Theorem plyexmo 24902

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) ∧ (𝑝 ↾ 𝐷) = 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.

Step	Hyp	Ref	Expression
1		simplr 767	. . . . . . . . 9 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → ¬ 𝐷 ∈ Fin)
2		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝐷 ⊆ ℂ)
3	2	sseld 3966	. . . . . . . . . . . . 13 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑏 ∈ 𝐷 → 𝑏 ∈ ℂ))
4		simprll 777	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 ∈ (Poly‘ℂ))
5		plyf 24788	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ (Poly‘ℂ) → 𝑝:ℂ⟶ℂ)
6	4, 5	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝:ℂ⟶ℂ)
7	6	ffnd 6515	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 Fn ℂ)
8	7	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑝 Fn ℂ)
9		simprrl 779	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑎 ∈ (Poly‘ℂ))
10		plyf 24788	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (Poly‘ℂ) → 𝑎:ℂ⟶ℂ)
11	9, 10	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑎:ℂ⟶ℂ)
12	11	ffnd 6515	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑎 Fn ℂ)
13	12	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑎 Fn ℂ)
14		cnex 10618	. . . . . . . . . . . . . . . . 17 ⊢ ℂ ∈ V
15	14	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → ℂ ∈ V)
16	2	sselda 3967	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ ℂ)
17		fnfvof 7423	. . . . . . . . . . . . . . . 16 ⊢ (((𝑝 Fn ℂ ∧ 𝑎 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑏 ∈ ℂ)) → ((𝑝 ∘f − 𝑎)‘𝑏) = ((𝑝‘𝑏) − (𝑎‘𝑏)))
18	8, 13, 15, 16, 17	syl22anc 836	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → ((𝑝 ∘f − 𝑎)‘𝑏) = ((𝑝‘𝑏) − (𝑎‘𝑏)))
19	6	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑝:ℂ⟶ℂ)
20	19, 16	ffvelrnd 6852	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → (𝑝‘𝑏) ∈ ℂ)
21		simprlr 778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ↾ 𝐷) = 𝐹)
22		simprrr 780	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑎 ↾ 𝐷) = 𝐹)
23	21, 22	eqtr4d 2859	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ↾ 𝐷) = (𝑎 ↾ 𝐷))
24	23	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → (𝑝 ↾ 𝐷) = (𝑎 ↾ 𝐷))
25	24	fveq1d 6672	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → ((𝑝 ↾ 𝐷)‘𝑏) = ((𝑎 ↾ 𝐷)‘𝑏))
26		fvres 6689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝐷 → ((𝑝 ↾ 𝐷)‘𝑏) = (𝑝‘𝑏))
27	26	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → ((𝑝 ↾ 𝐷)‘𝑏) = (𝑝‘𝑏))
28		fvres 6689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝐷 → ((𝑎 ↾ 𝐷)‘𝑏) = (𝑎‘𝑏))
29	28	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → ((𝑎 ↾ 𝐷)‘𝑏) = (𝑎‘𝑏))
30	25, 27, 29	3eqtr3d 2864	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → (𝑝‘𝑏) = (𝑎‘𝑏))
31	20, 30	subeq0bd 11066	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → ((𝑝‘𝑏) − (𝑎‘𝑏)) = 0)
32	18, 31	eqtrd 2856	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → ((𝑝 ∘f − 𝑎)‘𝑏) = 0)
33	32	ex 415	. . . . . . . . . . . . 13 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑏 ∈ 𝐷 → ((𝑝 ∘f − 𝑎)‘𝑏) = 0))
34	3, 33	jcad 515	. . . . . . . . . . . 12 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑏 ∈ 𝐷 → (𝑏 ∈ ℂ ∧ ((𝑝 ∘f − 𝑎)‘𝑏) = 0)))
35		plysubcl 24812	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ (Poly‘ℂ) ∧ 𝑎 ∈ (Poly‘ℂ)) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ))
36	4, 9, 35	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ))
37		plyf 24788	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ) → (𝑝 ∘f − 𝑎):ℂ⟶ℂ)
38		ffn 6514	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎):ℂ⟶ℂ → (𝑝 ∘f − 𝑎) Fn ℂ)
39		fniniseg 6830	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎) Fn ℂ → (𝑏 ∈ (◡(𝑝 ∘f − 𝑎) “ {0}) ↔ (𝑏 ∈ ℂ ∧ ((𝑝 ∘f − 𝑎)‘𝑏) = 0)))
40	36, 37, 38, 39	4syl 19	. . . . . . . . . . . 12 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑏 ∈ (◡(𝑝 ∘f − 𝑎) “ {0}) ↔ (𝑏 ∈ ℂ ∧ ((𝑝 ∘f − 𝑎)‘𝑏) = 0)))
41	34, 40	sylibrd 261	. . . . . . . . . . 11 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑏 ∈ 𝐷 → 𝑏 ∈ (◡(𝑝 ∘f − 𝑎) “ {0})))
42	41	ssrdv 3973	. . . . . . . . . 10 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0}))
43		ssfi 8738	. . . . . . . . . . 11 ⊢ (((◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin ∧ 𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0})) → 𝐷 ∈ Fin)
