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Theorem plyexmo 24817
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 765 . . . . . . . . 9 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ¬ 𝐷 ∈ Fin)
2 simpll 763 . . . . . . . . . . . . . 14 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝐷 ⊆ ℂ)
32sseld 3970 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷𝑏 ∈ ℂ))
4 simprll 775 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 ∈ (Poly‘ℂ))
5 plyf 24703 . . . . . . . . . . . . . . . . . . 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 6512 . . . . . . . . . . . . . . . . 17 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 Fn ℂ)
87adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑝 Fn ℂ)
9 simprrl 777 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎 ∈ (Poly‘ℂ))
10 plyf 24703 . . . . . . . . . . . . . . . . . . 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 6512 . . . . . . . . . . . . . . . . 17 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎 Fn ℂ)
1312adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑎 Fn ℂ)
14 cnex 10607 . . . . . . . . . . . . . . . . 17 ℂ ∈ V
1514a1i 11 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ℂ ∈ V)
162sselda 3971 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑏 ∈ ℂ)
17 fnfvof 7413 . . . . . . . . . . . . . . . 16 (((𝑝 Fn ℂ ∧ 𝑎 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑏 ∈ ℂ)) → ((𝑝f𝑎)‘𝑏) = ((𝑝𝑏) − (𝑎𝑏)))
188, 13, 15, 16, 17syl22anc 836 . . . . . . . . . . . . . . 15 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝f𝑎)‘𝑏) = ((𝑝𝑏) − (𝑎𝑏)))
196adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑝:ℂ⟶ℂ)
2019, 16ffvelrnd 6848 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝑏) ∈ ℂ)
21 simprlr 776 . . . . . . . . . . . . . . . . . . . 20 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝐷) = 𝐹)
22 simprrr 778 . . . . . . . . . . . . . . . . . . . 20 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑎𝐷) = 𝐹)
2321, 22eqtr4d 2864 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝐷) = (𝑎𝐷))
2423adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝐷) = (𝑎𝐷))
2524fveq1d 6669 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = ((𝑎𝐷)‘𝑏))
26 fvres 6686 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
2726adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
28 fvres 6686 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑎𝐷)‘𝑏) = (𝑎𝑏))
2928adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑎𝐷)‘𝑏) = (𝑎𝑏))
3025, 27, 293eqtr3d 2869 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝑏) = (𝑎𝑏))
3120, 30subeq0bd 11055 . . . . . . . . . . . . . . 15 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝑏) − (𝑎𝑏)) = 0)
3218, 31eqtrd 2861 . . . . . . . . . . . . . 14 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝f𝑎)‘𝑏) = 0)
3332ex 413 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → ((𝑝f𝑎)‘𝑏) = 0))
343, 33jcad 513 . . . . . . . . . . . 12 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → (𝑏 ∈ ℂ ∧ ((𝑝f𝑎)‘𝑏) = 0)))
35 plysubcl 24727 . . . . . . . . . . . . . 14 ((𝑝 ∈ (Poly‘ℂ) ∧ 𝑎 ∈ (Poly‘ℂ)) → (𝑝f𝑎) ∈ (Poly‘ℂ))
364, 9, 35syl2anc 584 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝f𝑎) ∈ (Poly‘ℂ))
37 plyf 24703 . . . . . . . . . . . . 13 ((𝑝f𝑎) ∈ (Poly‘ℂ) → (𝑝f𝑎):ℂ⟶ℂ)
38 ffn 6511 . . . . . . . . . . . . 13 ((𝑝f𝑎):ℂ⟶ℂ → (𝑝f𝑎) Fn ℂ)
39 fniniseg 6826 . . . . . . . . . . . . 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 260 . . . . . . . . . . 11 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷𝑏 ∈ ((𝑝f𝑎) “ {0})))
4241ssrdv 3977 . . . . . . . . . 10 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝐷 ⊆ ((𝑝f𝑎) “ {0}))
43 ssfi 8727 . . . . . . . . . . 11 ((((𝑝f𝑎) “ {0}) ∈ Fin ∧ 𝐷 ⊆ ((𝑝f𝑎) “ {0})) → 𝐷 ∈ Fin)
4443expcom 414 . . . . . . . . . 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 199 . . . . . . . 8 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ¬ ((𝑝f𝑎) “ {0}) ∈ Fin)
47 neqne 3029 . . . . . . . . . . 11 (¬ (𝑝f𝑎) = 0𝑝 → (𝑝f𝑎) ≠ 0𝑝)
48 eqid 2826 . . . . . . . . . . . 12 ((𝑝f𝑎) “ {0}) = ((𝑝f𝑎) “ {0})
4948fta1 24812 . . . . . . . . . . 11 (((𝑝f𝑎) ∈ (Poly‘ℂ) ∧ (𝑝f𝑎) ≠ 0𝑝) → (((𝑝f𝑎) “ {0}) ∈ Fin ∧ (♯‘((𝑝f𝑎) “ {0})) ≤ (deg‘(𝑝f𝑎))))
5036, 47, 49syl2an 595 . . . . . . . . . 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 495 . . . . . . . . 9 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ ¬ (𝑝f𝑎) = 0𝑝) → ((𝑝f𝑎) “ {0}) ∈ Fin)
5251ex 413 . . . . . . . 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 24186 . . . . . . 7 0𝑝 = (ℂ × {0})
5553, 54syl6eq 2877 . . . . . 6 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝f𝑎) = (ℂ × {0}))
56 ofsubeq0 11624 . . . . . . 7 ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
5714, 6, 11, 56mp3an2i 1459 . . . . . 6 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ((𝑝f𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
5855, 57mpbid 233 . . . . 5 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 = 𝑎)
5958ex 413 . . . 4 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → (((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
6059alrimivv 1922 . . 3 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
61 eleq1w 2900 . . . . 5 (𝑝 = 𝑎 → (𝑝 ∈ (Poly‘ℂ) ↔ 𝑎 ∈ (Poly‘ℂ)))
62 reseq1 5846 . . . . . 6 (𝑝 = 𝑎 → (𝑝𝐷) = (𝑎𝐷))
6362eqeq1d 2828 . . . . 5 (𝑝 = 𝑎 → ((𝑝𝐷) = 𝐹 ↔ (𝑎𝐷) = 𝐹))
6461, 63anbi12d 630 . . . 4 (𝑝 = 𝑎 → ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)))
6564mo4 2648 . . 3 (∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
6660, 65sylibr 235 . 2 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
67 plyssc 24705 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
6867sseli 3967 . . . 4 (𝑝 ∈ (Poly‘𝑆) → 𝑝 ∈ (Poly‘ℂ))
6968anim1i 614 . . 3 ((𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹) → (𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
7069moimi 2625 . 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 207  wa 396  wal 1528   = wceq 1530  wcel 2107  ∃*wmo 2618  wne 3021  Vcvv 3500  wss 3940  {csn 4564   class class class wbr 5063   × cxp 5552  ccnv 5553  cres 5556  cima 5557   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  f cof 7397  Fincfn 8498  cc 10524  0cc0 10526  cle 10665  cmin 10859  chash 13680  0𝑝c0p 24185  Polycply 24689  degcdgr 24692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7399  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-oi 8963  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-fl 13152  df-seq 13360  df-exp 13420  df-hash 13681  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-clim 14835  df-rlim 14836  df-sum 15033  df-0p 24186  df-ply 24693  df-idp 24694  df-coe 24695  df-dgr 24696  df-quot 24795
This theorem is referenced by: (None)
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