MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plyexmo Structured version   Visualization version   GIF version

Theorem plyexmo 24263
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) ∧ (𝑝𝐷) = 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.
StepHypRef Expression
1 simplr 809 . . . . . . . . 9 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ¬ 𝐷 ∈ Fin)
2 simpll 807 . . . . . . . . . . . . . 14 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝐷 ⊆ ℂ)
32sseld 3739 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷𝑏 ∈ ℂ))
4 simprll 821 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 ∈ (Poly‘ℂ))
5 plyf 24149 . . . . . . . . . . . . . . . . . . 19 (𝑝 ∈ (Poly‘ℂ) → 𝑝:ℂ⟶ℂ)
64, 5syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝:ℂ⟶ℂ)
7 ffn 6202 . . . . . . . . . . . . . . . . . 18 (𝑝:ℂ⟶ℂ → 𝑝 Fn ℂ)
86, 7syl 17 . . . . . . . . . . . . . . . . 17 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 Fn ℂ)
98adantr 472 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑝 Fn ℂ)
10 simprrl 823 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎 ∈ (Poly‘ℂ))
11 plyf 24149 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ (Poly‘ℂ) → 𝑎:ℂ⟶ℂ)
1210, 11syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎:ℂ⟶ℂ)
13 ffn 6202 . . . . . . . . . . . . . . . . . 18 (𝑎:ℂ⟶ℂ → 𝑎 Fn ℂ)
1412, 13syl 17 . . . . . . . . . . . . . . . . 17 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎 Fn ℂ)
1514adantr 472 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑎 Fn ℂ)
16 cnex 10205 . . . . . . . . . . . . . . . . 17 ℂ ∈ V
1716a1i 11 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ℂ ∈ V)
182sselda 3740 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑏 ∈ ℂ)
19 fnfvof 7072 . . . . . . . . . . . . . . . 16 (((𝑝 Fn ℂ ∧ 𝑎 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑏 ∈ ℂ)) → ((𝑝𝑓𝑎)‘𝑏) = ((𝑝𝑏) − (𝑎𝑏)))
209, 15, 17, 18, 19syl22anc 1478 . . . . . . . . . . . . . . 15 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝑓𝑎)‘𝑏) = ((𝑝𝑏) − (𝑎𝑏)))
216adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑝:ℂ⟶ℂ)
2221, 18ffvelrnd 6519 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝑏) ∈ ℂ)
23 simprlr 822 . . . . . . . . . . . . . . . . . . . 20 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝐷) = 𝐹)
24 simprrr 824 . . . . . . . . . . . . . . . . . . . 20 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑎𝐷) = 𝐹)
2523, 24eqtr4d 2793 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝐷) = (𝑎𝐷))
2625adantr 472 . . . . . . . . . . . . . . . . . 18 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝐷) = (𝑎𝐷))
2726fveq1d 6350 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = ((𝑎𝐷)‘𝑏))
28 fvres 6364 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
2928adantl 473 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
30 fvres 6364 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑎𝐷)‘𝑏) = (𝑎𝑏))
3130adantl 473 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑎𝐷)‘𝑏) = (𝑎𝑏))
3227, 29, 313eqtr3d 2798 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝑏) = (𝑎𝑏))
3322, 32subeq0bd 10644 . . . . . . . . . . . . . . 15 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝑏) − (𝑎𝑏)) = 0)
3420, 33eqtrd 2790 . . . . . . . . . . . . . 14 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝑓𝑎)‘𝑏) = 0)
3534ex 449 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → ((𝑝𝑓𝑎)‘𝑏) = 0))
363, 35jcad 556 . . . . . . . . . . . 12 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → (𝑏 ∈ ℂ ∧ ((𝑝𝑓𝑎)‘𝑏) = 0)))
37 plysubcl 24173 . . . . . . . . . . . . . 14 ((𝑝 ∈ (Poly‘ℂ) ∧ 𝑎 ∈ (Poly‘ℂ)) → (𝑝𝑓𝑎) ∈ (Poly‘ℂ))
384, 10, 37syl2anc 696 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝑓𝑎) ∈ (Poly‘ℂ))
39 plyf 24149 . . . . . . . . . . . . 13 ((𝑝𝑓𝑎) ∈ (Poly‘ℂ) → (𝑝𝑓𝑎):ℂ⟶ℂ)
40 ffn 6202 . . . . . . . . . . . . 13 ((𝑝𝑓𝑎):ℂ⟶ℂ → (𝑝𝑓𝑎) Fn ℂ)
41 fniniseg 6497 . . . . . . . . . . . . 13 ((𝑝𝑓𝑎) Fn ℂ → (𝑏 ∈ ((𝑝𝑓𝑎) “ {0}) ↔ (𝑏 ∈ ℂ ∧ ((𝑝𝑓𝑎)‘𝑏) = 0)))
4238, 39, 40, 414syl 19 . . . . . . . . . . . 12 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏 ∈ ((𝑝𝑓𝑎) “ {0}) ↔ (𝑏 ∈ ℂ ∧ ((𝑝𝑓𝑎)‘𝑏) = 0)))
4336, 42sylibrd 249 . . . . . . . . . . 11 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷𝑏 ∈ ((𝑝𝑓𝑎) “ {0})))
4443ssrdv 3746 . . . . . . . . . 10 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝐷 ⊆ ((𝑝𝑓𝑎) “ {0}))
