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Theorem fta1 26305
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 2725 . 2 (deg‘𝐹) = (deg‘𝐹)
2 dgrcl 26229 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
32adantr 479 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4 eqeq2 2737 . . . . . . 7 (𝑥 = 0 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 0))
54imbi1d 340 . . . . . 6 (𝑥 = 0 → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
65ralbidv 3167 . . . . 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 2737 . . . . . . 7 (𝑥 = 𝑑 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 𝑑))
87imbi1d 340 . . . . . 6 (𝑥 = 𝑑 → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
98ralbidv 3167 . . . . 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 2737 . . . . . . 7 (𝑥 = (𝑑 + 1) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (𝑑 + 1)))
1110imbi1d 340 . . . . . 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 3167 . . . . 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 2737 . . . . . . 7 (𝑥 = (deg‘𝐹) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (deg‘𝐹)))
1413imbi1d 340 . . . . . 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 3167 . . . . 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 4795 . . . . . . . . . . 11 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
1716adantr 479 . . . . . . . . . 10 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ≠ 0𝑝)
18 simplr 767 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (deg‘𝑓) = 0)
19 eldifi 4123 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℂ))
2019ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 ∈ (Poly‘ℂ))
21 0dgrb 26242 . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . 14 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = (ℂ × {(𝑓‘0)}))
2423fveq1d 6898 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
2519adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26 plyf 26194 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27 ffn 6723 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28 fniniseg 7068 . . . . . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓𝑥) = 0))
3130simprd 494 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = 0)
3230simpld 493 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑥 ∈ ℂ)
33 fvex 6909 . . . . . . . . . . . . . . . . . . 19 (𝑓‘0) ∈ V
3433fvconst2 7216 . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓‘0) = 0)
3736sneqd 4642 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → {(𝑓‘0)} = {0})
3837xpeq2d 5708 . . . . . . . . . . . . . 14 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (ℂ × {(𝑓‘0)}) = (ℂ × {0}))
3923, 38eqtrd 2765 . . . . . . . . . . . . 13 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = (ℂ × {0}))
40 df-0p 25660 . . . . . . . . . . . . 13 0𝑝 = (ℂ × {0})
4139, 40eqtr4di 2783 . . . . . . . . . . . 12 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = 0𝑝)
4241ex 411 . . . . . . . . . . 11 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (𝑓 “ {0}) → 𝑓 = 0𝑝))
4342necon3ad 2942 . . . . . . . . . 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 4406 . . . . . . . 8 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑓 “ {0}) = ∅)
4645ex 411 . . . . . . 7 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (𝑓 “ {0}) = ∅))
47 dgrcl 26229 . . . . . . . . 9 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48 nn0ge0 12535 . . . . . . . . 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 0fi 9070 . . . . . . . . . . 11 ∅ ∈ Fin
5250, 51eqeltrdi 2833 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (𝑓 “ {0}) ∈ Fin)
5352biantrurd 531 . . . . . . . . 9 ((𝑓 “ {0}) = ∅ → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
54 fveq2 6896 . . . . . . . . . . 11 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = (♯‘∅))
55 hash0 14370 . . . . . . . . . . 11 (♯‘∅) = 0
5654, 55eqtrdi 2781 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = 0)
5756breq1d 5159 . . . . . . . . 9 ((𝑓 “ {0}) = ∅ → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
5853, 57bitr3d 280 . . . . . . . 8 ((𝑓 “ {0}) = ∅ → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ 0 ≤ (deg‘𝑓)))
5949, 58syl5ibrcom 246 . . . . . . 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 3052 . . . . 5 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
62 fveqeq2 6905 . . . . . . . 8 (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63 cnveq 5876 . . . . . . . . . . 11 (𝑓 = 𝑔𝑓 = 𝑔)
6463imaeq1d 6063 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓 “ {0}) = (𝑔 “ {0}))
