Poly1β€˜β„‚fld))⟢(Baseβ€˜(β„‚fld ↑s β„‚)) β†’ 𝐸 Fn (Baseβ€˜(Poly1β€˜β„‚fld)))
9795, 96mp1i 13 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ 𝐸 Fn (Baseβ€˜(Poly1β€˜β„‚fld)))
9893subrgss 20239 . . . . . . . . . . . . . . 15 (𝐴 ∈ (SubRingβ€˜(Poly1β€˜β„‚fld)) β†’ 𝐴 βŠ† (Baseβ€˜(Poly1β€˜β„‚fld)))
9960, 98syl 17 . . . . . . . . . . . . . 14 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ 𝐴 βŠ† (Baseβ€˜(Poly1β€˜β„‚fld)))
10099ad2antrr 725 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ 𝐴 βŠ† (Baseβ€˜(Poly1β€˜β„‚fld)))
101 simpll 766 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ 𝑆 ∈ (SubRingβ€˜β„‚fld))
10254, 90, 57, 58, 101, 59, 4, 29subrg1asclcl 21647 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (((algScβ€˜(Poly1β€˜β„‚fld))β€˜(π‘Žβ€˜π‘˜)) ∈ 𝐴 ↔ (π‘Žβ€˜π‘˜) ∈ 𝑆))
10328, 102mpbird 257 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ ((algScβ€˜(Poly1β€˜β„‚fld))β€˜(π‘Žβ€˜π‘˜)) ∈ 𝐴)
104 fnfvima 7188 . . . . . . . . . . . . 13 ((𝐸 Fn (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ 𝐴 βŠ† (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ ((algScβ€˜(Poly1β€˜β„‚fld))β€˜(π‘Žβ€˜π‘˜)) ∈ 𝐴) β†’ (πΈβ€˜((algScβ€˜(Poly1β€˜β„‚fld))β€˜(π‘Žβ€˜π‘˜))) ∈ (𝐸 β€œ 𝐴))
10597, 100, 103, 104syl3anc 1372 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (πΈβ€˜((algScβ€˜(Poly1β€˜β„‚fld))β€˜(π‘Žβ€˜π‘˜))) ∈ (𝐸 β€œ 𝐴))
10692, 105eqeltrrd 2839 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (β„‚ Γ— {(π‘Žβ€˜π‘˜)}) ∈ (𝐸 β€œ 𝐴))
10767subrgss 20239 . . . . . . . . . . . . . . . . 17 ((𝐸 β€œ 𝐴) ∈ (SubRingβ€˜(β„‚fld ↑s β„‚)) β†’ (𝐸 β€œ 𝐴) βŠ† (Baseβ€˜(β„‚fld ↑s β„‚)))
10889, 107syl 17 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝐸 β€œ 𝐴) βŠ† (Baseβ€˜(β„‚fld ↑s β„‚)))
10960ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ 𝐴 ∈ (SubRingβ€˜(Poly1β€˜β„‚fld)))
110 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (mulGrpβ€˜(Poly1β€˜β„‚fld)) = (mulGrpβ€˜(Poly1β€˜β„‚fld))
111110subrgsubm 20251 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ (SubRingβ€˜(Poly1β€˜β„‚fld)) β†’ 𝐴 ∈ (SubMndβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld))))
112109, 111syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ 𝐴 ∈ (SubMndβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld))))
11326adantl 483 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ π‘˜ ∈ β„•0)
114 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (var1β€˜β„‚fld) = (var1β€˜β„‚fld)
115114, 101, 57, 58, 59subrgvr1cl 21649 . . . . . . . . . . . . . . . . . 18 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (var1β€˜β„‚fld) ∈ 𝐴)
116 eqid 2737 . . . . . . . . . . . . . . . . . . 19 (.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld))) = (.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))
117116submmulgcl 18926 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ (SubMndβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld))) ∧ π‘˜ ∈ β„•0 ∧ (var1β€˜β„‚fld) ∈ 𝐴) β†’ (π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ 𝐴)
118112, 113, 115, 117syl3anc 1372 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ 𝐴)
119 fnfvima 7188 . . . . . . . . . . . . . . . . 17 ((𝐸 Fn (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ 𝐴 βŠ† (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ (π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ 𝐴) β†’ (πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))) ∈ (𝐸 β€œ 𝐴))
12097, 100, 118, 119syl3anc 1372 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))) ∈ (𝐸 β€œ 𝐴))
121108, 120sseldd 3950 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))) ∈ (Baseβ€˜(β„‚fld ↑s β„‚)))
1223, 4, 67, 68, 69, 121pwselbas 17378 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))):β„‚βŸΆβ„‚)
123122feqmptd 6915 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))) = (𝑧 ∈ β„‚ ↦ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§)))
12452a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ β„‚fld ∈ CRing)
125 simpr 486 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ 𝑧 ∈ β„‚)
12653, 114, 4, 54, 93, 124, 125evl1vard 21719 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ ((var1β€˜β„‚fld) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ ((πΈβ€˜(var1β€˜β„‚fld))β€˜π‘§) = 𝑧))
127 eqid 2737 . . . . . . . . . . . . . . . . 17 (.gβ€˜(mulGrpβ€˜β„‚fld)) = (.gβ€˜(mulGrpβ€˜β„‚fld))
128113adantr 482 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ π‘˜ ∈ β„•0)
12953, 54, 4, 93, 124, 125, 126, 116, 127, 128evl1expd 21727 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ ((π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§) = (π‘˜(.gβ€˜(mulGrpβ€˜β„‚fld))𝑧)))
130129simprd 497 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§) = (π‘˜(.gβ€˜(mulGrpβ€˜β„‚fld))𝑧))
131 cnfldexp 20846 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ β„‚ ∧ π‘˜ ∈ β„•0) β†’ (π‘˜(.gβ€˜(mulGrpβ€˜β„‚fld))𝑧) = (π‘§β†‘π‘˜))
132125, 128, 131syl2anc 585 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ (π‘˜(.gβ€˜(mulGrpβ€˜β„‚fld))𝑧) = (π‘§β†‘π‘˜))