44	43	expcom 416	. . . . . . . . . 10 ⊢ (𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0}) → ((◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin → 𝐷 ∈ Fin))
45	42, 44	syl 17	. . . . . . . . 9 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → ((◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin → 𝐷 ∈ Fin))
46	1, 45	mtod 200	. . . . . . . 8 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → ¬ (◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin)
47		neqne 3024	. . . . . . . . . . 11 ⊢ (¬ (𝑝 ∘f − 𝑎) = 0𝑝 → (𝑝 ∘f − 𝑎) ≠ 0𝑝)
48		eqid 2821	. . . . . . . . . . . 12 ⊢ (◡(𝑝 ∘f − 𝑎) “ {0}) = (◡(𝑝 ∘f − 𝑎) “ {0})
49	48	fta1 24897	. . . . . . . . . . 11 ⊢ (((𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ) ∧ (𝑝 ∘f − 𝑎) ≠ 0𝑝) → ((◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin ∧ (♯‘(◡(𝑝 ∘f − 𝑎) “ {0})) ≤ (deg‘(𝑝 ∘f − 𝑎))))
50	36, 47, 49	syl2an 597	. . . . . . . . . 10 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ ¬ (𝑝 ∘f − 𝑎) = 0𝑝) → ((◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin ∧ (♯‘(◡(𝑝 ∘f − 𝑎) “ {0})) ≤ (deg‘(𝑝 ∘f − 𝑎))))
51	50	simpld 497	. . . . . . . . 9 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ ¬ (𝑝 ∘f − 𝑎) = 0𝑝) → (◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin)
52	51	ex 415	. . . . . . . 8 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (¬ (𝑝 ∘f − 𝑎) = 0𝑝 → (◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin))
53	46, 52	mt3d 150	. . . . . . 7 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ∘f − 𝑎) = 0𝑝)
54		df-0p 24271	. . . . . . 7 ⊢ 0𝑝 = (ℂ × {0})
55	53, 54	syl6eq 2872	. . . . . 6 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ∘f − 𝑎) = (ℂ × {0}))
56		ofsubeq0 11635	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
57	14, 6, 11, 56	mp3an2i 1462	. . . . . 6 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
58	55, 57	mpbid 234	. . . . 5 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 = 𝑎)
59	58	ex 415	. . . 4 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → (((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)) → 𝑝 = 𝑎))
60	59	alrimivv 1929	. . 3 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∀𝑝∀𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)) → 𝑝 = 𝑎))
61		eleq1w 2895	. . . . 5 ⊢ (𝑝 = 𝑎 → (𝑝 ∈ (Poly‘ℂ) ↔ 𝑎 ∈ (Poly‘ℂ)))
62		reseq1 5847	. . . . . 6 ⊢ (𝑝 = 𝑎 → (𝑝 ↾ 𝐷) = (𝑎 ↾ 𝐷))
63	62	eqeq1d 2823	. . . . 5 ⊢ (𝑝 = 𝑎 → ((𝑝 ↾ 𝐷) = 𝐹 ↔ (𝑎 ↾ 𝐷) = 𝐹))
64	61, 63	anbi12d 632	. . . 4 ⊢ (𝑝 = 𝑎 → ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ↔ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)))
65	64	mo4 2650	. . 3 ⊢ (∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ↔ ∀𝑝∀𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)) → 𝑝 = 𝑎))
66	60, 65	sylibr 236	. 2 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹))
67		plyssc 24790	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
68	67	sseli 3963	. . . 4 ⊢ (𝑝 ∈ (Poly‘𝑆) → 𝑝 ∈ (Poly‘ℂ))
69	68	anim1i 616	. . 3 ⊢ ((𝑝 ∈ (Poly‘𝑆) ∧ (𝑝 ↾ 𝐷) = 𝐹) → (𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹))
70	69	moimi 2627	. 2 ⊢ (∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) → ∃*𝑝(𝑝 ∈ (Poly‘𝑆) ∧ (𝑝 ↾ 𝐷) = 𝐹))
71	66, 70	syl 17	1 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘𝑆) ∧ (𝑝 ↾ 𝐷) = 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537   ∈ wcel 2114  ∃*wmo 2620   ≠ wne 3016  Vcvv 3494   ⊆ wss 3936  {csn 4567   class class class wbr 5066   × cxp 5553  ^◡ccnv 5554   ↾ cres 5557   “ cima 5558   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ∘_f cof 7407  Fincfn 8509  ℂcc 10535  0cc0 10537   ≤ cle 10676   − cmin 10870  ♯chash 13691  0_𝑝c0p 24270  Polycply 24774  degcdgr 24777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615  ax-addf 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-oi 8974  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-n0 11899  df-xnn0 11969  df-z 11983  df-uz 12245  df-rp 12391  df-fz 12894  df-fzo 13035  df-fl 13163  df-seq 13371  df-exp 13431  df-hash 13692  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-rlim 14846  df-sum 15043  df-0p 24271  df-ply 24778  df-idp 24779  df-coe 24780  df-dgr 24781  df-quot 24880

This theorem is referenced by: (None)