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Theorem fta1 25201
Description: The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg(𝐹) roots. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
fta1.1 𝑅 = (𝐹 “ {0})
Assertion
Ref Expression
fta1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))

Proof of Theorem fta1
Dummy variables 𝑥 𝑔 𝑓 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . 2 (deg‘𝐹) = (deg‘𝐹)
2 dgrcl 25127 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
32adantr 484 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4 eqeq2 2749 . . . . . . 7 (𝑥 = 0 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 0))
54imbi1d 345 . . . . . 6 (𝑥 = 0 → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
65ralbidv 3118 . . . . 5 (𝑥 = 0 → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
7 eqeq2 2749 . . . . . . 7 (𝑥 = 𝑑 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 𝑑))
87imbi1d 345 . . . . . 6 (𝑥 = 𝑑 → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
98ralbidv 3118 . . . . 5 (𝑥 = 𝑑 → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
10 eqeq2 2749 . . . . . . 7 (𝑥 = (𝑑 + 1) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (𝑑 + 1)))
1110imbi1d 345 . . . . . 6 (𝑥 = (𝑑 + 1) → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = (𝑑 + 1) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
1211ralbidv 3118 . . . . 5 (𝑥 = (𝑑 + 1) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (𝑑 + 1) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
13 eqeq2 2749 . . . . . . 7 (𝑥 = (deg‘𝐹) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (deg‘𝐹)))
1413imbi1d 345 . . . . . 6 (𝑥 = (deg‘𝐹) → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
1514ralbidv 3118 . . . . 5 (𝑥 = (deg‘𝐹) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
16 eldifsni 4703 . . . . . . . . . . 11 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
1716adantr 484 . . . . . . . . . 10 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ≠ 0𝑝)
18 simplr 769 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (deg‘𝑓) = 0)
19 eldifi 4041 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℂ))
2019ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 ∈ (Poly‘ℂ))
21 0dgrb 25140 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
2220, 21syl 17 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
2318, 22mpbid 235 . . . . . . . . . . . . . 14 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = (ℂ × {(𝑓‘0)}))
2423fveq1d 6719 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
2519adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26 plyf 25092 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27 ffn 6545 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28 fniniseg 6880 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn ℂ → (𝑥 ∈ (𝑓 “ {0}) ↔ (𝑥 ∈ ℂ ∧ (𝑓𝑥) = 0)))
2925, 26, 27, 284syl 19 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (𝑓 “ {0}) ↔ (𝑥 ∈ ℂ ∧ (𝑓𝑥) = 0)))
3029biimpa 480 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓𝑥) = 0))
3130simprd 499 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = 0)
3230simpld 498 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑥 ∈ ℂ)
33 fvex 6730 . . . . . . . . . . . . . . . . . . 19 (𝑓‘0) ∈ V
3433fvconst2 7019 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℂ → ((ℂ × {(𝑓‘0)})‘𝑥) = (𝑓‘0))
3532, 34syl 17 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → ((ℂ × {(𝑓‘0)})‘𝑥) = (𝑓‘0))
3624, 31, 353eqtr3rd 2786 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓‘0) = 0)
3736sneqd 4553 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → {(𝑓‘0)} = {0})
3837xpeq2d 5581 . . . . . . . . . . . . . 14 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (ℂ × {(𝑓‘0)}) = (ℂ × {0}))
3923, 38eqtrd 2777 . . . . . . . . . . . . 13 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = (ℂ × {0}))
40 df-0p 24567 . . . . . . . . . . . . 13 0𝑝 = (ℂ × {0})
4139, 40eqtr4di 2796 . . . . . . . . . . . 12 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = 0𝑝)
4241ex 416 . . . . . . . . . . 11 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (𝑓 “ {0}) → 𝑓 = 0𝑝))
4342necon3ad 2953 . . . . . . . . . 10 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑓 ≠ 0𝑝 → ¬ 𝑥 ∈ (𝑓 “ {0})))
4417, 43mpd 15 . . . . . . . . 9 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → ¬ 𝑥 ∈ (𝑓 “ {0}))
4544eq0rdv 4319 . . . . . . . 8 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑓 “ {0}) = ∅)
4645ex 416 . . . . . . 7 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (𝑓 “ {0}) = ∅))
47 dgrcl 25127 . . . . . . . . 9 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48 nn0ge0 12115 . . . . . . . . 9 ((deg‘𝑓) ∈ ℕ0 → 0 ≤ (deg‘𝑓))
4919, 47, 483syl 18 . . . . . . . 8 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 0 ≤ (deg‘𝑓))
50 id 22 . . . . . . . . . . 11 ((𝑓 “ {0}) = ∅ → (𝑓 “ {0}) = ∅)
51 0fin 8849 . . . . . . . . . . 11 ∅ ∈ Fin
5250, 51eqeltrdi 2846 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (𝑓 “ {0}) ∈ Fin)
5352biantrurd 536 . . . . . . . . 9 ((𝑓 “ {0}) = ∅ → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
54 fveq2 6717 . . . . . . . . . . 11 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = (♯‘∅))
55 hash0 13934 . . . . . . . . . . 11 (♯‘∅) = 0
5654, 55eqtrdi 2794 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = 0)
5756breq1d 5063 . . . . . . . . 9 ((𝑓 “ {0}) = ∅ → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
5853, 57bitr3d 284 . . . . . . . 8 ((𝑓 “ {0}) = ∅ → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ 0 ≤ (deg‘𝑓)))
5949, 58syl5ibrcom 250 . . . . . . 7 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((𝑓 “ {0}) = ∅ → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
6046, 59syld 47 . . . . . 6 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
6160rgen 3071 . . . . 5 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
62 fveqeq2 6726 . . . . . . . 8 (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63 cnveq 5742 . . . . . . . . . . 11 (𝑓 = 𝑔𝑓 = 𝑔)
6463imaeq1d 5928 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓 “ {0}) = (𝑔 “ {0}))
6564eleq1d 2822 . . . . . . . . 9 (𝑓 = 𝑔 → ((𝑓 “ {0}) ∈ Fin ↔ (𝑔 “ {0}) ∈ Fin))
