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Theorem fta1lem 25619
Description: Lemma for fta1 25620. (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 4746 . . . . . . . . . 10 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
31, 2sylib 217 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
43simpld 496 . . . . . . . 8 (𝜑𝐹 ∈ (Poly‘ℂ))
5 fta1.5 . . . . . . . . . 10 (𝜑𝐴 ∈ (𝐹 “ {0}))
6 plyf 25511 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7 ffn 6666 . . . . . . . . . . 11 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8 fniniseg 7008 . . . . . . . . . . 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 231 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0))
1110simpld 496 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
1210simprd 497 . . . . . . . 8 (𝜑 → (𝐹𝐴) = 0)
13 eqid 2738 . . . . . . . . 9 (Xpf − (ℂ × {𝐴})) = (Xpf − (ℂ × {𝐴}))
1413facth 25618 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
154, 11, 12, 14syl3anc 1372 . . . . . . 7 (𝜑𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
1615cnveqd 5830 . . . . . 6 (𝜑𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
1716imaeq1d 6011 . . . . 5 (𝜑 → (𝐹 “ {0}) = (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) “ {0}))
18 cnex 11091 . . . . . . 7 ℂ ∈ V
1918a1i 11 . . . . . 6 (𝜑 → ℂ ∈ V)
20 ssid 3965 . . . . . . . . 9 ℂ ⊆ ℂ
21 ax-1cn 11068 . . . . . . . . 9 1 ∈ ℂ
22 plyid 25522 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
2320, 21, 22mp2an 691 . . . . . . . 8 Xp ∈ (Poly‘ℂ)
24 plyconst 25519 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
2520, 11, 24sylancr 588 . . . . . . . 8 (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26 plysubcl 25535 . . . . . . . 8 ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
2723, 25, 26sylancr 588 . . . . . . 7 (𝜑 → (Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28 plyf 25511 . . . . . . 7 ((Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xpf − (ℂ × {𝐴})):ℂ⟶ℂ)
2927, 28syl 17 . . . . . 6 (𝜑 → (Xpf − (ℂ × {𝐴})):ℂ⟶ℂ)
3013plyremlem 25616 . . . . . . . . . . . 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 1144 . . . . . . . . . 10 (𝜑 → (deg‘(Xpf − (ℂ × {𝐴}))) = 1)
33 ax-1ne0 11079 . . . . . . . . . . 11 1 ≠ 0
3433a1i 11 . . . . . . . . . 10 (𝜑 → 1 ≠ 0)
3532, 34eqnetrd 3010 . . . . . . . . 9 (𝜑 → (deg‘(Xpf − (ℂ × {𝐴}))) ≠ 0)
36 fveq2 6840 . . . . . . . . . . 11 ((Xpf − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝐴}))) = (deg‘0𝑝))
37 dgr0 25575 . . . . . . . . . . 11 (deg‘0𝑝) = 0
3836, 37eqtrdi 2794 . . . . . . . . . 10 ((Xpf − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝐴}))) = 0)
3938necon3i 2975 . . . . . . . . 9 ((deg‘(Xpf − (ℂ × {𝐴}))) ≠ 0 → (Xpf − (ℂ × {𝐴})) ≠ 0𝑝)
4035, 39syl 17 . . . . . . . 8 (𝜑 → (Xpf − (ℂ × {𝐴})) ≠ 0𝑝)
41 quotcl2 25614 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝐴})) ≠ 0𝑝) → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
424, 27, 40, 41syl3anc 1372 . . . . . . 7 (𝜑 → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43 plyf 25511 . . . . . . 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 25594 . . . . . 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 1372 . . . . 5 (𝜑 → (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) “ {0}) = (((Xpf − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
4731simp3d 1145 . . . . . 6 (𝜑 → ((Xpf − (ℂ × {𝐴})) “ {0}) = {𝐴})
4847uneq1d 4121 . . . . 5 (𝜑 → (((Xpf − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
4917, 46, 483eqtrd 2782 . . . 4 (𝜑 → (𝐹 “ {0}) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
50 fta1.1 . . . 4 𝑅 = (𝐹 “ {0})
51 uncom 4112 . . . 4 (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}))
5249, 50, 513eqtr4g 2803 . . 3 (𝜑𝑅 = (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
5321a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℂ)
