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

Theorem fta1lem 24583
 Description: Lemma for fta1 24584. (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 4632 . . . . . . . . . 10 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
31, 2sylib 219 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
43simpld 495 . . . . . . . 8 (𝜑𝐹 ∈ (Poly‘ℂ))
5 fta1.5 . . . . . . . . . 10 (𝜑𝐴 ∈ (𝐹 “ {0}))
6 plyf 24475 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7 ffn 6389 . . . . . . . . . . 11 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8 fniniseg 6702 . . . . . . . . . . 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 233 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0))
1110simpld 495 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
1210simprd 496 . . . . . . . 8 (𝜑 → (𝐹𝐴) = 0)
13 eqid 2797 . . . . . . . . 9 (Xp𝑓 − (ℂ × {𝐴})) = (Xp𝑓 − (ℂ × {𝐴}))
1413facth 24582 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
154, 11, 12, 14syl3anc 1364 . . . . . . 7 (𝜑𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
1615cnveqd 5639 . . . . . 6 (𝜑𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
1716imaeq1d 5812 . . . . 5 (𝜑 → (𝐹 “ {0}) = (((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) “ {0}))
18 cnex 10471 . . . . . . 7 ℂ ∈ V
1918a1i 11 . . . . . 6 (𝜑 → ℂ ∈ V)
20 ssid 3916 . . . . . . . . 9 ℂ ⊆ ℂ
21 ax-1cn 10448 . . . . . . . . 9 1 ∈ ℂ
22 plyid 24486 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
2320, 21, 22mp2an 688 . . . . . . . 8 Xp ∈ (Poly‘ℂ)
24 plyconst 24483 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
2520, 11, 24sylancr 587 . . . . . . . 8 (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26 plysubcl 24499 . . . . . . . 8 ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
2723, 25, 26sylancr 587 . . . . . . 7 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28 plyf 24475 . . . . . . 7 ((Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xp𝑓 − (ℂ × {𝐴})):ℂ⟶ℂ)
2927, 28syl 17 . . . . . 6 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})):ℂ⟶ℂ)
3013plyremlem 24580 . . . . . . . . . . . 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 1136 . . . . . . . . . 10 (𝜑 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = 1)
33 ax-1ne0 10459 . . . . . . . . . . 11 1 ≠ 0
3433a1i 11 . . . . . . . . . 10 (𝜑 → 1 ≠ 0)
3532, 34eqnetrd 3053 . . . . . . . . 9 (𝜑 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) ≠ 0)
36 fveq2 6545 . . . . . . . . . . 11 ((Xp𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = (deg‘0𝑝))
37 dgr0 24539 . . . . . . . . . . 11 (deg‘0𝑝) = 0
3836, 37syl6eq 2849 . . . . . . . . . 10 ((Xp𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = 0)
3938necon3i 3018 . . . . . . . . 9 ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) ≠ 0 → (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝)
4035, 39syl 17 . . . . . . . 8 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝)
41 quotcl2 24578 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝) → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
424, 27, 40, 41syl3anc 1364 . . . . . . 7 (𝜑 → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43 plyf 24475 . . . . . . 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 24558 . . . . . 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 1364 . . . . 5 (𝜑 → (((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) “ {0}) = (((Xp𝑓 − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
4731simp3d 1137 . . . . . 6 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) “ {0}) = {𝐴})
4847uneq1d 4065 . . . . 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 4056 . . . 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 24510 . . . . . . . . 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 11811 . . . . . . 7 (𝜑 → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ∈ ℂ)
57 fta1.2 . . . . . . . 8 (𝜑𝐷 ∈ ℕ0)
5857nn0cnd 11811 . . . . . . 7 (𝜑𝐷 ∈ ℂ)
59 addcom 10679 . . . . . . . . 9 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
6021, 58, 59sylancr 587 . . . . . . . 8 (𝜑 → (1 + 𝐷) = (𝐷 + 1))
6115fveq2d 6549 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (deg‘((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
62 fta1.4 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (𝐷 + 1))
633simprd 496 . . . . . . . . . . . 12 (𝜑𝐹 ≠ 0𝑝)
6415eqcomd 2803 . . . . . . . . . . . 12 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐹)
65 0cnd 10487 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ ℂ)
66 mul01 10672 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
6766adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
6819, 29, 65, 65, 67caofid1 7304 . . . . . . . . . . . . 13 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0})) = (ℂ × {0}))
69 df-0p 23958 . . . . . . . . . . . . . 14 0𝑝 = (ℂ × {0})
7069oveq2i 7034 . . . . . . . . . . . . 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 3063 . . . . . . . . . . 11 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ≠ ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
73 oveq2 7031 . . . . . . . . . . . 12 ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) = 0𝑝 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
