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Theorem fta1 25668
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 2736 . 2 (deg‘𝐹) = (deg‘𝐹)
2 dgrcl 25594 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
32adantr 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4 eqeq2 2748 . . . . . . 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 3174 . . . . 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 2748 . . . . . . 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 3174 . . . . 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 2748 . . . . . . 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 3174 . . . . 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 2748 . . . . . . 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 3174 . . . . 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 4750 . . . . . . . . . . 11 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
1716adantr 481 . . . . . . . . . 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 4086 . . . . . . . . . . . . . . . . 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 25607 . . . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
2519adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26 plyf 25559 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27 ffn 6668 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28 fniniseg 7010 . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓𝑥) = 0))
3130simprd 496 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = 0)
3230simpld 495 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑥 ∈ ℂ)
33 fvex 6855 . . . . . . . . . . . . . . . . . . 19 (𝑓‘0) ∈ V
3433fvconst2 7153 . . . . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓‘0) = 0)
3736sneqd 4598 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → {(𝑓‘0)} = {0})
3837xpeq2d 5663 . . . . . . . . . . . . . 14 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (ℂ × {(𝑓‘0)}) = (ℂ × {0}))
3923, 38eqtrd 2776 . . . . . . . . . . . . 13 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = (ℂ × {0}))
40 df-0p 25034 . . . . . . . . . . . . 13 0𝑝 = (ℂ × {0})
4139, 40eqtr4di 2794 . . . . . . . . . . . 12 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = 0𝑝)
4241ex 413 . . . . . . . . . . 11 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (𝑓 “ {0}) → 𝑓 = 0𝑝))
4342necon3ad 2956 . . . . . . . . . 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 4364 . . . . . . . 8 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑓 “ {0}) = ∅)
4645ex 413 . . . . . . 7 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (𝑓 “ {0}) = ∅))
47 dgrcl 25594 . . . . . . . . 9 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48 nn0ge0 12438 . . . . . . . . 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 9115 . . . . . . . . . . 11 ∅ ∈ Fin
5250, 51eqeltrdi 2846 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (𝑓 “ {0}) ∈ Fin)
5352biantrurd 533 . . . . . . . . 9 ((𝑓 “ {0}) = ∅ → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
54 fveq2 6842 . . . . . . . . . . 11 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = (♯‘∅))
55 hash0 14267 . . . . . . . . . . 11 (♯‘∅) = 0
5654, 55eqtrdi 2792 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = 0)
5756breq1d 5115 . . . . . . . . 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 3066 . . . . 5 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
62 fveqeq2 6851 . . . . . . . 8 (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63 cnveq 5829 . . . . . . . . . . 11 (𝑓 = 𝑔𝑓 = 𝑔)
6463imaeq1d 6012 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓 “ {0}) = (𝑔 “ {0}))
6564eleq1d 2822 . . . . . . . . 9 (𝑓 = 𝑔 → ((𝑓 “ {0}) ∈ Fin ↔ (𝑔 “ {0}) ∈ Fin))
6664fveq2d 6846 . . . . . . . . . 10 (𝑓 = 𝑔 → (♯‘(𝑓 “ {0})) = (♯‘(𝑔 “ {0})))
67 fveq2 6842 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6866, 67breq12d 5118 . . . . . . . . 9 (𝑓 = 𝑔 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)))
6965, 68anbi12d 631 . . . . . . . 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 3225 . . . . . 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 4306 . . . . . . . . . . 11 ((𝑓 “ {0}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑓 “ {0}))
76 eqid 2736 . . . . . . . . . . . . . 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 25667 . . . . . . . . . . . . 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 421 . . . . . . . . . . . 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 1936 . . . . . . . . . . 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 3031 . . . . . . . . 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 413 . . . . . . . 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 3151 . . . . . 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 12598 . . . 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 25561 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
9493sseli 3940 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95 eldifsn 4747 . . . . 5 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96 fveqeq2 6851 . . . . . . 7 (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97 cnveq 5829 . . . . . . . . . . 11 (𝑓 = 𝐹𝑓 = 𝐹)
9897imaeq1d 6012 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓 “ {0}) = (𝐹 “ {0}))
99 fta1.1 . . . . . . . . . 10 𝑅 = (𝐹 “ {0})
10098, 99eqtr4di 2794 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ {0}) = 𝑅)
101100eleq1d 2822 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102100fveq2d 6846 . . . . . . . . 9 (𝑓 = 𝐹 → (♯‘(𝑓 “ {0})) = (♯‘𝑅))
103 fveq2 6842 . . . . . . . . 9 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104102, 103breq12d 5118 . . . . . . . 8 (𝑓 = 𝐹 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105101, 104anbi12d 631 . . . . . . 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 3577 . . . . 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 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 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  cdif 3907  c0 4282  {csn 4586   class class class wbr 5105   × cxp 5631  ccnv 5632  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  cc 11049  0cc0 11051  1c1 11052   + caddc 11054  cle 11190  0cn0 12413  chash 14230  0𝑝c0p 25033  Polycply 25545  degcdgr 25548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-0p 25034  df-ply 25549  df-idp 25550  df-coe 25551  df-dgr 25552  df-quot 25651
This theorem is referenced by:  vieta1lem2  25671  vieta1  25672  plyexmo  25673  aannenlem1  25688  aalioulem2  25693  basellem4  26433  dchrfi  26603
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