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Theorem fta1lem 25667
Description: Lemma for fta1 25668. (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 4747 . . . . . . . . . 10 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
31, 2sylib 217 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
43simpld 495 . . . . . . . 8 (𝜑𝐹 ∈ (Poly‘ℂ))
5 fta1.5 . . . . . . . . . 10 (𝜑𝐴 ∈ (𝐹 “ {0}))
6 plyf 25559 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7 ffn 6668 . . . . . . . . . . 11 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8 fniniseg 7010 . . . . . . . . . . 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 495 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
1210simprd 496 . . . . . . . 8 (𝜑 → (𝐹𝐴) = 0)
13 eqid 2736 . . . . . . . . 9 (Xpf − (ℂ × {𝐴})) = (Xpf − (ℂ × {𝐴}))
1413facth 25666 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
154, 11, 12, 14syl3anc 1371 . . . . . . 7 (𝜑𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
1615cnveqd 5831 . . . . . 6 (𝜑𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
1716imaeq1d 6012 . . . . 5 (𝜑 → (𝐹 “ {0}) = (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) “ {0}))
18 cnex 11132 . . . . . . 7 ℂ ∈ V
1918a1i 11 . . . . . 6 (𝜑 → ℂ ∈ V)
20 ssid 3966 . . . . . . . . 9 ℂ ⊆ ℂ
21 ax-1cn 11109 . . . . . . . . 9 1 ∈ ℂ
22 plyid 25570 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
2320, 21, 22mp2an 690 . . . . . . . 8 Xp ∈ (Poly‘ℂ)
24 plyconst 25567 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
2520, 11, 24sylancr 587 . . . . . . . 8 (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26 plysubcl 25583 . . . . . . . 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 25559 . . . . . . 7 ((Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xpf − (ℂ × {𝐴})):ℂ⟶ℂ)
2927, 28syl 17 . . . . . 6 (𝜑 → (Xpf − (ℂ × {𝐴})):ℂ⟶ℂ)
3013plyremlem 25664 . . . . . . . . . . . 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 1143 . . . . . . . . . 10 (𝜑 → (deg‘(Xpf − (ℂ × {𝐴}))) = 1)
33 ax-1ne0 11120 . . . . . . . . . . 11 1 ≠ 0
3433a1i 11 . . . . . . . . . 10 (𝜑 → 1 ≠ 0)
3532, 34eqnetrd 3011 . . . . . . . . 9 (𝜑 → (deg‘(Xpf − (ℂ × {𝐴}))) ≠ 0)
36 fveq2 6842 . . . . . . . . . . 11 ((Xpf − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝐴}))) = (deg‘0𝑝))
37 dgr0 25623 . . . . . . . . . . 11 (deg‘0𝑝) = 0
3836, 37eqtrdi 2792 . . . . . . . . . 10 ((Xpf − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝐴}))) = 0)
3938necon3i 2976 . . . . . . . . 9 ((deg‘(Xpf − (ℂ × {𝐴}))) ≠ 0 → (Xpf − (ℂ × {𝐴})) ≠ 0𝑝)
4035, 39syl 17 . . . . . . . 8 (𝜑 → (Xpf − (ℂ × {𝐴})) ≠ 0𝑝)
41 quotcl2 25662 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝐴})) ≠ 0𝑝) → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
424, 27, 40, 41syl3anc 1371 . . . . . . 7 (𝜑 → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43 plyf 25559 . . . . . . 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 25642 . . . . . 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 1371 . . . . 5 (𝜑 → (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) “ {0}) = (((Xpf − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
4731simp3d 1144 . . . . . 6 (𝜑 → ((Xpf − (ℂ × {𝐴})) “ {0}) = {𝐴})
4847uneq1d 4122 . . . . 5 (𝜑 → (((Xpf − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
4917, 46, 483eqtrd 2780 . . . 4 (𝜑 → (𝐹 “ {0}) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
50 fta1.1 . . . 4 𝑅 = (𝐹 “ {0})
51 uncom 4113 . . . 4 (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}))
5249, 50, 513eqtr4g 2801 . . 3 (𝜑𝑅 = (((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
5321a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℂ)
54 dgrcl 25594 . . . . . . . . 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 12475 . . . . . . 7 (𝜑 → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) ∈ ℂ)
57 fta1.2 . . . . . . . 8 (𝜑𝐷 ∈ ℕ0)
5857nn0cnd 12475 . . . . . . 7 (𝜑𝐷 ∈ ℂ)
59 addcom 11341 . . . . . . . . 9 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
6021, 58, 59sylancr 587 . . . . . . . 8 (𝜑 → (1 + 𝐷) = (𝐷 + 1))
6115fveq2d 6846 . . . . . . . . 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 2742 . . . . . . . . . . . 12 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐹)
65 0cnd 11148 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ ℂ)
66 mul01 11334 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
6766adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
6819, 29, 65, 65, 67caofid1 7650 . . . . . . . . . . . . 13 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (ℂ × {0})) = (ℂ × {0}))
69 df-0p 25034 . . . . . . . . . . . . . 14 0𝑝 = (ℂ × {0})
