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Theorem fta1lem 23866
Description: Lemma for fta1 23867. (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 4260 . . . . . . . . . 10 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
31, 2sylib 207 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
43simpld 474 . . . . . . . 8 (𝜑𝐹 ∈ (Poly‘ℂ))
5 fta1.5 . . . . . . . . . 10 (𝜑𝐴 ∈ (𝐹 “ {0}))
6 plyf 23758 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7 ffn 5958 . . . . . . . . . . 11 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8 fniniseg 6246 . . . . . . . . . . 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 221 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0))
1110simpld 474 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
1210simprd 478 . . . . . . . 8 (𝜑 → (𝐹𝐴) = 0)
13 eqid 2610 . . . . . . . . 9 (Xp𝑓 − (ℂ × {𝐴})) = (Xp𝑓 − (ℂ × {𝐴}))
1413facth 23865 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
154, 11, 12, 14syl3anc 1318 . . . . . . 7 (𝜑𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
1615cnveqd 5220 . . . . . 6 (𝜑𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
1716imaeq1d 5384 . . . . 5 (𝜑 → (𝐹 “ {0}) = (((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) “ {0}))
18 cnex 9896 . . . . . . 7 ℂ ∈ V
1918a1i 11 . . . . . 6 (𝜑 → ℂ ∈ V)
20 ssid 3587 . . . . . . . . 9 ℂ ⊆ ℂ
21 ax-1cn 9873 . . . . . . . . 9 1 ∈ ℂ
22 plyid 23769 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
2320, 21, 22mp2an 704 . . . . . . . 8 Xp ∈ (Poly‘ℂ)
24 plyconst 23766 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
2520, 11, 24sylancr 694 . . . . . . . 8 (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26 plysubcl 23782 . . . . . . . 8 ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
2723, 25, 26sylancr 694 . . . . . . 7 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28 plyf 23758 . . . . . . 7 ((Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xp𝑓 − (ℂ × {𝐴})):ℂ⟶ℂ)
2927, 28syl 17 . . . . . 6 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})):ℂ⟶ℂ)
3013plyremlem 23863 . . . . . . . . . . . 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 1067 . . . . . . . . . 10 (𝜑 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = 1)
33 ax-1ne0 9884 . . . . . . . . . . 11 1 ≠ 0
3433a1i 11 . . . . . . . . . 10 (𝜑 → 1 ≠ 0)
3532, 34eqnetrd 2849 . . . . . . . . 9 (𝜑 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) ≠ 0)
36 fveq2 6103 . . . . . . . . . . 11 ((Xp𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = (deg‘0𝑝))
37 dgr0 23822 . . . . . . . . . . 11 (deg‘0𝑝) = 0
3836, 37syl6eq 2660 . . . . . . . . . 10 ((Xp𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = 0)
3938necon3i 2814 . . . . . . . . 9 ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) ≠ 0 → (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝)
4035, 39syl 17 . . . . . . . 8 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝)
41 quotcl2 23861 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝) → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
424, 27, 40, 41syl3anc 1318 . . . . . . 7 (𝜑 → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43 plyf 23758 . . . . . . 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 23841 . . . . . 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 1318 . . . . 5 (𝜑 → (((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) “ {0}) = (((Xp𝑓 − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
4731simp3d 1068 . . . . . 6 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) “ {0}) = {𝐴})
4847uneq1d 3728 . . . . 5 (𝜑 → (((Xp𝑓 − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) = ({𝐴} ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
4917, 46, 483eqtrd 2648 . . . 4 (𝜑 → (𝐹 “ {0}) = ({𝐴} ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
50 fta1.1 . . . 4 𝑅 = (𝐹 “ {0})
51 uncom 3719 . . . 4 (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}))
5249, 50, 513eqtr4g 2669 . . 3 (𝜑𝑅 = (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
533simprd 478 . . . . . . . . 9 (𝜑𝐹 ≠ 0𝑝)
5415eqcomd 2616 . . . . . . . . 9 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐹)
55 0cnd 9912 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℂ)
56 mul01 10094 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
5756adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
5819, 29, 55, 55, 57caofid1 6825 . . . . . . . . . 10 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0})) = (ℂ × {0}))
59 df-0p 23243 . . . . . . . . . . 11 0𝑝 = (ℂ × {0})
6059oveq2i 6560 . . . . . . . . . 10 ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0}))
6158, 60, 593eqtr4g 2669 . . . . . . . . 9 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = 0𝑝)
6253, 54, 613netr4d 2859 . . . . . . . 8 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ≠ ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
63 oveq2 6557 . . . . . . . . 9 ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) = 0𝑝 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
6463necon3i 2814 . . . . . . . 8 (((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ≠ ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ≠ 0𝑝)
6562, 64syl 17 . . . . . . 7 (𝜑 → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ≠ 0𝑝)
66 eldifsn 4260 . . . . . . 7 ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) ∧ (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ≠ 0𝑝))
6742, 65, 66sylanbrc 695 . . . . . 6 (𝜑 → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
68 fta1.6 . . . . . 6 (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((𝑔 “ {0}) ∈ Fin ∧ (#‘(𝑔 “ {0})) ≤ (deg‘𝑔))))
6921a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℂ)
70 dgrcl 23793 . . . . . . . . 9 ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ) → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ∈ ℕ0)
