Theorem plyexmo 25673

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 3943	. . . . . . . . . . . . 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 25559	. . . . . . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 Fn ℂ)
8	7	adantr 481	. . . . . . . . . . . . . . . 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 25559	. . . . . . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑎 Fn ℂ)
13	12	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑎 Fn ℂ)
14		cnex 11132	. . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ ℂ)
17		fnfvof 7634	. . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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	ffvelcdmd 7036	. . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ↾ 𝐷) = (𝑎 ↾ 𝐷))
24	23	adantr 481	. . . . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝐷 → ((𝑝 ↾ 𝐷)‘𝑏) = (𝑝‘𝑏))
27	26	adantl 482	. . . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝐷 → ((𝑎 ↾ 𝐷)‘𝑏) = (𝑎‘𝑏))
29	28	adantl 482	. . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . 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 11581	. . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 513	. . . . . . . . . . . 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 25583	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ (Poly‘ℂ) ∧ 𝑎 ∈ (Poly‘ℂ)) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ))
36	4, 9, 35	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ))
37		plyf 25559	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ) → (𝑝 ∘f − 𝑎):ℂ⟶ℂ)
38		ffn 6668	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎):ℂ⟶ℂ → (𝑝 ∘f − 𝑎) Fn ℂ)
39		fniniseg 7010	. . . . . . . . . . . . 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 258	. . . . . . . . . . 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 3950	. . . . . . . . . 10 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0}))
43		ssfi 9117	. . . . . . . . . . 11 ⊢ (((◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin ∧ 𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0})) → 𝐷 ∈ Fin)
44	43	expcom 414	. . . . . . . . . 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 197	. . . . . . . 8 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → ¬ (◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin)
47		neqne 2951	. . . . . . . . . . 11 ⊢ (¬ (𝑝 ∘f − 𝑎) = 0𝑝 → (𝑝 ∘f − 𝑎) ≠ 0𝑝)
48		eqid 2736	. . . . . . . . . . . 12 ⊢ (◡(𝑝 ∘f − 𝑎) “ {0}) = (◡(𝑝 ∘f − 𝑎) “ {0})
49	48	fta1 25668	. . . . . . . . . . 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 596	. . . . . . . . . 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 495	. . . . . . . . 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 413	. . . . . . . 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 148	. . . . . . 7 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ∘f − 𝑎) = 0𝑝)
54		df-0p 25034	. . . . . . 7 ⊢ 0𝑝 = (ℂ × {0})
55	53, 54	eqtrdi 2792	. . . . . 6 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ∘f − 𝑎) = (ℂ × {0}))
56		ofsubeq0 12150	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
57	14, 6, 11, 56	mp3an2i 1466	. . . . . 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 231	. . . . 5 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 = 𝑎)
59	58	ex 413	. . . 4 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → (((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)) → 𝑝 = 𝑎))
60	59	alrimivv 1931	. . 3 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∀𝑝∀𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)) → 𝑝 = 𝑎))
61		eleq1w 2820	. . . . 5 ⊢ (𝑝 = 𝑎 → (𝑝 ∈ (Poly‘ℂ) ↔ 𝑎 ∈ (Poly‘ℂ)))
62		reseq1 5931	. . . . . 6 ⊢ (𝑝 = 𝑎 → (𝑝 ↾ 𝐷) = (𝑎 ↾ 𝐷))
63	62	eqeq1d 2738	. . . . 5 ⊢ (𝑝 = 𝑎 → ((𝑝 ↾ 𝐷) = 𝐹 ↔ (𝑎 ↾ 𝐷) = 𝐹))
64	61, 63	anbi12d 631	. . . 4 ⊢ (𝑝 = 𝑎 → ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ↔ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)))
65	64	mo4 2564	. . 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 233	. 2 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹))
67		plyssc 25561	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
68	67	sseli 3940	. . . 4 ⊢ (𝑝 ∈ (Poly‘𝑆) → 𝑝 ∈ (Poly‘ℂ))
69	68	anim1i 615	. . 3 ⊢ ((𝑝 ∈ (Poly‘𝑆) ∧ (𝑝 ↾ 𝐷) = 𝐹) → (𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹))
70	69	moimi 2543	. 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 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∃*wmo 2536   ≠ wne 2943  Vcvv 3445   ⊆ wss 3910  {csn 4586   class class class wbr 5105   × cxp 5631  ^◡ccnv 5632   ↾ cres 5635   “ cima 5636   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∘_f cof 7615  Fincfn 8883  ℂcc 11049  0cc0 11051   ≤ cle 11190   − cmin 11385  ♯chash 14230  0_𝑝c0p 25033  Polycply 25545  degcdgr 25548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-0p 25034  df-ply 25549  df-idp 25550  df-coe 25551  df-dgr 25552  df-quot 25651

This theorem is referenced by: (None)