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Theorem plyexmo 25206
Description: An infinite set of values can be extended to a polynomial in at most one way. (Contributed by Stefan O'Rear, 14-Nov-2014.)
Assertion
Ref Expression
plyexmo ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹))
Distinct variable groups:   𝑆,𝑝   𝐹,𝑝   𝐷,𝑝

Proof of Theorem plyexmo
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 769 . . . . . . . . 9 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ¬ 𝐷 ∈ Fin)
2 simpll 767 . . . . . . . . . . . . . 14 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝐷 ⊆ ℂ)
32sseld 3900 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷𝑏 ∈ ℂ))
4 simprll 779 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 ∈ (Poly‘ℂ))
5 plyf 25092 . . . . . . . . . . . . . . . . . . 19 (𝑝 ∈ (Poly‘ℂ) → 𝑝:ℂ⟶ℂ)
64, 5syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝:ℂ⟶ℂ)
76ffnd 6546 . . . . . . . . . . . . . . . . 17 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 Fn ℂ)
87adantr 484 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑝 Fn ℂ)
9 simprrl 781 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎 ∈ (Poly‘ℂ))
10 plyf 25092 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ (Poly‘ℂ) → 𝑎:ℂ⟶ℂ)
119, 10syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎:ℂ⟶ℂ)
1211ffnd 6546 . . . . . . . . . . . . . . . . 17 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎 Fn ℂ)
1312adantr 484 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑎 Fn ℂ)
14 cnex 10810 . . . . . . . . . . . . . . . . 17 ℂ ∈ V
1514a1i 11 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ℂ ∈ V)
162sselda 3901 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑏 ∈ ℂ)
17 fnfvof 7485 . . . . . . . . . . . . . . . 16 (((𝑝 Fn ℂ ∧ 𝑎 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑏 ∈ ℂ)) → ((𝑝f𝑎)‘𝑏) = ((𝑝𝑏) − (𝑎𝑏)))
188, 13, 15, 16, 17syl22anc 839 . . . . . . . . . . . . . . 15 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝f𝑎)‘𝑏) = ((𝑝𝑏) − (𝑎𝑏)))
196adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑝:ℂ⟶ℂ)
2019, 16ffvelrnd 6905 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝑏) ∈ ℂ)
21 simprlr 780 . . . . . . . . . . . . . . . . . . . 20 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝐷) = 𝐹)
22 simprrr 782 . . . . . . . . . . . . . . . . . . . 20 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑎𝐷) = 𝐹)
2321, 22eqtr4d 2780 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝐷) = (𝑎𝐷))
2423adantr 484 . . . . . . . . . . . . . . . . . 18 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝐷) = (𝑎𝐷))
2524fveq1d 6719 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = ((𝑎𝐷)‘𝑏))
26 fvres 6736 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
2726adantl 485 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
28 fvres 6736 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑎𝐷)‘𝑏) = (𝑎𝑏))
2928adantl 485 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑎𝐷)‘𝑏) = (𝑎𝑏))
3025, 27, 293eqtr3d 2785 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝑏) = (𝑎𝑏))
3120, 30subeq0bd 11258 . . . . . . . . . . . . . . 15 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝑏) − (𝑎𝑏)) = 0)
3218, 31eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝f𝑎)‘𝑏) = 0)
3332ex 416 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → ((𝑝f𝑎)‘𝑏) = 0))
343, 33jcad 516 . . . . . . . . . . . 12 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → (𝑏 ∈ ℂ ∧ ((𝑝f𝑎)‘𝑏) = 0)))
35 plysubcl 25116 . . . . . . . . . . . . . 14 ((𝑝 ∈ (Poly‘ℂ) ∧ 𝑎 ∈ (Poly‘ℂ)) → (𝑝f𝑎) ∈ (Poly‘ℂ))
364, 9, 35syl2anc 587 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝f𝑎) ∈ (Poly‘ℂ))
37 plyf 25092 . . . . . . . . . . . . 13 ((𝑝f𝑎) ∈ (Poly‘ℂ) → (𝑝f𝑎):ℂ⟶ℂ)
38 ffn 6545 . . . . . . . . . . . . 13 ((𝑝f𝑎):ℂ⟶ℂ → (𝑝f𝑎) Fn ℂ)
39 fniniseg 6880 . . . . . . . . . . . . 13 ((𝑝f𝑎) Fn ℂ → (𝑏 ∈ ((𝑝f𝑎) “ {0}) ↔ (𝑏 ∈ ℂ ∧ ((𝑝f𝑎)‘𝑏) = 0)))