133130, 132eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) ∧ 𝑧 ∈ β„‚) β†’ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§) = (π‘§β†‘π‘˜))
134133mpteq2dva 5210 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝑧 ∈ β„‚ ↦ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§)) = (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)))
135123, 134eqtrd 2777 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))) = (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)))
136135, 120eqeltrrd 2839 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)) ∈ (𝐸 β€œ 𝐴))
13781subrgmcl 20250 . . . . . . . . . . 11 (((𝐸 β€œ 𝐴) ∈ (SubRingβ€˜(β„‚fld ↑s β„‚)) ∧ (β„‚ Γ— {(π‘Žβ€˜π‘˜)}) ∈ (𝐸 β€œ 𝐴) ∧ (𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜)) ∈ (𝐸 β€œ 𝐴)) β†’ ((β„‚ Γ— {(π‘Žβ€˜π‘˜)})(.rβ€˜(β„‚fld ↑s β„‚))(𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜))) ∈ (𝐸 β€œ 𝐴))
13889, 106, 136, 137syl3anc 1372 . . . . . . . . . 10 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ ((β„‚ Γ— {(π‘Žβ€˜π‘˜)})(.rβ€˜(β„‚fld ↑s β„‚))(𝑧 ∈ β„‚ ↦ (π‘§β†‘π‘˜))) ∈ (𝐸 β€œ 𝐴))
13988, 138eqeltrrd 2839 . . . . . . . . 9 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ (𝐸 β€œ 𝐴))
140139fmpttd 7068 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (π‘˜ ∈ (0...𝑛) ↦ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))):(0...𝑛)⟢(𝐸 β€œ 𝐴))
14136, 8, 139, 40fsuppmptdm 9323 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (π‘˜ ∈ (0...𝑛) ↦ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜)))) finSupp (0gβ€˜(β„‚fld ↑s β„‚)))
1425, 51, 8, 66, 140, 141gsumsubmcl 19703 . . . . . . 7 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ ((β„‚fld ↑s β„‚) Ξ£g (π‘˜ ∈ (0...𝑛) ↦ (𝑧 ∈ β„‚ ↦ ((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))))) ∈ (𝐸 β€œ 𝐴))
14347, 142eqeltrrd 2839 . . . . . 6 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ (𝐸 β€œ 𝐴))
144 eleq1 2826 . . . . . 6 (𝑓 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) β†’ (𝑓 ∈ (𝐸 β€œ 𝐴) ↔ (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ (𝐸 β€œ 𝐴)))
145143, 144syl5ibrcom 247 . . . . 5 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ (𝑛 ∈ β„•0 ∧ π‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0))) β†’ (𝑓 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) β†’ 𝑓 ∈ (𝐸 β€œ 𝐴)))
146145rexlimdvva 3206 . . . 4 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ (βˆƒπ‘› ∈ β„•0 βˆƒπ‘Ž ∈ ((𝑆 βˆͺ {0}) ↑m β„•0)𝑓 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑛)((π‘Žβ€˜π‘˜) Β· (π‘§β†‘π‘˜))) β†’ 𝑓 ∈ (𝐸 β€œ 𝐴)))
1472, 146syl5 34 . . 3 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ (𝑓 ∈ (Polyβ€˜π‘†) β†’ 𝑓 ∈ (𝐸 β€œ 𝐴)))
148 ffun 6676 . . . . . 6 (𝐸:(Baseβ€˜(Poly1β€˜β„‚fld))⟢(Baseβ€˜(β„‚fld ↑s β„‚)) β†’ Fun 𝐸)
14995, 148ax-mp 5 . . . . 5 Fun 𝐸
150 fvelima 6913 . . . . 5 ((Fun 𝐸 ∧ 𝑓 ∈ (𝐸 β€œ 𝐴)) β†’ βˆƒπ‘Ž ∈ 𝐴 (πΈβ€˜π‘Ž) = 𝑓)
151149, 150mpan 689 . . . 4 (𝑓 ∈ (𝐸 β€œ 𝐴) β†’ βˆƒπ‘Ž ∈ 𝐴 (πΈβ€˜π‘Ž) = 𝑓)
15299sselda 3949 . . . . . . . . . . 11 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž ∈ (Baseβ€˜(Poly1β€˜β„‚fld)))
153 eqid 2737 . . . . . . . . . . . 12 ( ·𝑠 β€˜(Poly1β€˜β„‚fld)) = ( ·𝑠 β€˜(Poly1β€˜β„‚fld))
154 eqid 2737 . . . . . . . . . . . 12 (coe1β€˜π‘Ž) = (coe1β€˜π‘Ž)
15554, 114, 93, 153, 110, 116, 154ply1coe 21683 . . . . . . . . . . 11 ((β„‚fld ∈ Ring ∧ π‘Ž ∈ (Baseβ€˜(Poly1β€˜β„‚fld))) β†’ π‘Ž = ((Poly1β€˜β„‚fld) Ξ£g (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))))
1569, 152, 155sylancr 588 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž = ((Poly1β€˜β„‚fld) Ξ£g (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))))
157156fveq2d 6851 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (πΈβ€˜π‘Ž) = (πΈβ€˜((Poly1β€˜β„‚fld) Ξ£g (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))))))
158 eqid 2737 . . . . . . . . . 10 (0gβ€˜(Poly1β€˜β„‚fld)) = (0gβ€˜(Poly1β€˜β„‚fld))
15954ply1ring 21635 . . . . . . . . . . . 12 (β„‚fld ∈ Ring β†’ (Poly1β€˜β„‚fld) ∈ Ring)
1609, 159ax-mp 5 . . . . . . . . . . 11 (Poly1β€˜β„‚fld) ∈ Ring
161 ringcmn 20010 . . . . . . . . . . 11 ((Poly1β€˜β„‚fld) ∈ Ring β†’ (Poly1β€˜β„‚fld) ∈ CMnd)
162160, 161mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (Poly1β€˜β„‚fld) ∈ CMnd)
163 ringmnd 19981 . . . . . . . . . . 11 ((β„‚fld ↑s β„‚) ∈ Ring β†’ (β„‚fld ↑s β„‚) ∈ Mnd)
16449, 163mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (β„‚fld ↑s β„‚) ∈ Mnd)
165 nn0ex 12426 . . . . . . . . . . 11 β„•0 ∈ V
166165a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ β„•0 ∈ V)
167 rhmghm 20166 . . . . . . . . . . . 12 (𝐸 ∈ ((Poly1β€˜β„‚fld) RingHom (β„‚fld ↑s β„‚)) β†’ 𝐸 ∈ ((Poly1β€˜β„‚fld) GrpHom (β„‚fld ↑s β„‚)))
16856, 167ax-mp 5 . . . . . . . . . . 11 𝐸 ∈ ((Poly1β€˜β„‚fld) GrpHom (β„‚fld ↑s β„‚))
169 ghmmhm 19025 . . . . . . . . . . 11 (𝐸 ∈ ((Poly1β€˜β„‚fld) GrpHom (β„‚fld ↑s β„‚)) β†’ 𝐸 ∈ ((Poly1β€˜β„‚fld) MndHom (β„‚fld ↑s β„‚)))
