Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fta1lem Structured version   Visualization version   GIF version

Theorem fta1lem 24913
 Description: Lemma for fta1 24914. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
fta1.1 𝑅 = (𝐹 “ {0})
fta1.2 (𝜑𝐷 ∈ ℕ0)
fta1.3 (𝜑𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}))
fta1.4 (𝜑 → (deg‘𝐹) = (𝐷 + 1))
fta1.5 (𝜑𝐴 ∈ (𝐹 “ {0}))
fta1.6 (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
Assertion
Ref Expression
fta1lem (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
Distinct variable groups:   𝐴,𝑔   𝐷,𝑔   𝑔,𝐹
Allowed substitution hints:   𝜑(𝑔)   𝑅(𝑔)

Proof of Theorem fta1lem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fta1.3 . . . . . . . . . 10 (𝜑𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}))
2 eldifsn 4680 . . . . . . . . . 10 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
31, 2sylib 221 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
43simpld 498 . . . . . . . 8 (𝜑𝐹 ∈ (Poly‘ℂ))
5 fta1.5 . . . . . . . . . 10 (𝜑𝐴 ∈ (𝐹 “ {0}))
6 plyf 24805 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7 ffn 6488 . . . . . . . . . . 11 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8 fniniseg 6808 . . . . . . . . . . 11 (𝐹 Fn ℂ → (𝐴 ∈ (𝐹 “ {0}) ↔ (𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0)))
94, 6, 7, 84syl 19 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐹 “ {0}) ↔ (𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0)))
105, 9mpbid 235 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0))
1110simpld 498 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
1210simprd 499 . . . . . . . 8 (𝜑 → (𝐹𝐴) = 0)
13 eqid 2798 . . . . . . . . 9 (Xpf − (ℂ × {𝐴})) = (Xpf − (ℂ × {𝐴}))
1413facth 24912 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
154, 11, 12, 14syl3anc 1368 . . . . . . 7 (𝜑𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
1615cnveqd 5711 . . . . . 6 (𝜑𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
1716imaeq1d 5896 . . . . 5 (𝜑 → (𝐹 “ {0}) = (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) “ {0}))
18 cnex 10610 . . . . . . 7 ℂ ∈ V
1918a1i 11 . . . . . 6 (𝜑 → ℂ ∈ V)
20 ssid 3937 . . . . . . . . 9 ℂ ⊆ ℂ
21 ax-1cn 10587 . . . . . . . . 9 1 ∈ ℂ
22 plyid 24816 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
2320, 21, 22mp2an 691 . . . . . . . 8 Xp ∈ (Poly‘ℂ)
24 plyconst 24813 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
2520, 11, 24sylancr 590 . . . . . . . 8 (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26 plysubcl 24829 . . . . . . . 8 ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
2723, 25, 26sylancr 590 . . . . . . 7 (𝜑 → (Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28 plyf 24805 . . . . . . 7 ((Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xpf − (ℂ × {𝐴})):ℂ⟶ℂ)
2927, 28syl 17 . . . . . 6 (𝜑 → (Xpf − (ℂ × {𝐴})):ℂ⟶ℂ)
3013plyremlem 24910 . . . . . . . . . . . 12 (𝐴 ∈ ℂ → ((Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (deg‘(Xpf − (ℂ × {𝐴}))) = 1 ∧ ((Xpf − (ℂ × {𝐴})) “ {0}) = {𝐴}))
3111, 30syl 17 . . . . . . . . . . 11 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (deg‘(Xpf − (ℂ × {𝐴}))) = 1 ∧ ((Xpf − (ℂ × {𝐴})) “ {0}) = {𝐴}))
3231simp2d 1140 . . . . . . . . . 10 (𝜑 → (deg‘(Xpf − (ℂ × {𝐴}))) = 1)
33 ax-1ne0 10598 . . . . . . . . . . 11 1 ≠ 0
3433a1i 11 . . . . . . . . . 10 (𝜑 → 1 ≠ 0)
3532, 34eqnetrd 3054 . . . . . . . . 9 (𝜑 → (deg‘(Xpf − (ℂ × {𝐴}))) ≠ 0)
36 fveq2 6646 . . . . . . . . . . 11 ((Xpf − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝐴}))) = (deg‘0𝑝))
37 dgr0 24869 . . . . . . . . . . 11 (deg‘0𝑝) = 0
3836, 37eqtrdi 2849 . . . . . . . . . 10 ((Xpf − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝐴}))) = 0)
3938necon3i 3019 . . . . . . . . 9 ((deg‘(Xpf − (ℂ × {𝐴}))) ≠ 0 → (Xpf − (ℂ × {𝐴})) ≠ 0𝑝)
4035, 39syl 17 . . . . . . . 8 (𝜑 → (Xpf − (ℂ × {𝐴})) ≠ 0𝑝)
41 quotcl2 24908 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝐴})) ≠ 0𝑝) → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
424, 27, 40, 41syl3anc 1368 . . . . . . 7 (𝜑 → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43 plyf 24805 . . . . . . 7 ((𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) → (𝐹 quot (Xpf − (ℂ × {𝐴}))):ℂ⟶ℂ)
4442, 43syl 17 . . . . . 6 (𝜑 → (𝐹 quot (Xpf − (ℂ × {𝐴}))):ℂ⟶ℂ)
45 ofmulrt 24888 . . . . . 6 ((ℂ ∈ V ∧ (Xpf − (ℂ × {𝐴})):ℂ⟶ℂ ∧ (𝐹 quot (Xpf − (ℂ × {𝐴}))):ℂ⟶ℂ) → (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) “ {0}) = (((Xpf − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
4619, 29, 44, 45syl3anc 1368 . . . . 5 (𝜑 → (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) “ {0}) = (((Xpf − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
4731simp3d 1141 . . . . . 6 (𝜑 → ((Xpf − (ℂ × {𝐴})) “ {0}) = {𝐴})
