Theorem plyexmo 26301

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 775	. . . . . . . . 9 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → ¬ 𝐷 ∈ Fin)
2		simpll 773	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝐷 ⊆ ℂ)
3	2	sseld 3916	. . . . . . . . . . . . 13 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑏 ∈ 𝐷 → 𝑏 ∈ ℂ))
4		simprll 785	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 ∈ (Poly‘ℂ))
5		plyf 26185	. . . . . . . . . . . . . . . . . . 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 6660	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 Fn ℂ)
8	7	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑝 Fn ℂ)
9		simprrl 787	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑎 ∈ (Poly‘ℂ))
10		plyf 26185	. . . . . . . . . . . . . . . . . . 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 6660	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑎 Fn ℂ)
13	12	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑎 Fn ℂ)
14		cnex 11114	. . . . . . . . . . . . . . . . 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 3917	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ ℂ)
17		fnfvof 7641	. . . . . . . . . . . . . . . 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 845	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 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 7030	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → (𝑝‘𝑏) ∈ ℂ)
21		simprlr 786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ↾ 𝐷) = 𝐹)
22		simprrr 788	. . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 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 6833	. . . . . . . . . . . . . . . . 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 6850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝐷 → ((𝑝 ↾ 𝐷)‘𝑏) = (𝑝‘𝑏))
27	26	adantl 483	. . . . . . . . . . . . . . . . 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 6850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝐷 → ((𝑎 ↾ 𝐷)‘𝑏) = (𝑎‘𝑏))
29	28	adantl 483	. . . . . . . . . . . . . . . . 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 11571	. . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . 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 518	. . . . . . . . . . . 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 26209	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ (Poly‘ℂ) ∧ 𝑎 ∈ (Poly‘ℂ)) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ))
36	4, 9, 35	syl2anc 591	. . . . . . . . . . . . 13 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ))
37		plyf 26185	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ) → (𝑝 ∘f − 𝑎):ℂ⟶ℂ)
38		ffn 6659	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎):ℂ⟶ℂ → (𝑝 ∘f − 𝑎) Fn ℂ)
39		fniniseg 7005	. . . . . . . . . . . . 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 3923	. . . . . . . . . 10 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0}))
43		ssfi 9101	. . . . . . . . . . 11 ⊢ (((◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin ∧ 𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0})) → 𝐷 ∈ Fin)
44	43	expcom 415	. . . . . . . . . 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 2944	. . . . . . . . . . 11 ⊢ (¬ (𝑝 ∘f − 𝑎) = 0𝑝 → (𝑝 ∘f − 𝑎) ≠ 0𝑝)
48		eqid 2741	. . . . . . . . . . . 12 ⊢ (◡(𝑝 ∘f − 𝑎) “ {0}) = (◡(𝑝 ∘f − 𝑎) “ {0})
49	48	fta1 26296	. . . . . . . . . . 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 603	. . . . . . . . . 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 496	. . . . . . . . 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 414	. . . . . . . 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 25659	. . . . . . 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 12151	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
57	14, 6, 11, 56	mp3an2i 1475	. . . . . 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 414	. . . 4 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → (((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)) → 𝑝 = 𝑎))
60	59	alrimivv 1936	. . 3 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∀𝑝∀𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)) → 𝑝 = 𝑎))
61		eleq1w 2824	. . . . 5 ⊢ (𝑝 = 𝑎 → (𝑝 ∈ (Poly‘ℂ) ↔ 𝑎 ∈ (Poly‘ℂ)))
62		reseq1 5932	. . . . . 6 ⊢ (𝑝 = 𝑎 → (𝑝 ↾ 𝐷) = (𝑎 ↾ 𝐷))
63	62	eqeq1d 2743	. . . . 5 ⊢ (𝑝 = 𝑎 → ((𝑝 ↾ 𝐷) = 𝐹 ↔ (𝑎 ↾ 𝐷) = 𝐹))
64	61, 63	anbi12d 639	. . . 4 ⊢ (𝑝 = 𝑎 → ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ↔ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)))
65	64	mo4 2572	. . 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 26187	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
68	67	sseli 3913	. . . 4 ⊢ (𝑝 ∈ (Poly‘𝑆) → 𝑝 ∈ (Poly‘ℂ))
69	68	anim1i 622	. . 3 ⊢ ((𝑝 ∈ (Poly‘𝑆) ∧ (𝑝 ↾ 𝐷) = 𝐹) → (𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹))
70	69	moimi 2551	. 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 397  ∀wal 1546   = wceq 1548   ∈ wcel 2121  ∃*wmo 2543   ≠ wne 2936  Vcvv 3433   ⊆ wss 3885  {csn 4558   class class class wbr 5075   × cxp 5619  ^◡ccnv 5620   ↾ cres 5623   “ cima 5624   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360   ∘_f cof 7622  Fincfn 8887  ℂcc 11031  0cc0 11033   ≤ cle 11175   − cmin 11372  ♯chash 14287  0_𝑝c0p 25658  Polycply 26171  degcdgr 26174

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-oi 9419  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-xnn0 12506  df-z 12520  df-uz 12784  df-rp 12938  df-fz 13457  df-fzo 13604  df-fl 13746  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-rlim 15446  df-sum 15644  df-0p 25659  df-ply 26175  df-idp 26176  df-coe 26177  df-dgr 26178  df-quot 26279

This theorem is referenced by: (None)