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Theorem fta1 24368
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 2765 . 2 (deg‘𝐹) = (deg‘𝐹)
2 dgrcl 24294 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
32adantr 472 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (deg‘𝐹) ∈ ℕ0)
4 eqeq2 2776 . . . . . . 7 (𝑥 = 0 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 0))
54imbi1d 332 . . . . . 6 (𝑥 = 0 → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
65ralbidv 3133 . . . . 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 2776 . . . . . . 7 (𝑥 = 𝑑 → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = 𝑑))
87imbi1d 332 . . . . . 6 (𝑥 = 𝑑 → (((deg‘𝑓) = 𝑥 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))))
98ralbidv 3133 . . . . 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 2776 . . . . . . 7 (𝑥 = (𝑑 + 1) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (𝑑 + 1)))
1110imbi1d 332 . . . . . 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 3133 . . . . 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 2776 . . . . . . 7 (𝑥 = (deg‘𝐹) → ((deg‘𝑓) = 𝑥 ↔ (deg‘𝑓) = (deg‘𝐹)))
1413imbi1d 332 . . . . . 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 3133 . . . . 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 4478 . . . . . . . . . . 11 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ≠ 0𝑝)
1716adantr 472 . . . . . . . . . 10 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ≠ 0𝑝)
18 simplr 785 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (deg‘𝑓) = 0)
19 eldifi 3896 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℂ))
2019ad2antrr 717 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 ∈ (Poly‘ℂ))
21 0dgrb 24307 . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . 14 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = (ℂ × {(𝑓‘0)}))
2423fveq1d 6381 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = ((ℂ × {(𝑓‘0)})‘𝑥))
2519adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → 𝑓 ∈ (Poly‘ℂ))
26 plyf 24259 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
27 ffn 6225 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
28 fniniseg 6532 . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑥 ∈ ℂ ∧ (𝑓𝑥) = 0))
3130simprd 489 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓𝑥) = 0)
3230simpld 488 . . . . . . . . . . . . . . . . . 18 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑥 ∈ ℂ)
33 fvex 6392 . . . . . . . . . . . . . . . . . . 19 (𝑓‘0) ∈ V
3433fvconst2 6666 . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (𝑓‘0) = 0)
3736sneqd 4348 . . . . . . . . . . . . . . 15 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → {(𝑓‘0)} = {0})
3837xpeq2d 5309 . . . . . . . . . . . . . 14 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → (ℂ × {(𝑓‘0)}) = (ℂ × {0}))
3923, 38eqtrd 2799 . . . . . . . . . . . . 13 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = (ℂ × {0}))
40 df-0p 23742 . . . . . . . . . . . . 13 0𝑝 = (ℂ × {0})
4139, 40syl6eqr 2817 . . . . . . . . . . . 12 (((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) ∧ 𝑥 ∈ (𝑓 “ {0})) → 𝑓 = 0𝑝)
4241ex 401 . . . . . . . . . . 11 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑥 ∈ (𝑓 “ {0}) → 𝑓 = 0𝑝))
4342necon3ad 2950 . . . . . . . . . 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 4143 . . . . . . . 8 ((𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ∧ (deg‘𝑓) = 0) → (𝑓 “ {0}) = ∅)
4645ex 401 . . . . . . 7 (𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝}) → ((deg‘𝑓) = 0 → (𝑓 “ {0}) = ∅))
47 dgrcl 24294 . . . . . . . . 9 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
48 nn0ge0 11569 . . . . . . . . 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 8399 . . . . . . . . . . 11 ∅ ∈ Fin
5250, 51syl6eqel 2852 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (𝑓 “ {0}) ∈ Fin)
5352biantrurd 528 . . . . . . . . 9 ((𝑓 “ {0}) = ∅ → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))))
54 fveq2 6379 . . . . . . . . . . 11 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = (♯‘∅))
55 hash0 13365 . . . . . . . . . . 11 (♯‘∅) = 0
5654, 55syl6eq 2815 . . . . . . . . . 10 ((𝑓 “ {0}) = ∅ → (♯‘(𝑓 “ {0})) = 0)
5756breq1d 4821 . . . . . . . . 9 ((𝑓 “ {0}) = ∅ → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
5853, 57bitr3d 272 . . . . . . . 8 ((𝑓 “ {0}) = ∅ → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ 0 ≤ (deg‘𝑓)))
5949, 58syl5ibrcom 238 . . . . . . 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 3069 . . . . 5 𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑓) = 0 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