45 ssfi 8341 . . . . . . . . . . 11 ((((𝑝𝑓𝑎) “ {0}) ∈ Fin ∧ 𝐷 ⊆ ((𝑝𝑓𝑎) “ {0})) → 𝐷 ∈ Fin)
4645expcom 450 . . . . . . . . . 10 (𝐷 ⊆ ((𝑝𝑓𝑎) “ {0}) → (((𝑝𝑓𝑎) “ {0}) ∈ Fin → 𝐷 ∈ Fin))
4744, 46syl 17 . . . . . . . . 9 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (((𝑝𝑓𝑎) “ {0}) ∈ Fin → 𝐷 ∈ Fin))
481, 47mtod 189 . . . . . . . 8 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ¬ ((𝑝𝑓𝑎) “ {0}) ∈ Fin)
49 df-ne 2929 . . . . . . . . . . . 12 ((𝑝𝑓𝑎) ≠ 0𝑝 ↔ ¬ (𝑝𝑓𝑎) = 0𝑝)
5049biimpri 218 . . . . . . . . . . 11 (¬ (𝑝𝑓𝑎) = 0𝑝 → (𝑝𝑓𝑎) ≠ 0𝑝)
51 eqid 2756 . . . . . . . . . . . 12 ((𝑝𝑓𝑎) “ {0}) = ((𝑝𝑓𝑎) “ {0})
5251fta1 24258 . . . . . . . . . . 11 (((𝑝𝑓𝑎) ∈ (Poly‘ℂ) ∧ (𝑝𝑓𝑎) ≠ 0𝑝) → (((𝑝𝑓𝑎) “ {0}) ∈ Fin ∧ (♯‘((𝑝𝑓𝑎) “ {0})) ≤ (deg‘(𝑝𝑓𝑎))))
5338, 50, 52syl2an 495 . . . . . . . . . 10 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ ¬ (𝑝𝑓𝑎) = 0𝑝) → (((𝑝𝑓𝑎) “ {0}) ∈ Fin ∧ (♯‘((𝑝𝑓𝑎) “ {0})) ≤ (deg‘(𝑝𝑓𝑎))))
5453simpld 477 . . . . . . . . 9 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ ¬ (𝑝𝑓𝑎) = 0𝑝) → ((𝑝𝑓𝑎) “ {0}) ∈ Fin)
5554ex 449 . . . . . . . 8 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (¬ (𝑝𝑓𝑎) = 0𝑝 → ((𝑝𝑓𝑎) “ {0}) ∈ Fin))
5648, 55mt3d 140 . . . . . . 7 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝑓𝑎) = 0𝑝)
57 df-0p 23632 . . . . . . 7 0𝑝 = (ℂ × {0})
5856, 57syl6eq 2806 . . . . . 6 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝑓𝑎) = (ℂ × {0}))
5916a1i 11 . . . . . . 7 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ℂ ∈ V)
60 ofsubeq0 11205 . . . . . . 7 ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
6159, 6, 12, 60syl3anc 1477 . . . . . 6 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
6258, 61mpbid 222 . . . . 5 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 = 𝑎)
6362ex 449 . . . 4 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → (((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
6463alrimivv 2001 . . 3 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
65 eleq1w 2818 . . . . 5 (𝑝 = 𝑎 → (𝑝 ∈ (Poly‘ℂ) ↔ 𝑎 ∈ (Poly‘ℂ)))
66 reseq1 5541 . . . . . 6 (𝑝 = 𝑎 → (𝑝𝐷) = (𝑎𝐷))
6766eqeq1d 2758 . . . . 5 (𝑝 = 𝑎 → ((𝑝𝐷) = 𝐹 ↔ (𝑎𝐷) = 𝐹))
6865, 67anbi12d 749 . . . 4 (𝑝 = 𝑎 → ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)))
6968mo4 2651 . . 3 (∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
7064, 69sylibr 224 . 2 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
71 plyssc 24151 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
7271sseli 3736 . . . 4 (𝑝 ∈ (Poly‘𝑆) → 𝑝 ∈ (Poly‘ℂ))
7372anim1i 593 . . 3 ((𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹) → (𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
7473moimi 2654 . 2 (∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) → ∃*𝑝(𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹))
7570, 74syl 17 1 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1626   = wceq 1628  wcel 2135  ∃*wmo 2604  wne 2928  Vcvv 3336  wss 3711  {csn 4317   class class class wbr 4800   × cxp 5260  ccnv 5261  cres 5264  cima 5265   Fn wfn 6040  wf 6041  cfv 6045  (class class class)co 6809  𝑓 cof 7056  Fincfn 8117  cc 10122  0cc0 10124  cle 10263  cmin 10454  chash 13307  0𝑝c0p 23631  Polycply 24135  degcdgr 24138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-inf2 8707  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201  ax-pre-sup 10202  ax-addf 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-fal 1634  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-se 5222  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-isom 6054  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-of 7058  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-er 7907  df-map 8021  df-pm 8022  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-sup 8509  df-inf 8510  df-oi 8576  df-card 8951  df-cda 9178  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-div 10873  df-nn 11209  df-2 11267  df-3 11268  df-n0 11481  df-xnn0 11552  df-z 11566  df-uz 11876  df-rp 12022  df-fz 12516  df-fzo 12656  df-fl 12783  df-seq 12992  df-exp 13051  df-hash 13308  df-cj 14034  df-re 14035  df-im 14036  df-sqrt 14170  df-abs 14171  df-clim 14414  df-rlim 14415  df-sum 14612  df-0p 23632  df-ply 24139  df-idp 24140  df-coe 24141  df-dgr 24142  df-quot 24241
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator