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Theorem plyexmo 23785
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 787 . . . . . . . . 9 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ¬ 𝐷 ∈ Fin)
2 simpll 785 . . . . . . . . . . . . . 14 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝐷 ⊆ ℂ)
32sseld 3562 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷𝑏 ∈ ℂ))
4 simprll 797 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 ∈ (Poly‘ℂ))
5 plyf 23671 . . . . . . . . . . . . . . . . . . 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 5940 . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑝 Fn ℂ)
10 simprrl 799 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑎 ∈ (Poly‘ℂ))
11 plyf 23671 . . . . . . . . . . . . . . . . . . 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 5940 . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑎 Fn ℂ)
16 cnex 9869 . . . . . . . . . . . . . . . . 17 ℂ ∈ V
1716a1i 11 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ℂ ∈ V)
182sselda 3563 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑏 ∈ ℂ)
19 fnfvof 6782 . . . . . . . . . . . . . . . 16 (((𝑝 Fn ℂ ∧ 𝑎 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑏 ∈ ℂ)) → ((𝑝𝑓𝑎)‘𝑏) = ((𝑝𝑏) − (𝑎𝑏)))
209, 15, 17, 18, 19syl22anc 1318 . . . . . . . . . . . . . . 15 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝑓𝑎)‘𝑏) = ((𝑝𝑏) − (𝑎𝑏)))
216adantr 479 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → 𝑝:ℂ⟶ℂ)
2221, 18ffvelrnd 6249 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝑏) ∈ ℂ)
23 simprlr 798 . . . . . . . . . . . . . . . . . . . 20 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝐷) = 𝐹)
24 simprrr 800 . . . . . . . . . . . . . . . . . . . 20 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑎𝐷) = 𝐹)
2523, 24eqtr4d 2642 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝐷) = (𝑎𝐷))
2625adantr 479 . . . . . . . . . . . . . . . . . 18 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝐷) = (𝑎𝐷))
2726fveq1d 6086 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = ((𝑎𝐷)‘𝑏))
28 fvres 6098 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
2928adantl 480 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝐷)‘𝑏) = (𝑝𝑏))
30 fvres 6098 . . . . . . . . . . . . . . . . . 18 (𝑏𝐷 → ((𝑎𝐷)‘𝑏) = (𝑎𝑏))
3130adantl 480 . . . . . . . . . . . . . . . . 17 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑎𝐷)‘𝑏) = (𝑎𝑏))
3227, 29, 313eqtr3d 2647 . . . . . . . . . . . . . . . 16 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → (𝑝𝑏) = (𝑎𝑏))
3322, 32subeq0bd 10303 . . . . . . . . . . . . . . 15 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝑏) − (𝑎𝑏)) = 0)
3420, 33eqtrd 2639 . . . . . . . . . . . . . 14 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ 𝑏𝐷) → ((𝑝𝑓𝑎)‘𝑏) = 0)
3534ex 448 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → ((𝑝𝑓𝑎)‘𝑏) = 0))
363, 35jcad 553 . . . . . . . . . . . 12 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷 → (𝑏 ∈ ℂ ∧ ((𝑝𝑓𝑎)‘𝑏) = 0)))
37 plysubcl 23695 . . . . . . . . . . . . . 14 ((𝑝 ∈ (Poly‘ℂ) ∧ 𝑎 ∈ (Poly‘ℂ)) → (𝑝𝑓𝑎) ∈ (Poly‘ℂ))
384, 10, 37syl2anc 690 . . . . . . . . . . . . 13 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝑓𝑎) ∈ (Poly‘ℂ))
39 plyf 23671 . . . . . . . . . . . . 13 ((𝑝𝑓𝑎) ∈ (Poly‘ℂ) → (𝑝𝑓𝑎):ℂ⟶ℂ)
40 ffn 5940 . . . . . . . . . . . . 13 ((𝑝𝑓𝑎):ℂ⟶ℂ → (𝑝𝑓𝑎) Fn ℂ)
41 fniniseg 6227 . . . . . . . . . . . . 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 247 . . . . . . . . . . 11 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑏𝐷𝑏 ∈ ((𝑝𝑓𝑎) “ {0})))
4443ssrdv 3569 . . . . . . . . . 10 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝐷 ⊆ ((𝑝𝑓𝑎) “ {0}))
