Theorem plyexmo 26248

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 768	. . . . . . . . 9 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → ¬ 𝐷 ∈ Fin)
2		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝐷 ⊆ ℂ)
3	2	sseld 3928	. . . . . . . . . . . . 13 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑏 ∈ 𝐷 → 𝑏 ∈ ℂ))
4		simprll 778	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑝 ∈ (Poly‘ℂ))
5		plyf 26130	. . . . . . . . . . . . . . . . . . 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 6652	. . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝑎 ∈ (Poly‘ℂ))
10		plyf 26130	. . . . . . . . . . . . . . . . . . 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 6652	. . . . . . . . . . . . . . . . 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 11087	. . . . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ ℂ)
17		fnfvof 7627	. . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . 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 7018	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) ∧ 𝑏 ∈ 𝐷) → (𝑝‘𝑏) ∈ ℂ)
21		simprlr 779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ↾ 𝐷) = 𝐹)
22		simprrr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑎 ↾ 𝐷) = 𝐹)
23	21, 22	eqtr4d 2769	. . . . . . . . . . . . . . . . . . 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 6824	. . . . . . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . 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 11543	. . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . 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 26154	. . . . . . . . . . . . . 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 26130	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎) ∈ (Poly‘ℂ) → (𝑝 ∘f − 𝑎):ℂ⟶ℂ)
38		ffn 6651	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∘f − 𝑎):ℂ⟶ℂ → (𝑝 ∘f − 𝑎) Fn ℂ)
39		fniniseg 6993	. . . . . . . . . . . . 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 3935	. . . . . . . . . 10 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → 𝐷 ⊆ (◡(𝑝 ∘f − 𝑎) “ {0}))
43		ssfi 9082	. . . . . . . . . . 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 2936	. . . . . . . . . . 11 ⊢ (¬ (𝑝 ∘f − 𝑎) = 0𝑝 → (𝑝 ∘f − 𝑎) ≠ 0𝑝)
48		eqid 2731	. . . . . . . . . . . 12 ⊢ (◡(𝑝 ∘f − 𝑎) “ {0}) = (◡(𝑝 ∘f − 𝑎) “ {0})
49	48	fta1 26243	. . . . . . . . . . 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 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 25598	. . . . . . 7 ⊢ 0𝑝 = (ℂ × {0})
55	53, 54	eqtrdi 2782	. . . . . 6 ⊢ (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹))) → (𝑝 ∘f − 𝑎) = (ℂ × {0}))
56		ofsubeq0 12122	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝 ∘f − 𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
57	14, 6, 11, 56	mp3an2i 1468	. . . . . 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 1929	. . 3 ⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∀𝑝∀𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝 ↾ 𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎 ↾ 𝐷) = 𝐹)) → 𝑝 = 𝑎))
61		eleq1w 2814	. . . . 5 ⊢ (𝑝 = 𝑎 → (𝑝 ∈ (Poly‘ℂ) ↔ 𝑎 ∈ (Poly‘ℂ)))
62		reseq1 5921	. . . . . 6 ⊢ (𝑝 = 𝑎 → (𝑝 ↾ 𝐷) = (𝑎 ↾ 𝐷))
63	62	eqeq1d 2733	. . . . 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 2561	. . 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 26132	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
68	67	sseli 3925	. . . 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 2540	. 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 1539   = wceq 1541   ∈ wcel 2111  ∃*wmo 2533   ≠ wne 2928  Vcvv 3436   ⊆ wss 3897  {csn 4573   class class class wbr 5089   × cxp 5612  ^◡ccnv 5613   ↾ cres 5616   “ cima 5617   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∘_f cof 7608  Fincfn 8869  ℂcc 11004  0cc0 11006   ≤ cle 11147   − cmin 11344  ♯chash 14237  0_𝑝c0p 25597  Polycply 26116  degcdgr 26119

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-oi 9396  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-xnn0 12455  df-z 12469  df-uz 12733  df-rp 12891  df-fz 13408  df-fzo 13555  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-0p 25598  df-ply 26120  df-idp 26121  df-coe 26122  df-dgr 26123  df-quot 26226

This theorem is referenced by: (None)