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Theorem fta1 26350
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 26272 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
32adantr 480 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4 eqeq2 2749 . . . . . . 7 (𝑥 = 0 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 0))
54imbi1d 341 . . . . . 6 (𝑥 = 0 → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
65ralbidv 3178 . . . . 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 341 . . . . . 6 (𝑥 = 𝑑 → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
98ralbidv 3178 . . . . 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 341 . . . . . 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 3178 . . . . 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 341 . . . . . 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 3178 . . . . 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 4790 . . . . . . . . . . 11 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
1716adantr 480 . . . . . . . . . 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 4131 . . . . . . . . . . . . . . . . 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 26285 . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 14 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = (ℂ × {(𝑓‘0)}))
2423fveq1d 6908 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
2519adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26 plyf 26237 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27 ffn 6736 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28 fniniseg 7080 . . . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓𝑥) = 0))
3130simprd 495 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = 0)
3230simpld 494 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑥 ∈ ℂ)
33 fvex 6919 . . . . . . . . . . . . . . . . . . 19 (𝑓‘0) ∈ V
3433fvconst2 7224 . . . . . . . . . . . . . . . . . 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 4638 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → {(𝑓‘0)} = {0})
3837xpeq2d 5715 . . . . . . . . . . . . . 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 25705 . . . . . . . . . . . . 13 0𝑝 = (ℂ × {0})
4139, 40eqtr4di 2795 . . . . . . . . . . . 12 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = 0𝑝)
4241ex 412 . . . . . . . . . . 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 4407 . . . . . . . 8 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑓 “ {0}) = ∅)
4645ex 412 . . . . . . 7 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (𝑓 “ {0}) = ∅))
47 dgrcl 26272 . . . . . . . . 9 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48 nn0ge0 12551 . . . . . . . . 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 9082 . . . . . . . . . . 11 ∅ ∈ Fin
5250, 51eqeltrdi 2849 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (𝑓 “ {0}) ∈ Fin)
5352biantrurd 532 . . . . . . . . 9 ((𝑓 “ {0}) = ∅ → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
54 fveq2 6906 . . . . . . . . . . 11 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = (♯‘∅))
55 hash0 14406 . . . . . . . . . . 11 (♯‘∅) = 0
5654, 55eqtrdi 2793 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = 0)
5756breq1d 5153 . . . . . . . . 9 ((𝑓 “ {0}) = ∅ → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
5853, 57bitr3d 281 . . . . . . . 8 ((𝑓 “ {0}) = ∅ → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ 0 ≤ (deg‘𝑓)))
5949, 58syl5ibrcom 247 . . . . . . 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 3063 . . . . 5 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
62 fveqeq2 6915 . . . . . . . 8 (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63 cnveq 5884 . . . . . . . . . . 11 (𝑓 = 𝑔𝑓 = 𝑔)
6463imaeq1d 6077 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓 “ {0}) = (𝑔 “ {0}))
6564eleq1d 2826 . . . . . . . . 9 (𝑓 = 𝑔 → ((𝑓 “ {0}) ∈ Fin ↔ (𝑔 “ {0}) ∈ Fin))
6664fveq2d 6910 . . . . . . . . . 10 (𝑓 = 𝑔 → (♯‘(𝑓 “ {0})) = (♯‘(𝑔 “ {0})))
67 fveq2 6906 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6866, 67breq12d 5156 . . . . . . . . 9 (𝑓 = 𝑔 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)))
6965, 68anbi12d 632 . . . . . . . 8 (𝑓 = 𝑔 → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
7062, 69imbi12d 344 . . . . . . 7 (𝑓 = 𝑔 → (((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)))))
7170cbvralvw 3237 . . . . . 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 247 . . . . . . . . . . 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 4353 . . . . . . . . . . 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 26349 . . . . . . . . . . . . 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 420 . . . . . . . . . . . 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 1933 . . . . . . . . . . 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 242 . . . . . . . . . 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 412 . . . . . . . 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 3154 . . . . . 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 242 . . . . 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 12713 . . . 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 26239 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
9493sseli 3979 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95 eldifsn 4786 . . . . 5 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96 fveqeq2 6915 . . . . . . 7 (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97 cnveq 5884 . . . . . . . . . . 11 (𝑓 = 𝐹𝑓 = 𝐹)
9897imaeq1d 6077 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓 “ {0}) = (𝐹 “ {0}))
99 fta1.1 . . . . . . . . . 10 𝑅 = (𝐹 “ {0})
10098, 99eqtr4di 2795 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ {0}) = 𝑅)
101100eleq1d 2826 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102100fveq2d 6910 . . . . . . . . 9 (𝑓 = 𝐹 → (♯‘(𝑓 “ {0})) = (♯‘𝑅))
103 fveq2 6906 . . . . . . . . 9 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104102, 103breq12d 5156 . . . . . . . 8 (𝑓 = 𝐹 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105101, 104anbi12d 632 . . . . . . 7 (𝑓 = 𝐹 → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
10696, 105imbi12d 344 . . . . . 6 (𝑓 = 𝐹 → (((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
107106rspcv 3618 . . . . 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 235 . . . 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 580 . . 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 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  cdif 3948  c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  ccnv 5684  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  cc 11153  0cc0 11155  1c1 11156   + caddc 11158  cle 11296  0cn0 12526  chash 14369  0𝑝c0p 25704  Polycply 26223  degcdgr 26226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-0p 25705  df-ply 26227  df-idp 26228  df-coe 26229  df-dgr 26230  df-quot 26333
This theorem is referenced by:  vieta1lem2  26353  vieta1  26354  plyexmo  26355  aannenlem1  26370  aalioulem2  26375  basellem4  27127  dchrfi  27299
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