62 fveqeq2 6388 . . . . . . . 8 (𝑓 = 𝑔 → ((deg‘𝑓) = 𝑑 ↔ (deg‘𝑔) = 𝑑))
63 cnveq 5466 . . . . . . . . . . 11 (𝑓 = 𝑔𝑓 = 𝑔)
6463imaeq1d 5649 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓 “ {0}) = (𝑔 “ {0}))
6564eleq1d 2829 . . . . . . . . 9 (𝑓 = 𝑔 → ((𝑓 “ {0}) ∈ Fin ↔ (𝑔 “ {0}) ∈ Fin))
6664fveq2d 6383 . . . . . . . . . 10 (𝑓 = 𝑔 → (♯‘(𝑓 “ {0})) = (♯‘(𝑔 “ {0})))
67 fveq2 6379 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6866, 67breq12d 4824 . . . . . . . . 9 (𝑓 = 𝑔 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)))
6965, 68anbi12d 624 . . . . . . . 8 (𝑓 = 𝑔 → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
7062, 69imbi12d 335 . . . . . . 7 (𝑓 = 𝑔 → (((deg‘𝑓) = 𝑑 → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝑔) = 𝑑 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)))))
7170cbvralv 3319 . . . . . 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 718 . . . . . . . . . . . 12 (((𝑑 ∈ ℕ0𝑓 ∈ ((Poly‘ℂ) ∖ {0𝑝})) ∧ (deg‘𝑓) = (𝑑 + 1)) → 0 ≤ (deg‘𝑓))
7372, 58syl5ibrcom 238 . . . . . . . . . . 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 4097 . . . . . . . . . . 11 ((𝑓 “ {0}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑓 “ {0}))
76 eqid 2765 . . . . . . . . . . . . . 14 (𝑓 “ {0}) = (𝑓 “ {0})
77 simplll 791 . . . . . . . . . . . . . 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 793 . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . 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 24367 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 2028 . . . . . . . . . . 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 233 . . . . . . . . . 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 3023 . . . . . . . . 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 401 . . . . . . . 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 3116 . . . . . 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 233 . . . . 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 11724 . . . 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 24261 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
9493sseli 3759 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
95 eldifsn 4474 . . . . 5 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
96 fveqeq2 6388 . . . . . . 7 (𝑓 = 𝐹 → ((deg‘𝑓) = (deg‘𝐹) ↔ (deg‘𝐹) = (deg‘𝐹)))
97 cnveq 5466 . . . . . . . . . . 11 (𝑓 = 𝐹𝑓 = 𝐹)
9897imaeq1d 5649 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓 “ {0}) = (𝐹 “ {0}))
99 fta1.1 . . . . . . . . . 10 𝑅 = (𝐹 “ {0})
10098, 99syl6eqr 2817 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ {0}) = 𝑅)
101100eleq1d 2829 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓 “ {0}) ∈ Fin ↔ 𝑅 ∈ Fin))
102100fveq2d 6383 . . . . . . . . 9 (𝑓 = 𝐹 → (♯‘(𝑓 “ {0})) = (♯‘𝑅))
103 fveq2 6379 . . . . . . . . 9 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
104102, 103breq12d 4824 . . . . . . . 8 (𝑓 = 𝐹 → ((♯‘(𝑓 “ {0})) ≤ (deg‘𝑓) ↔ (♯‘𝑅) ≤ (deg‘𝐹)))
105101, 104anbi12d 624 . . . . . . 7 (𝑓 = 𝐹 → (((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)) ↔ (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹))))
10696, 105imbi12d 335 . . . . . 6 (𝑓 = 𝐹 → (((deg‘𝑓) = (deg‘𝐹) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓))) ↔ ((deg‘𝐹) = (deg‘𝐹) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))))
107106rspcv 3458 . . . . 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 226 . . . 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 575 . . 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 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  cdif 3731  c0 4081  {csn 4336   class class class wbr 4811   × cxp 5277  ccnv 5278  cima 5282   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6846  Fincfn 8164  cc 10191  0cc0 10193  1c1 10194   + caddc 10196  cle 10333  0cn0 11542  chash 13326  0𝑝c0p 23741  Polycply 24245  degcdgr 24248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-xnn0 11615  df-z 11629  df-uz 11892  df-rp 12034  df-fz 12539  df-fzo 12679  df-fl 12806  df-seq 13014  df-exp 13073  df-hash 13327  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-clim 14518  df-rlim 14519  df-sum 14716  df-0p 23742  df-ply 24249  df-idp 24250  df-coe 24251  df-dgr 24252  df-quot 24351
This theorem is referenced by:  vieta1lem2  24371  vieta1  24372  plyexmo  24373  aannenlem1  24388  aalioulem2  24393  basellem4  25115  basellem5  25116  dchrfi  25285
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