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Theorem fta1lem 24107
Description: Lemma for fta1 24108. (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 4350 . . . . . . . . . 10 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
31, 2sylib 208 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
43simpld 474 . . . . . . . 8 (𝜑𝐹 ∈ (Poly‘ℂ))
5 fta1.5 . . . . . . . . . 10 (𝜑𝐴 ∈ (𝐹 “ {0}))
6 plyf 23999 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7 ffn 6083 . . . . . . . . . . 11 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8 fniniseg 6378 . . . . . . . . . . 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 222 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0))
1110simpld 474 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
1210simprd 478 . . . . . . . 8 (𝜑 → (𝐹𝐴) = 0)
13 eqid 2651 . . . . . . . . 9 (Xp𝑓 − (ℂ × {𝐴})) = (Xp𝑓 − (ℂ × {𝐴}))
1413facth 24106 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
154, 11, 12, 14syl3anc 1366 . . . . . . 7 (𝜑𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
1615cnveqd 5330 . . . . . 6 (𝜑𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
1716imaeq1d 5500 . . . . 5 (𝜑 → (𝐹 “ {0}) = (((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) “ {0}))
18 cnex 10055 . . . . . . 7 ℂ ∈ V
1918a1i 11 . . . . . 6 (𝜑 → ℂ ∈ V)
20 ssid 3657 . . . . . . . . 9 ℂ ⊆ ℂ
21 ax-1cn 10032 . . . . . . . . 9 1 ∈ ℂ
22 plyid 24010 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
2320, 21, 22mp2an 708 . . . . . . . 8 Xp ∈ (Poly‘ℂ)
24 plyconst 24007 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
2520, 11, 24sylancr 696 . . . . . . . 8 (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26 plysubcl 24023 . . . . . . . 8 ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
2723, 25, 26sylancr 696 . . . . . . 7 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28 plyf 23999 . . . . . . 7 ((Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xp𝑓 − (ℂ × {𝐴})):ℂ⟶ℂ)
2927, 28syl 17 . . . . . 6 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})):ℂ⟶ℂ)
3013plyremlem 24104 . . . . . . . . . . . 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 1094 . . . . . . . . . 10 (𝜑 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = 1)
33 ax-1ne0 10043 . . . . . . . . . . 11 1 ≠ 0
3433a1i 11 . . . . . . . . . 10 (𝜑 → 1 ≠ 0)
3532, 34eqnetrd 2890 . . . . . . . . 9 (𝜑 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) ≠ 0)
36 fveq2 6229 . . . . . . . . . . 11 ((Xp𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = (deg‘0𝑝))
37 dgr0 24063 . . . . . . . . . . 11 (deg‘0𝑝) = 0
3836, 37syl6eq 2701 . . . . . . . . . 10 ((Xp𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = 0)
3938necon3i 2855 . . . . . . . . 9 ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) ≠ 0 → (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝)
4035, 39syl 17 . . . . . . . 8 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝)
41 quotcl2 24102 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝) → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
424, 27, 40, 41syl3anc 1366 . . . . . . 7 (𝜑 → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43 plyf 23999 . . . . . . 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 24082 . . . . . 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 1366 . . . . 5 (𝜑 → (((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) “ {0}) = (((Xp𝑓 − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
4731simp3d 1095 . . . . . 6 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) “ {0}) = {𝐴})
4847uneq1d 3799 . . . . 5 (𝜑 → (((Xp𝑓 − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) = ({𝐴} ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
4917, 46, 483eqtrd 2689 . . . 4 (𝜑 → (𝐹 “ {0}) = ({𝐴} ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
50 fta1.1 . . . 4 𝑅 = (𝐹 “ {0})
51 uncom 3790 . . . 4 (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}))
5249, 50, 513eqtr4g 2710 . . 3 (𝜑𝑅 = (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
533simprd 478 . . . . . . . . 9 (𝜑𝐹 ≠ 0𝑝)
5415eqcomd 2657 . . . . . . . . 9 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐹)
55 0cnd 10071 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℂ)
56 mul01 10253 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
5756adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
5819, 29, 55, 55, 57caofid1 6969 . . . . . . . . . 10 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0})) = (ℂ × {0}))
59 df-0p 23482 . . . . . . . . . . 11 0𝑝 = (ℂ × {0})
6059oveq2i 6701 . . . . . . . . . 10 ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0}))
6158, 60, 593eqtr4g 2710 . . . . . . . . 9 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = 0𝑝)
6253, 54, 613netr4d 2900 . . . . . . . 8 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ≠ ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
63 oveq2 6698 . . . . . . . . 9 ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) = 0𝑝 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