54 dgrcl 25546 . . . . . . . . 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 12434 . . . . . . 7 (𝜑 → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) ∈ ℂ)
57 fta1.2 . . . . . . . 8 (𝜑𝐷 ∈ ℕ0)
5857nn0cnd 12434 . . . . . . 7 (𝜑𝐷 ∈ ℂ)
59 addcom 11300 . . . . . . . . 9 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
6021, 58, 59sylancr 588 . . . . . . . 8 (𝜑 → (1 + 𝐷) = (𝐷 + 1))
6115fveq2d 6844 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (deg‘((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴}))))))
62 fta1.4 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (𝐷 + 1))
633simprd 497 . . . . . . . . . . . 12 (𝜑𝐹 ≠ 0𝑝)
6415eqcomd 2744 . . . . . . . . . . . 12 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐹)
65 0cnd 11107 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ ℂ)
66 mul01 11293 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
6766adantl 483 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
6819, 29, 65, 65, 67caofid1 7643 . . . . . . . . . . . . 13 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (ℂ × {0})) = (ℂ × {0}))
69 df-0p 24986 . . . . . . . . . . . . . 14 0𝑝 = (ℂ × {0})
7069oveq2i 7363 . . . . . . . . . . . . 13 ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝) = ((Xpf − (ℂ × {𝐴})) ∘f · (ℂ × {0}))
7168, 70, 693eqtr4g 2803 . . . . . . . . . . . 12 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝) = 0𝑝)
7263, 64, 713netr4d 3020 . . . . . . . . . . 11 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) ≠ ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝))
73 oveq2 7360 . . . . . . . . . . . 12 ((𝐹 quot (Xpf − (ℂ × {𝐴}))) = 0𝑝 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) = ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝))
7473necon3i 2975 . . . . . . . . . . 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 2738 . . . . . . . . . . 11 (deg‘(Xpf − (ℂ × {𝐴}))) = (deg‘(Xpf − (ℂ × {𝐴})))
77 eqid 2738 . . . . . . . . . . 11 (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))
7876, 77dgrmul 25583 . . . . . . . . . 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 838 . . . . . . . . 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 2786 . . . . . . . 8 (𝜑 → (𝐷 + 1) = ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
8132oveq1d 7367 . . . . . . . 8 (𝜑 → ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))) = (1 + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
8260, 80, 813eqtrrd 2783 . . . . . . 7 (𝜑 → (1 + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))) = (1 + 𝐷))
8353, 56, 58, 82addcanad 11319 . . . . . 6 (𝜑 → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷)
84 fveqeq2 6849 . . . . . . . 8 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷))
85 cnveq 5828 . . . . . . . . . . 11 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → 𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))))
8685imaeq1d 6011 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (𝑔 “ {0}) = ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}))
8786eleq1d 2823 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → ((𝑔 “ {0}) ∈ Fin ↔ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
8886fveq2d 6844 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (♯‘(𝑔 “ {0})) = (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
89 fveq2 6840 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))
9088, 89breq12d 5117 . . . . . . . . 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 632 . . . . . . . 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 345 . . . . . . 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 4746 . . . . . . . 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 584 . . . . . . 7 (𝜑 → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
9692, 93, 95rspcdva 3581 . . . . . 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 496 . . . 4 (𝜑 → ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99 snfi 8947 . . . 4 {𝐴} ∈ Fin