4036, 37, 38, 394syl 19 . . . . . . . . . . . 12 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏 ∈ ((𝑝f𝑎) “ {0}) ↔ (𝑏 ∈ ℂ ∧ ((𝑝f𝑎)‘𝑏) = 0)))
4134, 40sylibrd 262 . . . . . . . . . . 11 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷𝑏 ∈ ((𝑝f𝑎) “ {0})))
4241ssrdv 3907 . . . . . . . . . 10 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝐷 ⊆ ((𝑝f𝑎) “ {0}))
43 ssfi 8851 . . . . . . . . . . 11 ((((𝑝f𝑎) “ {0}) ∈ Fin ∧ 𝐷 ⊆ ((𝑝f𝑎) “ {0})) → 𝐷 ∈ Fin)
4443expcom 417 . . . . . . . . . 10 (𝐷 ⊆ ((𝑝f𝑎) “ {0}) → (((𝑝f𝑎) “ {0}) ∈ Fin → 𝐷 ∈ Fin))
4542, 44syl 17 . . . . . . . . 9 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (((𝑝f𝑎) “ {0}) ∈ Fin → 𝐷 ∈ Fin))
461, 45mtod 201 . . . . . . . 8 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ¬ ((𝑝f𝑎) “ {0}) ∈ Fin)
47 neqne 2948 . . . . . . . . . . 11 (¬ (𝑝f𝑎) = 0𝑝 → (𝑝f𝑎) ≠ 0𝑝)
48 eqid 2737 . . . . . . . . . . . 12 ((𝑝f𝑎) “ {0}) = ((𝑝f𝑎) “ {0})
4948fta1 25201 . . . . . . . . . . 11 (((𝑝f𝑎) ∈ (Poly‘ℂ) ∧ (𝑝f𝑎) ≠ 0𝑝) → (((𝑝f𝑎) “ {0}) ∈ Fin ∧ (♯‘((𝑝f𝑎) “ {0})) ≤ (deg‘(𝑝f𝑎))))
5036, 47, 49syl2an 599 . . . . . . . . . 10 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ ¬ (𝑝f𝑎) = 0𝑝) → (((𝑝f𝑎) “ {0}) ∈ Fin ∧ (♯‘((𝑝f𝑎) “ {0})) ≤ (deg‘(𝑝f𝑎))))
5150simpld 498 . . . . . . . . 9 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ ¬ (𝑝f𝑎) = 0𝑝) → ((𝑝f𝑎) “ {0}) ∈ Fin)
5251ex 416 . . . . . . . 8 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (¬ (𝑝f𝑎) = 0𝑝 → ((𝑝f𝑎) “ {0}) ∈ Fin))
5346, 52mt3d 150 . . . . . . 7 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝f𝑎) = 0𝑝)
54 df-0p 24567 . . . . . . 7 0𝑝 = (ℂ × {0})
5553, 54eqtrdi 2794 . . . . . 6 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝f𝑎) = (ℂ × {0}))
56 ofsubeq0 11827 . . . . . . 7 ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
5714, 6, 11, 56mp3an2i 1468 . . . . . 6 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
5855, 57mpbid 235 . . . . 5 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 = 𝑎)
5958ex 416 . . . 4 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → (((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
6059alrimivv 1936 . . 3 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
61 eleq1w 2820 . . . . 5 (𝑝 = 𝑎 → (𝑝 ∈ (Poly‘ℂ) ↔ 𝑎 ∈ (Poly‘ℂ)))
62 reseq1 5845 . . . . . 6 (𝑝 = 𝑎 → (𝑝𝐷) = (𝑎𝐷))
6362eqeq1d 2739 . . . . 5 (𝑝 = 𝑎 → ((𝑝𝐷) = 𝐹 ↔ (𝑎𝐷) = 𝐹))
6461, 63anbi12d 634 . . . 4 (𝑝 = 𝑎 → ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)))
6564mo4 2565 . . 3 (∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
6660, 65sylibr 237 . 2 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
67 plyssc 25094 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
6867sseli 3896 . . . 4 (𝑝 ∈ (Poly‘𝑆) → 𝑝 ∈ (Poly‘ℂ))
6968anim1i 618 . . 3 ((𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹) → (𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
7069moimi 2544 . 2 (∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) → ∃*𝑝(𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹))
7166, 70syl 17 1 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wcel 2110  ∃*wmo 2537  wne 2940  Vcvv 3408  wss 3866  {csn 4541   class class class wbr 5053   × cxp 5549  ccnv 5550  cres 5553  cima 5554   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  f cof 7467  Fincfn 8626  cc 10727  0cc0 10729  cle 10868  cmin 11062  chash 13896  0𝑝c0p 24566  Polycply 25078  degcdgr 25081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-map 8510  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-sup 9058  df-inf 9059  df-oi 9126  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-fl 13367  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-rlim 15050  df-sum 15250  df-0p 24567  df-ply 25082  df-idp 25083  df-coe 25084  df-dgr 25085  df-quot 25184
This theorem is referenced by: (None)
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