4847uneq1d 4089 . . . . 5 (𝜑 → (((Xpf − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
4917, 46, 483eqtrd 2837 . . . 4 (𝜑 → (𝐹 “ {0}) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
50 fta1.1 . . . 4 𝑅 = (𝐹 “ {0})
51 uncom 4080 . . . 4 (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}))
5249, 50, 513eqtr4g 2858 . . 3 (𝜑𝑅 = (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
5321a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℂ)
54 dgrcl 24840 . . . . . . . . 9 ((𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) ∈ ℕ0)
5542, 54syl 17 . . . . . . . 8 (𝜑 → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) ∈ ℕ0)
5655nn0cnd 11948 . . . . . . 7 (𝜑 → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) ∈ ℂ)
57 fta1.2 . . . . . . . 8 (𝜑𝐷 ∈ ℕ0)
5857nn0cnd 11948 . . . . . . 7 (𝜑𝐷 ∈ ℂ)
59 addcom 10818 . . . . . . . . 9 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
6021, 58, 59sylancr 590 . . . . . . . 8 (𝜑 → (1 + 𝐷) = (𝐷 + 1))
6115fveq2d 6650 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (deg‘((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴}))))))
62 fta1.4 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (𝐷 + 1))
633simprd 499 . . . . . . . . . . . 12 (𝜑𝐹 ≠ 0𝑝)
6415eqcomd 2804 . . . . . . . . . . . 12 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐹)
65 0cnd 10626 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ ℂ)
66 mul01 10811 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
6766adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
6819, 29, 65, 65, 67caofid1 7422 . . . . . . . . . . . . 13 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (ℂ × {0})) = (ℂ × {0}))
69 df-0p 24284 . . . . . . . . . . . . . 14 0𝑝 = (ℂ × {0})
7069oveq2i 7147 . . . . . . . . . . . . 13 ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝) = ((Xpf − (ℂ × {𝐴})) ∘f · (ℂ × {0}))
7168, 70, 693eqtr4g 2858 . . . . . . . . . . . 12 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝) = 0𝑝)
7263, 64, 713netr4d 3064 . . . . . . . . . . 11 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) ≠ ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝))
73 oveq2 7144 . . . . . . . . . . . 12 ((𝐹 quot (Xpf − (ℂ × {𝐴}))) = 0𝑝 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) = ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝))
7473necon3i 3019 . . . . . . . . . . 11 (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) ≠ ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝) → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ≠ 0𝑝)
7572, 74syl 17 . . . . . . . . . 10 (𝜑 → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ≠ 0𝑝)
76 eqid 2798 . . . . . . . . . . 11 (deg‘(Xpf − (ℂ × {𝐴}))) = (deg‘(Xpf − (ℂ × {𝐴})))
77 eqid 2798 . . . . . . . . . . 11 (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))
7876, 77dgrmul 24877 . . . . . . . . . 10 ((((Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝐴})) ≠ 0𝑝) ∧ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) ∧ (𝐹 quot (Xpf − (ℂ × {𝐴}))) ≠ 0𝑝)) → (deg‘((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴}))))) = ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
7927, 40, 42, 75, 78syl22anc 837 . . . . . . . . 9 (𝜑 → (deg‘((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴}))))) = ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
8061, 62, 793eqtr3d 2841 . . . . . . . 8 (𝜑 → (𝐷 + 1) = ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
8132oveq1d 7151 . . . . . . . 8 (𝜑 → ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))) = (1 + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
8260, 80, 813eqtrrd 2838 . . . . . . 7 (𝜑 → (1 + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))) = (1 + 𝐷))
8353, 56, 58, 82addcanad 10837 . . . . . 6 (𝜑 → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷)
84 fveqeq2 6655 . . . . . . . 8 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷))
85 cnveq 5709 . . . . . . . . . . 11 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → 𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))))
8685imaeq1d 5896 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (𝑔 “ {0}) = ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}))
8786eleq1d 2874 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → ((𝑔 “ {0}) ∈ Fin ↔ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
8886fveq2d 6650 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (♯‘(𝑔 “ {0})) = (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
89 fveq2 6646 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))
9088, 89breq12d 5044 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → ((♯‘(𝑔 “ {0})) ≤ (deg‘𝑔) ↔ (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
9187, 90anbi12d 633 . . . . . . . 8 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔)) ↔ (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))))