100 unfi 9075 . . . 4 ((((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ {𝐴} ∈ Fin) → (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10198, 99, 100sylancl 587 . . 3 (𝜑 → (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10252, 101eqeltrd 2839 . 2 (𝜑𝑅 ∈ Fin)
10352fveq2d 6844 . . 3 (𝜑 → (♯‘𝑅) = (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104 hashcl 14210 . . . . . 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 12433 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107 hashcl 14210 . . . . . . 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 12433 . . . . 5 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110 peano2re 11287 . . . . 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 25546 . . . . . 6 (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
1134, 112syl 17 . . . . 5 (𝜑 → (deg‘𝐹) ∈ ℕ0)
114113nn0red 12433 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℝ)
115 hashun2 14237 . . . . . 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 587 . . . . 5 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
117 hashsng 14223 . . . . . . 7 (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
11811, 117syl 17 . . . . . 6 (𝜑 → (♯‘{𝐴}) = 1)
119118oveq2d 7368 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1))
120116, 119breqtrd 5130 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1))
12157nn0red 12433 . . . . . 6 (𝜑𝐷 ∈ ℝ)
122 1red 11115 . . . . . 6 (𝜑 → 1 ∈ ℝ)
12397simprd 497 . . . . . . 7 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))
124123, 83breqtrd 5130 . . . . . 6 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125109, 121, 122, 124leadd1dd 11728 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126125, 62breqtrrd 5132 . . . 4 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127106, 111, 114, 120, 126letrd 11271 . . 3 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128103, 127eqbrtrd 5126 . 2 (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129102, 128jca 513 1 (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2942  wral 3063  Vcvv 3444  cdif 3906  cun 3907  wss 3909  {csn 4585   class class class wbr 5104   × cxp 5630  ccnv 5631  cima 5635   Fn wfn 6489  wf 6490  cfv 6494  (class class class)co 7352  f cof 7608  Fincfn 8842  cc 11008  cr 11009  0cc0 11010  1c1 11011   + caddc 11013   · cmul 11015  cle 11149  cmin 11344  0cn0 12372  chash 14184  0𝑝c0p 24985  Polycply 25497  Xpcidp 25498  degcdgr 25500   quot cquot 25602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665  ax-inf2 9536  ax-cnex 11066  ax-resscn 11067  ax-1cn 11068  ax-icn 11069  ax-addcl 11070  ax-addrcl 11071  ax-mulcl 11072  ax-mulrcl 11073  ax-mulcom 11074  ax-addass 11075  ax-mulass 11076  ax-distr 11077  ax-i2m1 11078  ax-1ne0 11079  ax-1rid 11080  ax-rnegex 11081  ax-rrecex 11082  ax-cnre 11083  ax-pre-lttri 11084  ax-pre-lttrn 11085  ax-pre-ltadd 11086  ax-pre-mulgt0 11087  ax-pre-sup 11088
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-se 5588  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7308  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7610  df-om 7796  df-1st 7914  df-2nd 7915  df-frecs 8205  df-wrecs 8236  df-recs 8310  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8607  df-map 8726  df-pm 8727  df-en 8843  df-dom 8844  df-sdom 8845  df-fin 8846  df-sup 9337  df-inf 9338  df-oi 9405  df-dju 9796  df-card 9834  df-pnf 11150  df-mnf 11151  df-xr 11152  df-ltxr 11153  df-le 11154  df-sub 11346  df-neg 11347  df-div 11772  df-nn 12113  df-2 12175  df-3 12176  df-n0 12373  df-xnn0 12445  df-z 12459  df-uz 12723  df-rp 12871  df-fz 13380  df-fzo 13523  df-fl 13652  df-seq 13862  df-exp 13923  df-hash 14185  df-cj 14944  df-re 14945  df-im 14946  df-sqrt 15080  df-abs 15081  df-clim 15330  df-rlim 15331  df-sum 15531  df-0p 24986  df-ply 25501  df-idp 25502  df-coe 25503  df-dgr 25504  df-quot 25603
This theorem is referenced by:  fta1  25620
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