6664fveq2d 6721 . . . . . . . . . 10 (𝑓 = 𝑔 → (♯‘(𝑓 “ {0})) = (♯‘(𝑔 “ {0})))
67 fveq2 6717 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6866, 67breq12d 5066 . . . . . . . . 9 (𝑓 = 𝑔 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)))
6965, 68anbi12d 634 . . . . . . . 8 (𝑓 = 𝑔 → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
7062, 69imbi12d 348 . . . . . . 7 (𝑓 = 𝑔 → (((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)))))
7170cbvralvw 3358 . . . . . 6 (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
7249ad2antlr 727 . . . . . . . . . . . 12 (((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → 0 ≤ (deg‘𝑓))
7372, 58syl5ibrcom 250 . . . . . . . . . . 11 (((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → ((𝑓 “ {0}) = ∅ → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
7473a1dd 50 . . . . . . . . . 10 (((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → ((𝑓 “ {0}) = ∅ → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
75 n0 4261 . . . . . . . . . . 11 ((𝑓 “ {0}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑓 “ {0}))
76 eqid 2737 . . . . . . . . . . . . . 14 (𝑓 “ {0}) = (𝑓 “ {0})
77 simplll 775 . . . . . . . . . . . . . 14 ((((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))) → 𝑑 ∈ ℕ0)
78 simpllr 776 . . . . . . . . . . . . . 14 ((((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))) → 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}))
79 simplr 769 . . . . . . . . . . . . . 14 ((((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))) → (deg‘𝑓) = (𝑑 + 1))
80 simprl 771 . . . . . . . . . . . . . 14 ((((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))) → 𝑥 ∈ (𝑓 “ {0}))
81 simprr 773 . . . . . . . . . . . . . 14 ((((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))) → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
8276, 77, 78, 79, 80, 81fta1lem 25200 . . . . . . . . . . . . 13 ((((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) ∧ (𝑥 ∈ (𝑓 “ {0}) ∧ ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
8382exp32 424 . . . . . . . . . . . 12 (((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → (𝑥 ∈ (𝑓 “ {0}) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
8483exlimdv 1941 . . . . . . . . . . 11 (((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → (∃𝑥 𝑥 ∈ (𝑓 “ {0}) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
8575, 84syl5bi 245 . . . . . . . . . 10 (((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → ((𝑓 “ {0}) ≠ ∅ → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
8674, 85pm2.61dne 3028 . . . . . . . . 9 (((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
8786ex 416 . . . . . . . 8 ((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → ((deg‘𝑓) = (𝑑 + 1) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
8887com23 86 . . . . . . 7 ((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))) → ((deg‘𝑓) = (𝑑 + 1) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
8988ralrimdva 3110 . . . . . 6 (𝑑 ∈ ℕ0 → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))) → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (𝑑 + 1) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
9071, 89syl5bi 245 . . . . 5 (𝑑 ∈ ℕ0 → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (𝑑 + 1) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
916, 9, 12, 15, 61, 90nn0ind 12272 . . . 4 ((deg‘𝐹) ∈ ℕ0 → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
923, 91syl 17 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → ∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
93 plyssc 25094 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
9493sseli 3896 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95 eldifsn 4700 . . . . 5 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96 fveqeq2 6726 . . . . . . 7 (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97 cnveq 5742 . . . . . . . . . . 11 (𝑓 = 𝐹𝑓 = 𝐹)
9897imaeq1d 5928 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓 “ {0}) = (𝐹 “ {0}))
99 fta1.1 . . . . . . . . . 10 𝑅 = (𝐹 “ {0})
10098, 99eqtr4di 2796 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ {0}) = 𝑅)
101100eleq1d 2822 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102100fveq2d 6721 . . . . . . . . 9 (𝑓 = 𝐹 → (♯‘(𝑓 “ {0})) = (♯‘𝑅))
103 fveq2 6717 . . . . . . . . 9 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104102, 103breq12d 5066 . . . . . . . 8 (𝑓 = 𝐹 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105101, 104anbi12d 634 . . . . . . 7 (𝑓 = 𝐹 → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
10696, 105imbi12d 348 . . . . . 6 (𝑓 = 𝐹 → (((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
107106rspcv 3532 . . . . 5 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) → ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
10895, 107sylbir 238 . . . 4 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) → ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
10994, 108sylan 583 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (∀𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) → ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
11092, 109mpd 15 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
1111, 110mpi 20 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2110  wne 2940  wral 3061  cdif 3863  c0 4237  {csn 4541   class class class wbr 5053   × cxp 5549  ccnv 5550  cima 5554   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  Fincfn 8626  cc 10727  0cc0 10729  1c1 10730   + caddc 10732  cle 10868  0cn0 12090  chash 13896  0𝑝c0p 24566  Polycply 25078  degcdgr 25081
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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-map 8510  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-sup 9058  df-inf 9059  df-oi 9126  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-fl 13367  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-rlim 15050  df-sum 15250  df-0p 24567  df-ply 25082  df-idp 25083  df-coe 25084  df-dgr 25085  df-quot 25184
This theorem is referenced by:  vieta1lem2  25204  vieta1  25205  plyexmo  25206  aannenlem1  25221  aalioulem2  25226  basellem4  25966  dchrfi  26136
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