170168, 169mp1i 13 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ 𝐸 ∈ ((Poly1β€˜β„‚fld) MndHom (β„‚fld ↑s β„‚)))
17154ply1lmod 21639 . . . . . . . . . . . . 13 (β„‚fld ∈ Ring β†’ (Poly1β€˜β„‚fld) ∈ LMod)
1729, 171mp1i 13 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (Poly1β€˜β„‚fld) ∈ LMod)
17312ad2antrr 725 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ 𝑆 βŠ† β„‚)
174 eqid 2737 . . . . . . . . . . . . . . . . 17 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
175154, 59, 58, 174coe1f 21598 . . . . . . . . . . . . . . . 16 (π‘Ž ∈ 𝐴 β†’ (coe1β€˜π‘Ž):β„•0⟢(Baseβ€˜π‘…))
17657subrgbas 20247 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ 𝑆 = (Baseβ€˜π‘…))
177176feq3d 6660 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ ((coe1β€˜π‘Ž):β„•0βŸΆπ‘† ↔ (coe1β€˜π‘Ž):β„•0⟢(Baseβ€˜π‘…)))
178175, 177syl5ibr 246 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ (π‘Ž ∈ 𝐴 β†’ (coe1β€˜π‘Ž):β„•0βŸΆπ‘†))
179178imp 408 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (coe1β€˜π‘Ž):β„•0βŸΆπ‘†)
180179ffvelcdmda 7040 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ ((coe1β€˜π‘Ž)β€˜π‘˜) ∈ 𝑆)
181173, 180sseldd 3950 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ ((coe1β€˜π‘Ž)β€˜π‘˜) ∈ β„‚)
182110, 93mgpbas 19909 . . . . . . . . . . . . 13 (Baseβ€˜(Poly1β€˜β„‚fld)) = (Baseβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))
183110ringmgp 19977 . . . . . . . . . . . . . 14 ((Poly1β€˜β„‚fld) ∈ Ring β†’ (mulGrpβ€˜(Poly1β€˜β„‚fld)) ∈ Mnd)
184160, 183mp1i 13 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (mulGrpβ€˜(Poly1β€˜β„‚fld)) ∈ Mnd)
185 simpr 486 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ π‘˜ ∈ β„•0)
186114, 54, 93vr1cl 21604 . . . . . . . . . . . . . 14 (β„‚fld ∈ Ring β†’ (var1β€˜β„‚fld) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)))
1879, 186mp1i 13 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (var1β€˜β„‚fld) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)))
188182, 116, 184, 185, 187mulgnn0cld 18904 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)))
18954ply1sca 21640 . . . . . . . . . . . . . 14 (β„‚fld ∈ Ring β†’ β„‚fld = (Scalarβ€˜(Poly1β€˜β„‚fld)))
1909, 189ax-mp 5 . . . . . . . . . . . . 13 β„‚fld = (Scalarβ€˜(Poly1β€˜β„‚fld))
19193, 190, 153, 4lmodvscl 20355 . . . . . . . . . . . 12 (((Poly1β€˜β„‚fld) ∈ LMod ∧ ((coe1β€˜π‘Ž)β€˜π‘˜) ∈ β„‚ ∧ (π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ (Baseβ€˜(Poly1β€˜β„‚fld))) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)))
192172, 181, 188, 191syl3anc 1372 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)))
193192fmpttd 7068 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))):β„•0⟢(Baseβ€˜(Poly1β€˜β„‚fld)))
194165mptex 7178 . . . . . . . . . . . . 13 (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) ∈ V
195 funmpt 6544 . . . . . . . . . . . . 13 Fun (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))
196 fvex 6860 . . . . . . . . . . . . 13 (0gβ€˜(Poly1β€˜β„‚fld)) ∈ V
197194, 195, 1963pm3.2i 1340 . . . . . . . . . . . 12 ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) ∈ V ∧ Fun (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) ∧ (0gβ€˜(Poly1β€˜β„‚fld)) ∈ V)
198197a1i 11 . . . . . . . . . . 11 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) ∈ V ∧ Fun (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) ∧ (0gβ€˜(Poly1β€˜β„‚fld)) ∈ V))
199154, 93, 54, 17coe1sfi 21600 . . . . . . . . . . . . 13 (π‘Ž ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) β†’ (coe1β€˜π‘Ž) finSupp 0)
200152, 199syl 17 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (coe1β€˜π‘Ž) finSupp 0)
201200fsuppimpd 9319 . . . . . . . . . . 11 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((coe1β€˜π‘Ž) supp 0) ∈ Fin)
202179feqmptd 6915 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (coe1β€˜π‘Ž) = (π‘˜ ∈ β„•0 ↦ ((coe1β€˜π‘Ž)β€˜π‘˜)))
203202oveq1d 7377 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((coe1β€˜π‘Ž) supp 0) = ((π‘˜ ∈ β„•0 ↦ ((coe1β€˜π‘Ž)β€˜π‘˜)) supp 0))
204 eqimss2 4006 . . . . . . . . . . . . 13 (((coe1β€˜π‘Ž) supp 0) = ((π‘˜ ∈ β„•0 ↦ ((coe1β€˜π‘Ž)β€˜π‘˜)) supp 0) β†’ ((π‘˜ ∈ β„•0 ↦ ((coe1β€˜π‘Ž)β€˜π‘˜)) supp 0) βŠ† ((coe1β€˜π‘Ž) supp 0))
205203, 204syl 17 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((π‘˜ ∈ β„•0 ↦ ((coe1β€˜π‘Ž)β€˜π‘˜)) supp 0) βŠ† ((coe1β€˜π‘Ž) supp 0))
2069, 171mp1i 13 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (Poly1β€˜β„‚fld) ∈ LMod)
20793, 190, 153, 17, 158lmod0vs 20371 . . . . . . . . . . . . 13 (((Poly1β€˜β„‚fld) ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Poly1β€˜β„‚fld))) β†’ (0( ·𝑠 β€˜(Poly1β€˜β„‚fld))π‘₯) = (0gβ€˜(Poly1β€˜β„‚fld)))
208206, 207sylan 581 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘₯ ∈ (Baseβ€˜(Poly1β€˜β„‚fld))) β†’ (0( ·𝑠 β€˜(Poly1β€˜β„‚fld))π‘₯) = (0gβ€˜(Poly1β€˜β„‚fld)))
209 c0ex 11156 . . . . . . . . . . . . 13 0 ∈ V
210209a1i 11 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ 0 ∈ V)