7142, 70syl 17 . . . . . . . 8 (𝜑 → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ∈ ℕ0)
7271nn0cnd 11230 . . . . . . 7 (𝜑 → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ∈ ℂ)
73 fta1.2 . . . . . . . 8 (𝜑𝐷 ∈ ℕ0)
7473nn0cnd 11230 . . . . . . 7 (𝜑𝐷 ∈ ℂ)
75 addcom 10101 . . . . . . . . 9 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
7621, 74, 75sylancr 694 . . . . . . . 8 (𝜑 → (1 + 𝐷) = (𝐷 + 1))
7715fveq2d 6107 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (deg‘((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
78 fta1.4 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (𝐷 + 1))
79 eqid 2610 . . . . . . . . . . 11 (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = (deg‘(Xp𝑓 − (ℂ × {𝐴})))
80 eqid 2610 . . . . . . . . . . 11 (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))
8179, 80dgrmul 23830 . . . . . . . . . 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}))))))
8227, 40, 42, 65, 81syl22anc 1319 . . . . . . . . 9 (𝜑 → (deg‘((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))) = ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
8377, 78, 823eqtr3d 2652 . . . . . . . 8 (𝜑 → (𝐷 + 1) = ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
8432oveq1d 6564 . . . . . . . 8 (𝜑 → ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))) = (1 + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
8576, 83, 843eqtrrd 2649 . . . . . . 7 (𝜑 → (1 + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))) = (1 + 𝐷))
8669, 72, 74, 85addcanad 10120 . . . . . 6 (𝜑 → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐷)
87 fveq2 6103 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
8887eqeq1d 2612 . . . . . . . 8 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐷))
89 cnveq 5218 . . . . . . . . . . 11 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → 𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))
9089imaeq1d 5384 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (𝑔 “ {0}) = ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}))
9190eleq1d 2672 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → ((𝑔 “ {0}) ∈ Fin ↔ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
9290fveq2d 6107 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (#‘(𝑔 “ {0})) = (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
9392, 87breq12d 4596 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → ((#‘(𝑔 “ {0})) ≤ (deg‘𝑔) ↔ (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
9491, 93anbi12d 743 . . . . . . . 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})))))))
9588, 94imbi12d 333 . . . . . . 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}))))))))
9695rspcv 3278 . . . . . 6 ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}) → (∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((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}))))))))
9767, 68, 86, 96syl3c 64 . . . . 5 (𝜑 → (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
9897simpld 474 . . . 4 (𝜑 → ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99 snfi 7923 . . . 4 {𝐴} ∈ Fin
100 unfi 8112 . . . 4 ((((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ {𝐴} ∈ Fin) → (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10198, 99, 100sylancl 693 . . 3 (𝜑 → (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10252, 101eqeltrd 2688 . 2 (𝜑𝑅 ∈ Fin)
10352fveq2d 6107 . . 3 (𝜑 → (#‘𝑅) = (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104 hashcl 13009 . . . . . 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 11229 . . . 4 (𝜑 → (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107 hashcl 13009 . . . . . . 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 11229 . . . . 5 (𝜑 → (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110 peano2re 10088 . . . . 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 23793 . . . . . 6 (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
1134, 112syl 17 . . . . 5 (𝜑 → (deg‘𝐹) ∈ ℕ0)
114113nn0red 11229 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℝ)
115 hashun2 13033 . . . . . 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 693 . . . . 5 (𝜑 → (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + (#‘{𝐴})))
117 hashsng 13020 . . . . . . 7 (𝐴 ∈ ℂ → (#‘{𝐴}) = 1)
11811, 117syl 17 . . . . . 6 (𝜑 → (#‘{𝐴}) = 1)
119118oveq2d 6565 . . . . 5 (𝜑 → ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + (#‘{𝐴})) = ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
120116, 119breqtrd 4609 . . . 4 (𝜑 → (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
12173nn0red 11229 . . . . . 6 (𝜑𝐷 ∈ ℝ)
122 1red 9934 . . . . . 6 (𝜑 → 1 ∈ ℝ)
12397simprd 478 . . . . . . 7 (𝜑 → (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
124123, 86breqtrd 4609 . . . . . 6 (𝜑 → (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125109, 121, 122, 124leadd1dd 10520 . . . . 5 (𝜑 → ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126125, 78breqtrrd 4611 . . . 4 (𝜑 → ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127106, 111, 114, 120, 126letrd 10073 . . 3 (𝜑 → (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128103, 127eqbrtrd 4605 . 2 (𝜑 → (#‘𝑅) ≤ (deg‘𝐹))
129102, 128jca 553 1 (𝜑 → (𝑅 ∈ Fin ∧ (#‘𝑅) ≤ (deg‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  Vcvv 3173  cdif 3537  cun 3538  wss 3540  {csn 4125   class class class wbr 4583   × cxp 5036  ccnv 5037  cima 5041   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  𝑓 cof 6793  Fincfn 7841  cc 9813  cr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  cle 9954  cmin 10145  0cn0 11169  #chash 12979  0𝑝c0p 23242  Polycply 23744  Xpcidp 23745  degcdgr 23747   quot cquot 23849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-idp 23749  df-coe 23750  df-dgr 23751  df-quot 23850
This theorem is referenced by:  fta1  23867
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