9284, 91imbi12d 348 . . . . . . 7 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (((deg‘𝑔) = 𝐷 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))) ↔ ((deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷 → (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))))
93 fta1.6 . . . . . . 7 (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((𝑔 “ {0}) ∈ Fin ∧ (♯‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
94 eldifsn 4680 . . . . . . . 8 ((𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) ∧ (𝐹 quot (Xpf − (ℂ × {𝐴}))) ≠ 0𝑝))
9542, 75, 94sylanbrc 586 . . . . . . 7 (𝜑 → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
9692, 93, 95rspcdva 3573 . . . . . 6 (𝜑 → ((deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷 → (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))))
9783, 96mpd 15 . . . . 5 (𝜑 → (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
9897simpld 498 . . . 4 (𝜑 → ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99 snfi 8580 . . . 4 {𝐴} ∈ Fin
100 unfi 8772 . . . 4 ((((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ {𝐴} ∈ Fin) → (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10198, 99, 100sylancl 589 . . 3 (𝜑 → (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10252, 101eqeltrd 2890 . 2 (𝜑𝑅 ∈ Fin)
10352fveq2d 6650 . . 3 (𝜑 → (♯‘𝑅) = (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104 hashcl 13716 . . . . . 6 ((((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℕ0)
105101, 104syl 17 . . . . 5 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℕ0)
106105nn0red 11947 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107 hashcl 13716 . . . . . . 7 (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ∈ ℕ0)
10898, 107syl 17 . . . . . 6 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ∈ ℕ0)
109108nn0red 11947 . . . . 5 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110 peano2re 10805 . . . . 5 ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ∈ ℝ → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ∈ ℝ)
111109, 110syl 17 . . . 4 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ∈ ℝ)
112 dgrcl 24840 . . . . . 6 (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
1134, 112syl 17 . . . . 5 (𝜑 → (deg‘𝐹) ∈ ℕ0)
114113nn0red 11947 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℝ)
115 hashun2 13743 . . . . . 6 ((((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ {𝐴} ∈ Fin) → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
11698, 99, 115sylancl 589 . . . . 5 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
117 hashsng 13729 . . . . . . 7 (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
11811, 117syl 17 . . . . . 6 (𝜑 → (♯‘{𝐴}) = 1)
119118oveq2d 7152 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1))
120116, 119breqtrd 5057 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1))
12157nn0red 11947 . . . . . 6 (𝜑𝐷 ∈ ℝ)
122 1red 10634 . . . . . 6 (𝜑 → 1 ∈ ℝ)
12397simprd 499 . . . . . . 7 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))
124123, 83breqtrd 5057 . . . . . 6 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125109, 121, 122, 124leadd1dd 11246 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126125, 62breqtrrd 5059 . . . 4 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127106, 111, 114, 120, 126letrd 10789 . . 3 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128103, 127eqbrtrd 5053 . 2 (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129102, 128jca 515 1 (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ⊆ wss 3881  {csn 4525   class class class wbr 5031   × cxp 5518  ◡ccnv 5519   “ cima 5523   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136   ∘f cof 7389  Fincfn 8495  ℂcc 10527  ℝcr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534   ≤ cle 10668   − cmin 10862  ℕ0cn0 11888  ♯chash 13689  0𝑝c0p 24283  Polycply 24791  Xpcidp 24792  degcdgr 24794   quot cquot 24896 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-inf2 9091  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-of 7391  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-map 8394  df-pm 8395  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-sup 8893  df-inf 8894  df-oi 8961  df-dju 9317  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11629  df-2 11691  df-3 11692  df-n0 11889  df-xnn0 11959  df-z 11973  df-uz 12235  df-rp 12381  df-fz 12889  df-fzo 13032  df-fl 13160  df-seq 13368  df-exp 13429  df-hash 13690  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-rlim 14841  df-sum 15038  df-0p 24284  df-ply 24795  df-idp 24796  df-coe 24797  df-dgr 24798  df-quot 24897 This theorem is referenced by:  fta1  24914
 Copyright terms: Public domain W3C validator