211205, 208, 180, 188, 210suppssov1 8134 . . . . . . . . . . 11 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) supp (0gβ€˜(Poly1β€˜β„‚fld))) βŠ† ((coe1β€˜π‘Ž) supp 0))
212 suppssfifsupp 9327 . . . . . . . . . . 11 ((((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) ∈ V ∧ Fun (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) ∧ (0gβ€˜(Poly1β€˜β„‚fld)) ∈ V) ∧ (((coe1β€˜π‘Ž) supp 0) ∈ Fin ∧ ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) supp (0gβ€˜(Poly1β€˜β„‚fld))) βŠ† ((coe1β€˜π‘Ž) supp 0))) β†’ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) finSupp (0gβ€˜(Poly1β€˜β„‚fld)))
213198, 201, 211, 212syl12anc 836 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) finSupp (0gβ€˜(Poly1β€˜β„‚fld)))
21493, 158, 162, 164, 166, 170, 193, 213gsummhm 19722 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((β„‚fld ↑s β„‚) Ξ£g (𝐸 ∘ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))))) = (πΈβ€˜((Poly1β€˜β„‚fld) Ξ£g (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))))))
21595a1i 11 . . . . . . . . . . . 12 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ 𝐸:(Baseβ€˜(Poly1β€˜β„‚fld))⟢(Baseβ€˜(β„‚fld ↑s β„‚)))
216215, 192cofmpt 7083 . . . . . . . . . . 11 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (𝐸 ∘ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))) = (π‘˜ ∈ β„•0 ↦ (πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))))
2179a1i 11 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ β„‚fld ∈ Ring)
2186a1i 11 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ β„‚ ∈ V)
21995ffvelcdmi 7039 . . . . . . . . . . . . . . . 16 ((((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) β†’ (πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) ∈ (Baseβ€˜(β„‚fld ↑s β„‚)))
220192, 219syl 17 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) ∈ (Baseβ€˜(β„‚fld ↑s β„‚)))
2213, 4, 67, 217, 218, 220pwselbas 17378 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))):β„‚βŸΆβ„‚)
222221feqmptd 6915 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) = (𝑧 ∈ β„‚ ↦ ((πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))β€˜π‘§)))
22352a1i 11 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ β„‚fld ∈ CRing)
224 simpr 486 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ 𝑧 ∈ β„‚)
22553, 114, 4, 54, 93, 223, 224evl1vard 21719 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ ((var1β€˜β„‚fld) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ ((πΈβ€˜(var1β€˜β„‚fld))β€˜π‘§) = 𝑧))
226185adantr 482 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ π‘˜ ∈ β„•0)
22753, 54, 4, 93, 223, 224, 225, 116, 127, 226evl1expd 21727 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ ((π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§) = (π‘˜(.gβ€˜(mulGrpβ€˜β„‚fld))𝑧)))
228224, 226, 131syl2anc 585 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ (π‘˜(.gβ€˜(mulGrpβ€˜β„‚fld))𝑧) = (π‘§β†‘π‘˜))
229228eqeq2d 2748 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ (((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§) = (π‘˜(.gβ€˜(mulGrpβ€˜β„‚fld))𝑧) ↔ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§) = (π‘§β†‘π‘˜)))
230229anbi2d 630 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ (((π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§) = (π‘˜(.gβ€˜(mulGrpβ€˜β„‚fld))𝑧)) ↔ ((π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§) = (π‘§β†‘π‘˜))))
231227, 230mpbid 231 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ ((π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ ((πΈβ€˜(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))β€˜π‘§) = (π‘§β†‘π‘˜)))
232181adantr 482 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ ((coe1β€˜π‘Ž)β€˜π‘˜) ∈ β„‚)
23353, 54, 4, 93, 223, 224, 231, 232, 153, 80evl1vsd 21726 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ ((((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))) ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ ((πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))β€˜π‘§) = (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
234233simprd 497 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) ∧ 𝑧 ∈ β„‚) β†’ ((πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))β€˜π‘§) = (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))
235234mpteq2dva 5210 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (𝑧 ∈ β„‚ ↦ ((πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))β€˜π‘§)) = (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
236222, 235eqtrd 2777 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ β„•0) β†’ (πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))) = (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
237236mpteq2dva 5210 . . . . . . . . . . 11 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (π‘˜ ∈ β„•0 ↦ (πΈβ€˜(((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))) = (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
