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Theorem fta1lem 25200
Description: Lemma for fta1 25201. (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 4700 . . . . . . . . . 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 25092 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7 ffn 6545 . . . . . . . . . . 11 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8 fniniseg 6880 . . . . . . . . . . 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 2737 . . . . . . . . 9 (Xpf − (ℂ × {𝐴})) = (Xpf − (ℂ × {𝐴}))
1413facth 25199 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
154, 11, 12, 14syl3anc 1373 . . . . . . 7 (𝜑𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
1615cnveqd 5744 . . . . . 6 (𝜑𝐹 = ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))))
1716imaeq1d 5928 . . . . 5 (𝜑 → (𝐹 “ {0}) = (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) “ {0}))
18 cnex 10810 . . . . . . 7 ℂ ∈ V
1918a1i 11 . . . . . 6 (𝜑 → ℂ ∈ V)
20 ssid 3923 . . . . . . . . 9 ℂ ⊆ ℂ
21 ax-1cn 10787 . . . . . . . . 9 1 ∈ ℂ
22 plyid 25103 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
2320, 21, 22mp2an 692 . . . . . . . 8 Xp ∈ (Poly‘ℂ)
24 plyconst 25100 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
2520, 11, 24sylancr 590 . . . . . . . 8 (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26 plysubcl 25116 . . . . . . . 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 25092 . . . . . . 7 ((Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xpf − (ℂ × {𝐴})):ℂ⟶ℂ)
2927, 28syl 17 . . . . . 6 (𝜑 → (Xpf − (ℂ × {𝐴})):ℂ⟶ℂ)
3013plyremlem 25197 . . . . . . . . . . . 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 1145 . . . . . . . . . 10 (𝜑 → (deg‘(Xpf − (ℂ × {𝐴}))) = 1)
33 ax-1ne0 10798 . . . . . . . . . . 11 1 ≠ 0
3433a1i 11 . . . . . . . . . 10 (𝜑 → 1 ≠ 0)
3532, 34eqnetrd 3008 . . . . . . . . 9 (𝜑 → (deg‘(Xpf − (ℂ × {𝐴}))) ≠ 0)
36 fveq2 6717 . . . . . . . . . . 11 ((Xpf − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝐴}))) = (deg‘0𝑝))
37 dgr0 25156 . . . . . . . . . . 11 (deg‘0𝑝) = 0
3836, 37eqtrdi 2794 . . . . . . . . . 10 ((Xpf − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝐴}))) = 0)
3938necon3i 2973 . . . . . . . . 9 ((deg‘(Xpf − (ℂ × {𝐴}))) ≠ 0 → (Xpf − (ℂ × {𝐴})) ≠ 0𝑝)
4035, 39syl 17 . . . . . . . 8 (𝜑 → (Xpf − (ℂ × {𝐴})) ≠ 0𝑝)
41 quotcl2 25195 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝐴})) ≠ 0𝑝) → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
424, 27, 40, 41syl3anc 1373 . . . . . . 7 (𝜑 → (𝐹 quot (Xpf − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43 plyf 25092 . . . . . . 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 25175 . . . . . 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 1373 . . . . 5 (𝜑 → (((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) “ {0}) = (((Xpf − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
4731simp3d 1146 . . . . . 6 (𝜑 → ((Xpf − (ℂ × {𝐴})) “ {0}) = {𝐴})
4847uneq1d 4076 . . . . 5 (𝜑 → (((Xpf − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
4917, 46, 483eqtrd 2781 . . . 4 (𝜑 → (𝐹 “ {0}) = ({𝐴} ∪ ((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
50 fta1.1 . . . 4 𝑅 = (𝐹 “ {0})
51 uncom 4067 . . . 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 25127 . . . . . . . . 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 12152 . . . . . . 7 (𝜑 → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) ∈ ℂ)
57 fta1.2 . . . . . . . 8 (𝜑𝐷 ∈ ℕ0)
5857nn0cnd 12152 . . . . . . 7 (𝜑𝐷 ∈ ℂ)
59 addcom 11018 . . . . . . . . 9 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
6021, 58, 59sylancr 590 . . . . . . . 8 (𝜑 → (1 + 𝐷) = (𝐷 + 1))
6115fveq2d 6721 . . . . . . . . 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 2743 . . . . . . . . . . . 12 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐹)
65 0cnd 10826 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ ℂ)
66 mul01 11011 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
6766adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
6819, 29, 65, 65, 67caofid1 7501 . . . . . . . . . . . . 13 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (ℂ × {0})) = (ℂ × {0}))
69 df-0p 24567 . . . . . . . . . . . . . 14 0𝑝 = (ℂ × {0})
7069oveq2i 7224 . . . . . . . . . . . . 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 3018 . . . . . . . . . . 11 (𝜑 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) ≠ ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝))
73 oveq2 7221 . . . . . . . . . . . 12 ((𝐹 quot (Xpf − (ℂ × {𝐴}))) = 0𝑝 → ((Xpf − (ℂ × {𝐴})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝐴})))) = ((Xpf − (ℂ × {𝐴})) ∘f · 0𝑝))
