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Theorem plyexmo 26355
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 3982 . . . . . . . . . . . . 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 26237 . . . . . . . . . . . . . . . . . . 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 6737 . . . . . . . . . . . . . . . . 17 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 Fn ℂ)
87adantr 480 . . . . . . . . . . . . . . . 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 26237 . . . . . . . . . . . . . . . . . . 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 6737 . . . . . . . . . . . . . . . . 17 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎 Fn ℂ)
1312adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑎 Fn ℂ)
14 cnex 11236 . . . . . . . . . . . . . . . . 17 ℂ ∈ V
1514a1i 11 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ℂ ∈ V)
162sselda 3983 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑏 ∈ ℂ)
17 fnfvof 7714 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑝:ℂ⟶ℂ)
2019, 16ffvelcdmd 7105 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 18 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝐷) = (𝑎𝐷))
2524fveq1d 6908 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = ((𝑎𝐷)‘𝑏))
26 fvres 6925 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
2726adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
28 fvres 6925 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑎𝐷)‘𝑏) = (𝑎𝑏))
2928adantl 481 . . . . . . . . . . . . . . . . 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 11689 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → ((𝑝f𝑎)‘𝑏) = 0))
343, 33jcad 512 . . . . . . . . . . . 12 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → (𝑏 ∈ ℂ ∧ ((𝑝f𝑎)‘𝑏) = 0)))
35 plysubcl 26261 . . . . . . . . . . . . . 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 26237 . . . . . . . . . . . . 13 ((𝑝f𝑎) ∈ (Poly‘ℂ) → (𝑝f𝑎):ℂ⟶ℂ)
38 ffn 6736 . . . . . . . . . . . . 13 ((𝑝f𝑎):ℂ⟶ℂ → (𝑝f𝑎) Fn ℂ)
39 fniniseg 7080 . . . . . . . . . . . . 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 259 . . . . . . . . . . 11 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷𝑏 ∈ ((𝑝f𝑎) “ {0})))
4241ssrdv 3989 . . . . . . . . . 10 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝐷 ⊆ ((𝑝f𝑎) “ {0}))
43 ssfi 9213 . . . . . . . . . . 11 ((((𝑝f𝑎) “ {0}) ∈ Fin ∧ 𝐷 ⊆ ((𝑝f𝑎) “ {0})) → 𝐷 ∈ Fin)
4443expcom 413 . . . . . . . . . 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 198 . . . . . . . 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 26350 . . . . . . . . . . 11 (((𝑝f𝑎) ∈ (Poly‘ℂ) ∧ (𝑝f𝑎) ≠ 0𝑝) → (((𝑝f𝑎) “ {0}) ∈ Fin ∧ (♯‘((𝑝f𝑎) “ {0})) ≤ (deg‘(𝑝f𝑎))))
5036, 47, 49syl2an 596 . . . . . . . . . 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 494 . . . . . . . . 9 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ ¬ (𝑝f𝑎) = 0𝑝) → ((𝑝f𝑎) “ {0}) ∈ Fin)
5251ex 412 . . . . . . . 8 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (¬ (𝑝f𝑎) = 0𝑝 → ((𝑝f𝑎) “ {0}) ∈ Fin))
5346, 52mt3d 148 . . . . . . 7 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝f𝑎) = 0𝑝)
54 df-0p 25705 . . . . . . 7 0𝑝 = (ℂ × {0})
5553, 54eqtrdi 2793 . . . . . 6 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝f𝑎) = (ℂ × {0}))
56 ofsubeq0 12263 . . . . . . 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 232 . . . . 5 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 = 𝑎)
5958ex 412 . . . 4 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → (((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
6059alrimivv 1928 . . 3 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
61 eleq1w 2824 . . . . 5 (𝑝 = 𝑎 → (𝑝 ∈ (Poly‘ℂ) ↔ 𝑎 ∈ (Poly‘ℂ)))
62 reseq1 5991 . . . . . 6 (𝑝 = 𝑎 → (𝑝𝐷) = (𝑎𝐷))
6362eqeq1d 2739 . . . . 5 (𝑝 = 𝑎 → ((𝑝𝐷) = 𝐹 ↔ (𝑎𝐷) = 𝐹))
6461, 63anbi12d 632 . . . 4 (𝑝 = 𝑎 → ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)))
6564mo4 2566 . . 3 (∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
6660, 65sylibr 234 . 2 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
67 plyssc 26239 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
6867sseli 3979 . . . 4 (𝑝 ∈ (Poly‘𝑆) → 𝑝 ∈ (Poly‘ℂ))
6968anim1i 615 . . 3 ((𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹) → (𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
7069moimi 2545 . 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 206  wa 395  wal 1538   = wceq 1540  wcel 2108  ∃*wmo 2538  wne 2940  Vcvv 3480  wss 3951  {csn 4626   class class class wbr 5143   × cxp 5683  ccnv 5684  cres 5687  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  Fincfn 8985  cc 11153  0cc0 11155  cle 11296  cmin 11492  chash 14369  0𝑝c0p 25704  Polycply 26223  degcdgr 26226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-0p 25705  df-ply 26227  df-idp 26228  df-coe 26229  df-dgr 26230  df-quot 26333
This theorem is referenced by: (None)
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