7473necon3i 3018 . . . . . . . . . . 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 2797 . . . . . . . . . . 11 (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = (deg‘(Xp𝑓 − (ℂ × {𝐴})))
77 eqid 2797 . . . . . . . . . . 11 (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))
7876, 77dgrmul 24547 . . . . . . . . . 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 835 . . . . . . . . 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 7038 . . . . . . . 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 10698 . . . . . 6 (𝜑 → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐷)
84 fveqeq2 6554 . . . . . . . 8 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐷))
85 cnveq 5637 . . . . . . . . . . 11 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → 𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))
8685imaeq1d 5812 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (𝑔 “ {0}) = ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}))
8786eleq1d 2869 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → ((𝑔 “ {0}) ∈ Fin ↔ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
8886fveq2d 6549 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (♯‘(𝑔 “ {0})) = (♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
89 fveq2 6545 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
9088, 89breq12d 4981 . . . . . . . . 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 630 . . . . . . . 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 346 . . . . . . 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 4632 . . . . . . . 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 583 . . . . . . 7 (𝜑 → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
9692, 93, 95rspcdva 3567 . . . . . 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 495 . . . 4 (𝜑 → ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99 snfi 8449 . . . 4 {𝐴} ∈ Fin
100 unfi 8638 . . . 4 ((((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ {𝐴} ∈ Fin) → (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10198, 99, 100sylancl 586 . . 3 (𝜑 → (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10252, 101eqeltrd 2885 . 2 (𝜑𝑅 ∈ Fin)
10352fveq2d 6549 . . 3 (𝜑 → (♯‘𝑅) = (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104 hashcl 13571 . . . . . 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 11810 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107 hashcl 13571 . . . . . . 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 11810 . . . . 5 (𝜑 → (♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110 peano2re 10666 . . . . 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 24510 . . . . . 6 (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
1134, 112syl 17 . . . . 5 (𝜑 → (deg‘𝐹) ∈ ℕ0)
114113nn0red 11810 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℝ)
115 hashun2 13596 . . . . . 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 586 . . . . 5 (𝜑 → (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
117 hashsng 13583 . . . . . . 7 (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
11811, 117syl 17 . . . . . 6 (𝜑 → (♯‘{𝐴}) = 1)
119118oveq2d 7039 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
120116, 119breqtrd 4994 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
12157nn0red 11810 . . . . . 6 (𝜑𝐷 ∈ ℝ)
122 1red 10495 . . . . . 6 (𝜑 → 1 ∈ ℝ)
12397simprd 496 . . . . . . 7 (𝜑 → (♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
124123, 83breqtrd 4994 . . . . . 6 (𝜑 → (♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125109, 121, 122, 124leadd1dd 11108 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126125, 62breqtrrd 4996 . . . 4 (𝜑 → ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127106, 111, 114, 120, 126letrd 10650 . . 3 (𝜑 → (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128103, 127eqbrtrd 4990 . 2 (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129102, 128jca 512 1 (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083   ≠ wne 2986  ∀wral 3107  Vcvv 3440   ∖ cdif 3862   ∪ cun 3863   ⊆ wss 3865  {csn 4478   class class class wbr 4968   × cxp 5448  ◡ccnv 5449   “ cima 5453   Fn wfn 6227  ⟶wf 6228  ‘cfv 6232  (class class class)co 7023   ∘𝑓 cof 7272  Fincfn 8364  ℂcc 10388  ℝcr 10389  0cc0 10390  1c1 10391   + caddc 10393   · cmul 10395   ≤ cle 10529   − cmin 10723  ℕ0cn0 11751  ♯chash 13544  0𝑝c0p 23957  Polycply 24461  Xpcidp 24462  degcdgr 24464   quot cquot 24566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468  ax-addf 10469 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-map 8265  df-pm 8266  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-inf 8760  df-oi 8827  df-dju 9183  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-n0 11752  df-xnn0 11822  df-z 11836  df-uz 12098  df-rp 12244  df-fz 12747  df-fzo 12888  df-fl 13016  df-seq 13224  df-exp 13284  df-hash 13545  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-clim 14683  df-rlim 14684  df-sum 14881  df-0p 23958  df-ply 24465  df-idp 24466  df-coe 24467  df-dgr 24468  df-quot 24567 This theorem is referenced by:  fta1  24584
 Copyright terms: Public domain W3C validator