238216, 237eqtrd 2777 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (𝐸 ∘ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld))))) = (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
239238oveq2d 7378 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((β„‚fld ↑s β„‚) Ξ£g (𝐸 ∘ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜)( ·𝑠 β€˜(Poly1β€˜β„‚fld))(π‘˜(.gβ€˜(mulGrpβ€˜(Poly1β€˜β„‚fld)))(var1β€˜β„‚fld)))))) = ((β„‚fld ↑s β„‚) Ξ£g (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))))
240157, 214, 2393eqtr2d 2783 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (πΈβ€˜π‘Ž) = ((β„‚fld ↑s β„‚) Ξ£g (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))))
2416a1i 11 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ β„‚ ∈ V)
2429, 10mp1i 13 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ β„‚fld ∈ CMnd)
243181adantlr 714 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ β„•0) β†’ ((coe1β€˜π‘Ž)β€˜π‘˜) ∈ β„‚)
24433adantll 713 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ β„•0) β†’ (π‘§β†‘π‘˜) ∈ β„‚)
245243, 244mulcld 11182 . . . . . . . . . 10 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ β„•0) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)) ∈ β„‚)
246245anasss 468 . . . . . . . . 9 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ (𝑧 ∈ β„‚ ∧ π‘˜ ∈ β„•0)) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)) ∈ β„‚)
247165mptex 7178 . . . . . . . . . . . 12 (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ∈ V
248 funmpt 6544 . . . . . . . . . . . 12 Fun (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
249247, 248, 393pm3.2i 1340 . . . . . . . . . . 11 ((π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ∈ V ∧ Fun (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ∧ (0gβ€˜(β„‚fld ↑s β„‚)) ∈ V)
250249a1i 11 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ∈ V ∧ Fun (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ∧ (0gβ€˜(β„‚fld ↑s β„‚)) ∈ V))
251 fzfid 13885 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ∈ Fin)
252 eldifn 4092 . . . . . . . . . . . . . . . . . 18 (π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))) β†’ Β¬ π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))
253252adantl 483 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ Β¬ π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))
254152ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ π‘Ž ∈ (Baseβ€˜(Poly1β€˜β„‚fld)))
255 eldifi 4091 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))) β†’ π‘˜ ∈ β„•0)
256255adantl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ π‘˜ ∈ β„•0)
257 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 ( deg1 β€˜β„‚fld) = ( deg1 β€˜β„‚fld)
258257, 54, 93, 17, 154deg1ge 25479 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘Ž ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ π‘˜ ∈ β„•0 ∧ ((coe1β€˜π‘Ž)β€˜π‘˜) β‰  0) β†’ π‘˜ ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž))
2592583expia 1122 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘Ž ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) ∧ π‘˜ ∈ β„•0) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) β‰  0 β†’ π‘˜ ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž)))
260254, 256, 259syl2anc 585 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) β‰  0 β†’ π‘˜ ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž)))
261 0xr 11209 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℝ*
262257, 54, 93deg1xrcl 25463 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘Ž ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) β†’ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ ℝ*)
263152, 262syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ ℝ*)
264263ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ ℝ*)
265 xrmax2 13102 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℝ* ∧ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ ℝ*) β†’ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))
266261, 264, 265sylancr 588 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))
267256nn0red 12481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ π‘˜ ∈ ℝ)
268267rexrd 11212 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ π‘˜ ∈ ℝ*)
269 ifcl 4536 . . . . . . . . . . . . . . . . . . . . . . . 24 (((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ ℝ*)
270264, 261, 269sylancl 587 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ ℝ*)
271 xrletr 13084 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘˜ ∈ ℝ* ∧ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ ℝ* ∧ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ ℝ*) β†’ ((π‘˜ ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∧ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) β†’ π‘˜ ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))
272268, 264, 270, 271syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ ((π‘˜ ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∧ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) β†’ π‘˜ ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))