6463necon3i 2855 . . . . . . . 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 4350 . . . . . . 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 699 . . . . . 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 24034 . . . . . . . . 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 11391 . . . . . . 7 (𝜑 → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ∈ ℂ)
73 fta1.2 . . . . . . . 8 (𝜑𝐷 ∈ ℕ0)
7473nn0cnd 11391 . . . . . . 7 (𝜑𝐷 ∈ ℂ)
75 addcom 10260 . . . . . . . . 9 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
7621, 74, 75sylancr 696 . . . . . . . 8 (𝜑 → (1 + 𝐷) = (𝐷 + 1))
7715fveq2d 6233 . . . . . . . . 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 2651 . . . . . . . . . . 11 (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = (deg‘(Xp𝑓 − (ℂ × {𝐴})))
80 eqid 2651 . . . . . . . . . . 11 (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))
8179, 80dgrmul 24071 . . . . . . . . . 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 1367 . . . . . . . . 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 2693 . . . . . . . 8 (𝜑 → (𝐷 + 1) = ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
8432oveq1d 6705 . . . . . . . 8 (𝜑 → ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))) = (1 + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
8576, 83, 843eqtrrd 2690 . . . . . . 7 (𝜑 → (1 + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))) = (1 + 𝐷))
8669, 72, 74, 85addcanad 10279 . . . . . 6 (𝜑 → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐷)
87 fveq2 6229 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
8887eqeq1d 2653 . . . . . . . 8 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐷))
89 cnveq 5328 . . . . . . . . . . 11 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → 𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))
9089imaeq1d 5500 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (𝑔 “ {0}) = ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}))
9190eleq1d 2715 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → ((𝑔 “ {0}) ∈ Fin ↔ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
9290fveq2d 6233 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (#‘(𝑔 “ {0})) = (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
9392, 87breq12d 4698 . . . . . . . . 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 747 . . . . . . . 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 3336 . . . . . 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 66 . . . . 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 8079 . . . 4 {𝐴} ∈ Fin
100 unfi 8268 . . . 4 ((((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ {𝐴} ∈ Fin) → (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10198, 99, 100sylancl 695 . . 3 (𝜑 → (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10252, 101eqeltrd 2730 . 2 (𝜑𝑅 ∈ Fin)
10352fveq2d 6233 . . 3 (𝜑 → (#‘𝑅) = (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104 hashcl 13185 . . . . . 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 11390 . . . 4 (𝜑 → (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107 hashcl 13185 . . . . . . 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 11390 . . . . 5 (𝜑 → (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110 peano2re 10247 . . . . 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 24034 . . . . . 6 (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
1134, 112syl 17 . . . . 5 (𝜑 → (deg‘𝐹) ∈ ℕ0)
114113nn0red 11390 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℝ)
115 hashun2 13210 . . . . . 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 695 . . . . 5 (𝜑 → (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + (#‘{𝐴})))
117 hashsng 13197 . . . . . . 7 (𝐴 ∈ ℂ → (#‘{𝐴}) = 1)
11811, 117syl 17 . . . . . 6 (𝜑 → (#‘{𝐴}) = 1)
119118oveq2d 6706 . . . . 5 (𝜑 → ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + (#‘{𝐴})) = ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
120116, 119breqtrd 4711 . . . 4 (𝜑 → (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
12173nn0red 11390 . . . . . 6 (𝜑𝐷 ∈ ℝ)
122 1red 10093 . . . . . 6 (𝜑 → 1 ∈ ℝ)
12397simprd 478 . . . . . . 7 (𝜑 → (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
124123, 86breqtrd 4711 . . . . . 6 (𝜑 → (#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125109, 121, 122, 124leadd1dd 10679 . . . . 5 (𝜑 → ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126125, 78breqtrrd 4713 . . . 4 (𝜑 → ((#‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127106, 111, 114, 120, 126letrd 10232 . . 3 (𝜑 → (#‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128103, 127eqbrtrd 4707 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  cun 3605  wss 3607  {csn 4210   class class class wbr 4685   × cxp 5141  ccnv 5142  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  cle 10113  cmin 10304  0cn0 11330  #chash 13157  0𝑝c0p 23481  Polycply 23985  Xpcidp 23986  degcdgr 23988   quot cquot 24090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-0p 23482  df-ply 23989  df-idp 23990  df-coe 23991  df-dgr 23992  df-quot 24091
This theorem is referenced by:  fta1  24108
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