6564eleq1d 2810 . . . . . . . . 9 (𝑓 = 𝑔 → ((𝑓 “ {0}) ∈ Fin ↔ (𝑔 “ {0}) ∈ Fin))
6664fveq2d 6900 . . . . . . . . . 10 (𝑓 = 𝑔 → (♯‘(𝑓 “ {0})) = (♯‘(𝑔 “ {0})))
67 fveq2 6896 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6866, 67breq12d 5162 . . . . . . . . 9 (𝑓 = 𝑔 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)))
6965, 68anbi12d 630 . . . . . . . 8 (𝑓 = 𝑔 → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
7062, 69imbi12d 343 . . . . . . 7 (𝑓 = 𝑔 → (((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)))))
7170cbvralvw 3224 . . . . . 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 725 . . . . . . . . . . . 12 (((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → 0 ≤ (deg‘𝑓))
7372, 58syl5ibrcom 246 . . . . . . . . . . 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 4346 . . . . . . . . . . 11 ((𝑓 “ {0}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑓 “ {0}))
76 eqid 2725 . . . . . . . . . . . . . 14 (𝑓 “ {0}) = (𝑓 “ {0})
77 simplll 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))))) → 𝑑 ∈ ℕ0)
78 simpllr 774 . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . 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 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))))) → 𝑥 ∈ (𝑓 “ {0}))
81 simprr 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))))) → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
8276, 77, 78, 79, 80, 81fta1lem 26304 . . . . . . . . . . . . 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 419 . . . . . . . . . . . 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 1928 . . . . . . . . . . 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, 84biimtrid 241 . . . . . . . . . 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 3017 . . . . . . . . 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 411 . . . . . . . 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 3143 . . . . . 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, 89biimtrid 241 . . . . 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 12695 . . . 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 26196 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
9493sseli 3972 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95 eldifsn 4792 . . . . 5 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96 fveqeq2 6905 . . . . . . 7 (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97 cnveq 5876 . . . . . . . . . . 11 (𝑓 = 𝐹𝑓 = 𝐹)
9897imaeq1d 6063 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓 “ {0}) = (𝐹 “ {0}))
99 fta1.1 . . . . . . . . . 10 𝑅 = (𝐹 “ {0})
10098, 99eqtr4di 2783 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ {0}) = 𝑅)
101100eleq1d 2810 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102100fveq2d 6900 . . . . . . . . 9 (𝑓 = 𝐹 → (♯‘(𝑓 “ {0})) = (♯‘𝑅))
103 fveq2 6896 . . . . . . . . 9 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104102, 103breq12d 5162 . . . . . . . 8 (𝑓 = 𝐹 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105101, 104anbi12d 630 . . . . . . 7 (𝑓 = 𝐹 → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
10696, 105imbi12d 343 . . . . . 6 (𝑓 = 𝐹 → (((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
107106rspcv 3602 . . . . 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 234 . . . 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 578 . . 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 205  wa 394   = wceq 1533  wex 1773  wcel 2098  wne 2929  wral 3050  cdif 3941  c0 4322  {csn 4630   class class class wbr 5149   × cxp 5676  ccnv 5677  cima 5681   Fn wfn 6544  wf 6545  cfv 6549  (class class class)co 7419  Fincfn 8964  cc 11143  0cc0 11145  1c1 11146   + caddc 11148  cle 11286  0cn0 12510  chash 14333  0𝑝c0p 25659  Polycply 26180  degcdgr 26183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9671  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222  ax-pre-sup 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9472  df-inf 9473  df-oi 9540  df-dju 9931  df-card 9969  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-xnn0 12583  df-z 12597  df-uz 12861  df-rp 13015  df-fz 13525  df-fzo 13668  df-fl 13798  df-seq 14008  df-exp 14068  df-hash 14334  df-cj 15090  df-re 15091  df-im 15092  df-sqrt 15226  df-abs 15227  df-clim 15476  df-rlim 15477  df-sum 15677  df-0p 25660  df-ply 26184  df-idp 26185  df-coe 26186  df-dgr 26187  df-quot 26288
This theorem is referenced by:  vieta1lem2  26308  vieta1  26309  plyexmo  26310  aannenlem1  26325  aalioulem2  26330  basellem4  27081  dchrfi  27253
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