7473necon3i 2973 . . . . . . . . . . 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 2737 . . . . . . . . . . 11 (deg‘(Xpf − (ℂ × {𝐴}))) = (deg‘(Xpf − (ℂ × {𝐴})))
77 eqid 2737 . . . . . . . . . . 11 (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))
7876, 77dgrmul 25164 . . . . . . . . . 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 839 . . . . . . . . 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 2785 . . . . . . . 8 (𝜑 → (𝐷 + 1) = ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
8132oveq1d 7228 . . . . . . . 8 (𝜑 → ((deg‘(Xpf − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))) = (1 + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))))
8260, 80, 813eqtrrd 2782 . . . . . . 7 (𝜑 → (1 + (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴}))))) = (1 + 𝐷))
8353, 56, 58, 82addcanad 11037 . . . . . 6 (𝜑 → (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷)
84 fveqeq2 6726 . . . . . . . 8 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))) = 𝐷))
85 cnveq 5742 . . . . . . . . . . 11 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → 𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))))
8685imaeq1d 5928 . . . . . . . . . 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 6721 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (♯‘(𝑔 “ {0})) = (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})))
89 fveq2 6717 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xpf − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))
9088, 89breq12d 5066 . . . . . . . . 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 634 . . . . . . . 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 4700 . . . . . . . 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 3539 . . . . . 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 8721 . . . 4 {𝐴} ∈ Fin
100 unfi 8850 . . . 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 2838 . 2 (𝜑𝑅 ∈ Fin)
10352fveq2d 6721 . . 3 (𝜑 → (♯‘𝑅) = (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104 hashcl 13923 . . . . . 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 12151 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107 hashcl 13923 . . . . . . 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 12151 . . . . 5 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110 peano2re 11005 . . . . 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 25127 . . . . . 6 (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
1134, 112syl 17 . . . . 5 (𝜑 → (deg‘𝐹) ∈ ℕ0)
114113nn0red 12151 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℝ)
115 hashun2 13950 . . . . . 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 13936 . . . . . . 7 (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
11811, 117syl 17 . . . . . 6 (𝜑 → (♯‘{𝐴}) = 1)
119118oveq2d 7229 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1))
120116, 119breqtrd 5079 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1))
12157nn0red 12151 . . . . . 6 (𝜑𝐷 ∈ ℝ)
122 1red 10834 . . . . . 6 (𝜑 → 1 ∈ ℝ)
12397simprd 499 . . . . . . 7 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xpf − (ℂ × {𝐴})))))
124123, 83breqtrd 5079 . . . . . 6 (𝜑 → (♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125109, 121, 122, 124leadd1dd 11446 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126125, 62breqtrrd 5081 . . . 4 (𝜑 → ((♯‘((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127106, 111, 114, 120, 126letrd 10989 . . 3 (𝜑 → (♯‘(((𝐹 quot (Xpf − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128103, 127eqbrtrd 5075 . 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 1089   = wceq 1543  wcel 2110  wne 2940  wral 3061  Vcvv 3408  cdif 3863  cun 3864  wss 3866  {csn 4541   class class class wbr 5053   × cxp 5549  ccnv 5550  cima 5554   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  f cof 7467  Fincfn 8626  cc 10727  cr 10728  0cc0 10729  1c1 10730   + caddc 10732   · cmul 10734  cle 10868  cmin 11062  0cn0 12090  chash 13896  0𝑝c0p 24566  Polycply 25078  Xpcidp 25079  degcdgr 25081   quot cquot 25183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-map 8510  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-sup 9058  df-inf 9059  df-oi 9126  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-fl 13367  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-rlim 15050  df-sum 15250  df-0p 24567  df-ply 25082  df-idp 25083  df-coe 25084  df-dgr 25085  df-quot 25184
This theorem is referenced by:  fta1  25201
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