7069oveq2i 7368 . . . . . . . . . . . . 13 ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝) = ((Xpf − (ℂ × {𝐴})) ∘f · (ℂ × {0}))
7168, 70, 693eqtr4g 2801 . . . . . . . . . . . 12 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝) = 0𝑝)
7263, 64, 713netr4d 3021 . . . . . . . . . . 11 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) ≠ ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝))
73 oveq2 7365 . . . . . . . . . . . 12 ((𝐹 quot (Xpf − (ℂ × {𝐴}))) = 0𝑝 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) = ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝))
7473necon3i 2976 . . . . . . . . . . 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 2736 . . . . . . . . . . 11 (deg‘(Xpf − (ℂ × {𝐴}))) = (deg‘(Xpf − (ℂ × {𝐴})))
77 eqid 2736 . . . . . . . . . . 11 (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))
7876, 77dgrmul 25631 . . . . . . . . . 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 2784 . . . . . . . 8 (𝜑 → (𝐷 + 1) = ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
8132oveq1d 7372 . . . . . . . 8 (𝜑 → ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))) = (1 + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
8260, 80, 813eqtrrd 2781 . . . . . . 7 (𝜑 → (1 + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))) = (1 + 𝐷))
8353, 56, 58, 82addcanad 11360 . . . . . 6 (𝜑 → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷)
84 fveqeq2 6851 . . . . . . . 8 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷))
85 cnveq 5829 . . . . . . . . . . 11 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → 𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))))
8685imaeq1d 6012 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (𝑔 “ {0}) = ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}))
8786eleq1d 2822 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → ((𝑔 “ {0}) ∈ Fin ↔ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
8886fveq2d 6846 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (♯‘(𝑔 “ {0})) = (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
89 fveq2 6842 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))
9088, 89breq12d 5118 . . . . . . . . 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 631 . . . . . . . 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 344 . . . . . . 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 4747 . . . . . . . 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 3582 . . . . . 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 8988 . . . 4 {𝐴} ∈ Fin
100 unfi 9116 . . . 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 2838 . 2 (𝜑𝑅 ∈ Fin)
10352fveq2d 6846 . . 3 (𝜑 → (♯‘𝑅) = (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104 hashcl 14256 . . . . . 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 12474 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107 hashcl 14256 . . . . . . 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 12474 . . . . 5 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110 peano2re 11328 . . . . 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 25594 . . . . . 6 (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
1134, 112syl 17 . . . . 5 (𝜑 → (deg‘𝐹) ∈ ℕ0)
114113nn0red 12474 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℝ)
115 hashun2 14283 . . . . . 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 14269 . . . . . . 7 (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
11811, 117syl 17 . . . . . 6 (𝜑 → (♯‘{𝐴}) = 1)
119118oveq2d 7373 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1))
120116, 119breqtrd 5131 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1))
12157nn0red 12474 . . . . . 6 (𝜑𝐷 ∈ ℝ)
122 1red 11156 . . . . . 6 (𝜑 → 1 ∈ ℝ)
12397simprd 496 . . . . . . 7 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))
124123, 83breqtrd 5131 . . . . . 6 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125109, 121, 122, 124leadd1dd 11769 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126125, 62breqtrrd 5133 . . . 4 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127106, 111, 114, 120, 126letrd 11312 . . 3 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128103, 127eqbrtrd 5127 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  cdif 3907  cun 3908  wss 3910  {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  f cof 7615  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  cle 11190  cmin 11385  0cn0 12413  chash 14230  0𝑝c0p 25033  Polycply 25545  Xpcidp 25546  degcdgr 25548   quot cquot 25650
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:  fta1  25668
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