Theorem plyexmo 26294

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 769	. . . . . . . . 9 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → ¬ 𝐷 ∈ Fin)
2		simpll 767	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝐷 ⊆ ℂ)
3	2	sseld 3921	. . . . . . . . . . . . 13 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑏 ∈ 𝐷 → 𝑏 ∈ ℂ))
4		simprll 779	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 ∈ (Poly‘ℂ))
5		plyf 26177	. . . . . . . . . . . . . . . . . . 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 6665	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 Fn ℂ)
8	7	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑝 Fn ℂ)
9		simprrl 781	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑎 ∈ (Poly‘ℂ))
10		plyf 26177	. . . . . . . . . . . . . . . . . . 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 6665	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑎 Fn ℂ)
13	12	adantr 480	. . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ ℂ)
17		fnfvof 7643	. . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 7033	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → (𝑝‘𝑏) ∈ ℂ)
21		simprlr 780	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ↾ 𝐷) = 𝐹)
22		simprrr 782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑎 ↾ 𝐷) = 𝐹)
23	21, 22	eqtr4d 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ↾ 𝐷) = (𝑎 ↾ 𝐷))
24	23	adantr 480	. . . . . . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝐷 → ((𝑝 ↾ 𝐷)‘𝑏) = (𝑝‘𝑏))
27	26	adantl 481	. . . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ 𝐷 → ((𝑎 ↾ 𝐷)‘𝑏) = (𝑎‘𝑏))
29	28	adantl 481	. . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 512	. . . . . . . . . . . 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 26201	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ (Poly‘ℂ) ∧ 𝑎 ∈ (Poly‘ℂ)) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ))
36	4, 9, 35	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ))
37		plyf 26177	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ) → (𝑝 ∘f − 𝑎):ℂ⟶ℂ)
38		ffn 6664	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎):ℂ⟶ℂ → (𝑝 ∘f − 𝑎) Fn ℂ)
39		fniniseg 7008	. . . . . . . . . . . . 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 259	. . . . . . . . . . 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 3928	. . . . . . . . . 10 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0}))
43		ssfi 9102	. . . . . . . . . . 11 ⊢ (((◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin ∧ 𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0})) → 𝐷 ∈ Fin)
44	43	expcom 413	. . . . . . . . . 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 198	. . . . . . . 8 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → ¬ (◡(𝑝 ∘f − 𝑎) “ {0}) ∈ Fin)
47		neqne 2941	. . . . . . . . . . 11 ⊢ (¬ (𝑝 ∘f − 𝑎) = 0𝑝 → (𝑝 ∘f − 𝑎) ≠ 0𝑝)
48		eqid 2737	. . . . . . . . . . . 12 ⊢ (◡(𝑝 ∘f − 𝑎) “ {0}) = (◡(𝑝 ∘f − 𝑎) “ {0})
49	48	fta1 26289	. . . . . . . . . . 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 494	. . . . . . . . 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 412	. . . . . . . 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 25651	. . . . . . 7 ⊢ 0𝑝 = (ℂ × {0})
55	53, 54	eqtrdi 2788	. . . . . 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 1469	. . . . . 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 232	. . . . 5 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 = 𝑎)
59	58	ex 412	. . . 4 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → (((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)) → 𝑝 = 𝑎))
60	59	alrimivv 1930	. . 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 5934	. . . . . 6 ⊢ (𝑝 = 𝑎 → (𝑝 ↾ 𝐷) = (𝑎 ↾ 𝐷))
63	62	eqeq1d 2739	. . . . 5 ⊢ (𝑝 = 𝑎 → ((𝑝 ↾ 𝐷) = 𝐹 ↔ (𝑎 ↾ 𝐷) = 𝐹))
64	61, 63	anbi12d 633	. . . 4 ⊢ (𝑝 = 𝑎 → ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ↔ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)))
65	64	mo4 2567	. . 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 234	. 2 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹))
67		plyssc 26179	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
68	67	sseli 3918	. . . 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 2546	. 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 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∃*wmo 2538   ≠ wne 2933  Vcvv 3430   ⊆ wss 3890  {csn 4568   class class class wbr 5086   × cxp 5624  ^◡ccnv 5625   ↾ cres 5628   “ cima 5629   Fn wfn 6489  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   ∘_f cof 7624  Fincfn 8888  ℂcc 11031  0cc0 11033   ≤ cle 11175   − cmin 11372  ♯chash 14287  0_𝑝c0p 25650  Polycply 26163  degcdgr 26166

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  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 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7626  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-oadd 8404  df-er 8638  df-map 8770  df-pm 8771  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-oi 9420  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 25651  df-ply 26167  df-idp 26168  df-coe 26169  df-dgr 26170  df-quot 26272

This theorem is referenced by: (None)