45 ssfi 8038 . . . . . . . . . . 11 ((((𝑝𝑓𝑎) “ {0}) ∈ Fin ∧ 𝐷 ⊆ ((𝑝𝑓𝑎) “ {0})) → 𝐷 ∈ Fin)
4645expcom 449 . . . . . . . . . 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 187 . . . . . . . 8 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ¬ ((𝑝𝑓𝑎) “ {0}) ∈ Fin)
49 df-ne 2777 . . . . . . . . . . . 12 ((𝑝𝑓𝑎) ≠ 0𝑝 ↔ ¬ (𝑝𝑓𝑎) = 0𝑝)
5049biimpri 216 . . . . . . . . . . 11 (¬ (𝑝𝑓𝑎) = 0𝑝 → (𝑝𝑓𝑎) ≠ 0𝑝)
51 eqid 2605 . . . . . . . . . . . 12 ((𝑝𝑓𝑎) “ {0}) = ((𝑝𝑓𝑎) “ {0})
5251fta1 23780 . . . . . . . . . . 11 (((𝑝𝑓𝑎) ∈ (Poly‘ℂ) ∧ (𝑝𝑓𝑎) ≠ 0𝑝) → (((𝑝𝑓𝑎) “ {0}) ∈ Fin ∧ (#‘((𝑝𝑓𝑎) “ {0})) ≤ (deg‘(𝑝𝑓𝑎))))
5338, 50, 52syl2an 492 . . . . . . . . . 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 473 . . . . . . . . 9 ((((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) ∧ ¬ (𝑝𝑓𝑎) = 0𝑝) → ((𝑝𝑓𝑎) “ {0}) ∈ Fin)
5554ex 448 . . . . . . . 8 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (¬ (𝑝𝑓𝑎) = 0𝑝 → ((𝑝𝑓𝑎) “ {0}) ∈ Fin))
5648, 55mt3d 138 . . . . . . 7 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → (𝑝𝑓𝑎) = 0𝑝)
57 df-0p 23156 . . . . . . 7 0𝑝 = (ℂ × {0})
5856, 57syl6eq 2655 . . . . . 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 10860 . . . . . . 7 ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑎:ℂ⟶ℂ) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
6159, 6, 12, 60syl3anc 1317 . . . . . 6 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → ((𝑝𝑓𝑎) = (ℂ × {0}) ↔ 𝑝 = 𝑎))
6258, 61mpbid 220 . . . . 5 (((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) ∧ ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹))) → 𝑝 = 𝑎)
6362ex 448 . . . 4 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → (((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
6463alrimivv 1841 . . 3 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
65 eleq1 2671 . . . . 5 (𝑝 = 𝑎 → (𝑝 ∈ (Poly‘ℂ) ↔ 𝑎 ∈ (Poly‘ℂ)))
66 reseq1 5294 . . . . . 6 (𝑝 = 𝑎 → (𝑝𝐷) = (𝑎𝐷))
6766eqeq1d 2607 . . . . 5 (𝑝 = 𝑎 → ((𝑝𝐷) = 𝐹 ↔ (𝑎𝐷) = 𝐹))
6865, 67anbi12d 742 . . . 4 (𝑝 = 𝑎 → ((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)))
6968mo4 2500 . . 3 (∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ↔ ∀𝑝𝑎(((𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹) ∧ (𝑎 ∈ (Poly‘ℂ) ∧ (𝑎𝐷) = 𝐹)) → 𝑝 = 𝑎))
7064, 69sylibr 222 . 2 ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
71 plyssc 23673 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
7271sseli 3559 . . . 4 (𝑝 ∈ (Poly‘𝑆) → 𝑝 ∈ (Poly‘ℂ))
7372anim1i 589 . . 3 ((𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹) → (𝑝 ∈ (Poly‘ℂ) ∧ (𝑝𝐷) = 𝐹))
7473moimi 2503 . 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 194  wa 382  wal 1472   = wceq 1474  wcel 1975  ∃*wmo 2454  wne 2775  Vcvv 3168  wss 3535  {csn 4120   class class class wbr 4573   × cxp 5022  ccnv 5023  cres 5026  cima 5027   Fn wfn 5781  wf 5782  cfv 5786  (class class class)co 6523  𝑓 cof 6766  Fincfn 7814  cc 9786  0cc0 9788  cle 9927  cmin 10113  #chash 12930  0𝑝c0p 23155  Polycply 23657  degcdgr 23660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866  ax-addf 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-of 6768  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-inf 8205  df-oi 8271  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-fl 12406  df-seq 12615  df-exp 12674  df-hash 12931  df-cj 13629  df-re 13630  df-im 13631  df-sqrt 13765  df-abs 13766  df-clim 14009  df-rlim 14010  df-sum 14207  df-0p 23156  df-ply 23661  df-idp 23662  df-coe 23663  df-dgr 23664  df-quot 23763
This theorem is referenced by: (None)
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