273266, 272mpan2d 693 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (π‘˜ ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž) β†’ π‘˜ ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))
274260, 273syld 47 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) β‰  0 β†’ π‘˜ ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))
275274, 256jctild 527 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) β‰  0 β†’ (π‘˜ ∈ β„•0 ∧ π‘˜ ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))))
276257, 54, 93deg1cl 25464 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘Ž ∈ (Baseβ€˜(Poly1β€˜β„‚fld)) β†’ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ (β„•0 βˆͺ {-∞}))
277152, 276syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ (β„•0 βˆͺ {-∞}))
278 elun 4113 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ (β„•0 βˆͺ {-∞}) ↔ ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ β„•0 ∨ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ {-∞}))
279277, 278sylib 217 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ β„•0 ∨ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ {-∞}))
280 nn0ge0 12445 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ β„•0 β†’ 0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž))
281280iftrued 4499 . . . . . . . . . . . . . . . . . . . . . . . 24 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ β„•0 β†’ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) = (( deg1 β€˜β„‚fld)β€˜π‘Ž))
282 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ β„•0 β†’ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ β„•0)
283281, 282eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ β„•0 β†’ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ β„•0)
284 mnflt0 13053 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -∞ < 0
285 mnfxr 11219 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 -∞ ∈ ℝ*
286 xrltnle 11229 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) β†’ (-∞ < 0 ↔ Β¬ 0 ≀ -∞))
287285, 261, 286mp2an 691 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (-∞ < 0 ↔ Β¬ 0 ≀ -∞)
288284, 287mpbi 229 . . . . . . . . . . . . . . . . . . . . . . . . . 26 Β¬ 0 ≀ -∞
289 elsni 4608 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ {-∞} β†’ (( deg1 β€˜β„‚fld)β€˜π‘Ž) = -∞)
290289breq2d 5122 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ {-∞} β†’ (0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ↔ 0 ≀ -∞))
291288, 290mtbiri 327 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ {-∞} β†’ Β¬ 0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž))
292291iffalsed 4502 . . . . . . . . . . . . . . . . . . . . . . . 24 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ {-∞} β†’ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) = 0)
293 0nn0 12435 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ β„•0
294292, 293eqeltrdi 2846 . . . . . . . . . . . . . . . . . . . . . . 23 ((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ {-∞} β†’ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ β„•0)
295283, 294jaoi 856 . . . . . . . . . . . . . . . . . . . . . 22 (((( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ β„•0 ∨ (( deg1 β€˜β„‚fld)β€˜π‘Ž) ∈ {-∞}) β†’ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ β„•0)
296279, 295syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ β„•0)
297296ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ β„•0)
298 fznn0 13540 . . . . . . . . . . . . . . . . . . . 20 (if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ β„•0 β†’ (π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ↔ (π‘˜ ∈ β„•0 ∧ π‘˜ ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))))
299297, 298syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ↔ (π‘˜ ∈ β„•0 ∧ π‘˜ ≀ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))))
300275, 299sylibrd 259 . . . . . . . . . . . . . . . . . 18 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) β‰  0 β†’ π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))))
301300necon1bd 2962 . . . . . . . . . . . . . . . . 17 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (Β¬ π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) β†’ ((coe1β€˜π‘Ž)β€˜π‘˜) = 0))
302253, 301mpd 15 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ ((coe1β€˜π‘Ž)β€˜π‘˜) = 0)
303302oveq1d 7377 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)) = (0 Β· (π‘§β†‘π‘˜)))
304255, 244sylan2 594 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (π‘§β†‘π‘˜) ∈ β„‚)
305304mul02d 11360 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (0 Β· (π‘§β†‘π‘˜)) = 0)
306303, 305eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)) = 0)
307306an32s 651 . . . . . . . . . . . . 13 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) ∧ 𝑧 ∈ β„‚) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)) = 0)
308307mpteq2dva 5210 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) = (𝑧 ∈ β„‚ ↦ 0))
309 fconstmpt 5699 . . . . . . . . . . . . 13 (β„‚ Γ— {0}) = (𝑧 ∈ β„‚ ↦ 0)
310 ringmnd 19981 . . . . . . . . . . . . . . 15 (β„‚fld ∈ Ring β†’ β„‚fld ∈ Mnd)
3119, 310ax-mp 5 . . . . . . . . . . . . . 14 β„‚fld ∈ Mnd
3123, 17pws0g 18599 . . . . . . . . . . . . . 14 ((β„‚fld ∈ Mnd ∧ β„‚ ∈ V) β†’ (β„‚ Γ— {0}) = (0gβ€˜(β„‚fld ↑s β„‚)))
313311, 6, 312mp2an 691 . . . . . . . . . . . . 13 (β„‚ Γ— {0}) = (0gβ€˜(β„‚fld ↑s β„‚))
314309, 313eqtr3i 2767 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ ↦ 0) = (0gβ€˜(β„‚fld ↑s β„‚))
315308, 314eqtrdi 2793 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ π‘˜ ∈ (β„•0 βˆ– (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) = (0gβ€˜(β„‚fld ↑s β„‚)))
316315, 166suppss2 8136 . . . . . . . . . 10 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) supp (0gβ€˜(β„‚fld ↑s β„‚))) βŠ† (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))
317 suppssfifsupp 9327 . . . . . . . . . 10 ((((π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ∈ V ∧ Fun (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) ∧ (0gβ€˜(β„‚fld ↑s β„‚)) ∈ V) ∧ ((0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ∈ Fin ∧ ((π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) supp (0gβ€˜(β„‚fld ↑s β„‚))) βŠ† (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) finSupp (0gβ€˜(β„‚fld ↑s β„‚)))
318250, 251, 316, 317syl12anc 836 . . . . . . . . 9 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) finSupp (0gβ€˜(β„‚fld ↑s β„‚)))
3193, 4, 5, 241, 166, 242, 246, 318pwsgsum 19766 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((β„‚fld ↑s β„‚) Ξ£g (π‘˜ ∈ β„•0 ↦ (𝑧 ∈ β„‚ ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))) = (𝑧 ∈ β„‚ ↦ (β„‚fld Ξ£g (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))))
320 fz0ssnn0 13543 . . . . . . . . . . . 12 (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) βŠ† β„•0
321 resmpt 5996 . . . . . . . . . . . 12 ((0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) βŠ† β„•0 β†’ ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) β†Ύ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))) = (π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
322320, 321ax-mp 5 . . . . . . . . . . 11 ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) β†Ύ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))) = (π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))
323322oveq2i 7373 . . . . . . . . . 10 (β„‚fld Ξ£g ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) β†Ύ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) = (β„‚fld Ξ£g (π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
3249, 10mp1i 13 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ β„‚fld ∈ CMnd)
325165a1i 11 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ β„•0 ∈ V)
326245fmpttd 7068 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))):β„•0βŸΆβ„‚)
327306, 325suppss2 8136 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) supp 0) βŠ† (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))
328165mptex 7178 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ V
329 funmpt 6544 . . . . . . . . . . . . . 14 Fun (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))
330328, 329, 2093pm3.2i 1340 . . . . . . . . . . . . 13 ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ V ∧ Fun (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∧ 0 ∈ V)
331330a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ V ∧ Fun (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∧ 0 ∈ V))
332 fzfid 13885 . . . . . . . . . . . 12 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ∈ Fin)
333 suppssfifsupp 9327 . . . . . . . . . . . 12 ((((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ V ∧ Fun (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∧ 0 ∈ V) ∧ ((0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ∈ Fin ∧ ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) supp 0) βŠ† (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) β†’ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) finSupp 0)
334331, 332, 327, 333syl12anc 836 . . . . . . . . . . 11 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) finSupp 0)
3354, 17, 324, 325, 326, 327, 334gsumres 19697 . . . . . . . . . 10 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ (β„‚fld Ξ£g ((π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) β†Ύ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)))) = (β„‚fld Ξ£g (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))))
336 elfznn0 13541 . . . . . . . . . . . 12 (π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) β†’ π‘˜ ∈ β„•0)
337336, 245sylan2 594 . . . . . . . . . . 11 ((((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) ∧ π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))) β†’ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)) ∈ β„‚)
338332, 337gsumfsum 20880 . . . . . . . . . 10 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ (β„‚fld Ξ£g (π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0)) ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) = Ξ£π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))(((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))
339323, 335, 3383eqtr3a 2801 . . . . . . . . 9 (((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) ∧ 𝑧 ∈ β„‚) β†’ (β„‚fld Ξ£g (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))) = Ξ£π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))(((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜)))
340339mpteq2dva 5210 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (𝑧 ∈ β„‚ ↦ (β„‚fld Ξ£g (π‘˜ ∈ β„•0 ↦ (((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))) = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))(((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
341240, 319, 3403eqtrd 2781 . . . . . . 7 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (πΈβ€˜π‘Ž) = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))(((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
34212adantr 482 . . . . . . . 8 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ 𝑆 βŠ† β„‚)
343 elplyr 25578 . . . . . . . 8 ((𝑆 βŠ† β„‚ ∧ if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0) ∈ β„•0 ∧ (coe1β€˜π‘Ž):β„•0βŸΆπ‘†) β†’ (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))(((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ (Polyβ€˜π‘†))
344342, 296, 179, 343syl3anc 1372 . . . . . . 7 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...if(0 ≀ (( deg1 β€˜β„‚fld)β€˜π‘Ž), (( deg1 β€˜β„‚fld)β€˜π‘Ž), 0))(((coe1β€˜π‘Ž)β€˜π‘˜) Β· (π‘§β†‘π‘˜))) ∈ (Polyβ€˜π‘†))
345341, 344eqeltrd 2838 . . . . . 6 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ (πΈβ€˜π‘Ž) ∈ (Polyβ€˜π‘†))
346 eleq1 2826 . . . . . 6 ((πΈβ€˜π‘Ž) = 𝑓 β†’ ((πΈβ€˜π‘Ž) ∈ (Polyβ€˜π‘†) ↔ 𝑓 ∈ (Polyβ€˜π‘†)))
347345, 346syl5ibcom 244 . . . . 5 ((𝑆 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐴) β†’ ((πΈβ€˜π‘Ž) = 𝑓 β†’ 𝑓 ∈ (Polyβ€˜π‘†)))
348347rexlimdva 3153 . . . 4 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ (βˆƒπ‘Ž ∈ 𝐴 (πΈβ€˜π‘Ž) = 𝑓 β†’ 𝑓 ∈ (Polyβ€˜π‘†)))
349151, 348syl5 34 . . 3 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ (𝑓 ∈ (𝐸 β€œ 𝐴) β†’ 𝑓 ∈ (Polyβ€˜π‘†)))
350147, 349impbid 211 . 2 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ (𝑓 ∈ (Polyβ€˜π‘†) ↔ 𝑓 ∈ (𝐸 β€œ 𝐴)))
351350eqrdv 2735 1 (𝑆 ∈ (SubRingβ€˜β„‚fld) β†’ (Polyβ€˜π‘†) = (𝐸 β€œ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074  Vcvv 3448   βˆ– cdif 3912   βˆͺ cun 3913   βŠ† wss 3915  ifcif 4491  {csn 4591   class class class wbr 5110   ↦ cmpt 5193   Γ— cxp 5636   β†Ύ cres 5640   β€œ cima 5641   ∘ ccom 5642  Fun wfun 6495   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ∘f cof 7620   supp csupp 8097   ↑m cmap 8772  Fincfn 8890   finSupp cfsupp 9312  β„‚cc 11056  0cc0 11058   Β· cmul 11063  -∞cmnf 11194  β„*cxr 11195   < clt 11196   ≀ cle 11197  β„•0cn0 12420  ...cfz 13431  β†‘cexp 13974  Ξ£csu 15577  Basecbs 17090   β†Ύs cress 17119  .rcmulr 17141  Scalarcsca 17143   ·𝑠 cvsca 17144  0gc0g 17328   Ξ£g cgsu 17329   ↑s cpws 17335  Mndcmnd 18563   MndHom cmhm 18606  SubMndcsubmnd 18607  .gcmg 18879  SubGrpcsubg 18929   GrpHom cghm 19012  CMndccmn 19569  mulGrpcmgp 19903  Ringcrg 19971  CRingccrg 19972   RingHom crh 20152  SubRingcsubrg 20234  LModclmod 20338  β„‚fldccnfld 20812  algSccascl 21274  var1cv1 21563  Poly1cpl1 21564  coe1cco1 21565  eval1ce1 21696   deg1 cdg1 25432  Polycply 25561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136  ax-addf 11137  ax-mulf 11138
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-ofr 7623  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-pm 8775  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-sup 9385  df-oi 9453  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-rp 12923  df-fz 13432  df-fzo 13575  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-sum 15578  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-starv 17155  df-sca 17156  df-vsca 17157  df-ip 17158  df-tset 17159  df-ple 17160  df-ds 17162  df-unif 17163  df-hom 17164  df-cco 17165  df-0g 17330  df-gsum 17331  df-prds 17336  df-pws 17338  df-mre 17473  df-mrc 17474  df-acs 17476  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mhm 18608  df-submnd 18609  df-grp 18758  df-minusg 18759  df-sbg 18760  df-mulg 18880  df-subg 18932  df-ghm 19013  df-cntz 19104  df-cmn 19571  df-abl 19572  df-mgp 19904  df-ur 19921  df-srg 19925  df-ring 19973  df-cring 19974  df-rnghom 20155  df-subrg 20236  df-lmod 20340  df-lss 20409  df-lsp 20449  df-cnfld 20813  df-assa 21275  df-asp 21276  df-ascl 21277  df-psr 21327  df-mvr 21328  df-mpl 21329  df-opsr 21331  df-evls 21498  df-evl 21499  df-psr1 21567  df-vr1 21568  df-ply1 21569  df-coe1 21570  df-evl1 21698  df-mdeg 25433  df-deg1 25434  df-ply 25565
This theorem is referenced by: (None)
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