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Theorem fta1lem 24356
Description: Lemma for fta1 24357. (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 4474 . . . . . . . . . 10 (𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
31, 2sylib 209 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (Poly‘ℂ) ∧ 𝐹 ≠ 0𝑝))
43simpld 488 . . . . . . . 8 (𝜑𝐹 ∈ (Poly‘ℂ))
5 fta1.5 . . . . . . . . . 10 (𝜑𝐴 ∈ (𝐹 “ {0}))
6 plyf 24248 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘ℂ) → 𝐹:ℂ⟶ℂ)
7 ffn 6225 . . . . . . . . . . 11 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
8 fniniseg 6530 . . . . . . . . . . 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 223 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0))
1110simpld 488 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
1210simprd 489 . . . . . . . 8 (𝜑 → (𝐹𝐴) = 0)
13 eqid 2765 . . . . . . . . 9 (Xp𝑓 − (ℂ × {𝐴})) = (Xp𝑓 − (ℂ × {𝐴}))
1413facth 24355 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
154, 11, 12, 14syl3anc 1490 . . . . . . 7 (𝜑𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
1615cnveqd 5468 . . . . . 6 (𝜑𝐹 = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
1716imaeq1d 5649 . . . . 5 (𝜑 → (𝐹 “ {0}) = (((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) “ {0}))
18 cnex 10272 . . . . . . 7 ℂ ∈ V
1918a1i 11 . . . . . 6 (𝜑 → ℂ ∈ V)
20 ssid 3785 . . . . . . . . 9 ℂ ⊆ ℂ
21 ax-1cn 10249 . . . . . . . . 9 1 ∈ ℂ
22 plyid 24259 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
2320, 21, 22mp2an 683 . . . . . . . 8 Xp ∈ (Poly‘ℂ)
24 plyconst 24256 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
2520, 11, 24sylancr 581 . . . . . . . 8 (𝜑 → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26 plysubcl 24272 . . . . . . . 8 ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝐴}) ∈ (Poly‘ℂ)) → (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
2723, 25, 26sylancr 581 . . . . . . 7 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ))
28 plyf 24248 . . . . . . 7 ((Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ) → (Xp𝑓 − (ℂ × {𝐴})):ℂ⟶ℂ)
2927, 28syl 17 . . . . . 6 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})):ℂ⟶ℂ)
3013plyremlem 24353 . . . . . . . . . . . 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 1173 . . . . . . . . . 10 (𝜑 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = 1)
33 ax-1ne0 10260 . . . . . . . . . . 11 1 ≠ 0
3433a1i 11 . . . . . . . . . 10 (𝜑 → 1 ≠ 0)
3532, 34eqnetrd 3004 . . . . . . . . 9 (𝜑 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) ≠ 0)
36 fveq2 6377 . . . . . . . . . . 11 ((Xp𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = (deg‘0𝑝))
37 dgr0 24312 . . . . . . . . . . 11 (deg‘0𝑝) = 0
3836, 37syl6eq 2815 . . . . . . . . . 10 ((Xp𝑓 − (ℂ × {𝐴})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = 0)
3938necon3i 2969 . . . . . . . . 9 ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) ≠ 0 → (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝)
4035, 39syl 17 . . . . . . . 8 (𝜑 → (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝)
41 quotcl2 24351 . . . . . . . 8 ((𝐹 ∈ (Poly‘ℂ) ∧ (Xp𝑓 − (ℂ × {𝐴})) ∈ (Poly‘ℂ) ∧ (Xp𝑓 − (ℂ × {𝐴})) ≠ 0𝑝) → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
424, 27, 40, 41syl3anc 1490 . . . . . . 7 (𝜑 → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ (Poly‘ℂ))
43 plyf 24248 . . . . . . 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 24331 . . . . . 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 1490 . . . . 5 (𝜑 → (((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) “ {0}) = (((Xp𝑓 − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
4731simp3d 1174 . . . . . 6 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) “ {0}) = {𝐴})
4847uneq1d 3930 . . . . 5 (𝜑 → (((Xp𝑓 − (ℂ × {𝐴})) “ {0}) ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) = ({𝐴} ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
4917, 46, 483eqtrd 2803 . . . 4 (𝜑 → (𝐹 “ {0}) = ({𝐴} ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
50 fta1.1 . . . 4 𝑅 = (𝐹 “ {0})
51 uncom 3921 . . . 4 (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) = ({𝐴} ∪ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}))
5249, 50, 513eqtr4g 2824 . . 3 (𝜑𝑅 = (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}))
5321a1i 11 . . . . . . 7 (𝜑 → 1 ∈ ℂ)
54 dgrcl 24283 . . . . . . . . 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 11602 . . . . . . 7 (𝜑 → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ∈ ℂ)
57 fta1.2 . . . . . . . 8 (𝜑𝐷 ∈ ℕ0)
5857nn0cnd 11602 . . . . . . 7 (𝜑𝐷 ∈ ℂ)
59 addcom 10478 . . . . . . . . 9 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
6021, 58, 59sylancr 581 . . . . . . . 8 (𝜑 → (1 + 𝐷) = (𝐷 + 1))
6115fveq2d 6381 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (deg‘((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
62 fta1.4 . . . . . . . . 9 (𝜑 → (deg‘𝐹) = (𝐷 + 1))
633simprd 489 . . . . . . . . . . . 12 (𝜑𝐹 ≠ 0𝑝)
6415eqcomd 2771 . . . . . . . . . . . 12 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐹)
65 0cnd 10288 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ ℂ)
66 mul01 10471 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
6766adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
6819, 29, 65, 65, 67caofid1 7127 . . . . . . . . . . . . 13 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0})) = (ℂ × {0}))
69 df-0p 23731 . . . . . . . . . . . . . 14 0𝑝 = (ℂ × {0})
7069oveq2i 6855 . . . . . . . . . . . . 13 ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (ℂ × {0}))
7168, 70, 693eqtr4g 2824 . . . . . . . . . . . 12 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝) = 0𝑝)
7263, 64, 713netr4d 3014 . . . . . . . . . . 11 (𝜑 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) ≠ ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
73 oveq2 6852 . . . . . . . . . . . 12 ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) = 0𝑝 → ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = ((Xp𝑓 − (ℂ × {𝐴})) ∘𝑓 · 0𝑝))
7473necon3i 2969 . . . . . . . . . . 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 2765 . . . . . . . . . . 11 (deg‘(Xp𝑓 − (ℂ × {𝐴}))) = (deg‘(Xp𝑓 − (ℂ × {𝐴})))
77 eqid 2765 . . . . . . . . . . 11 (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))
7876, 77dgrmul 24320 . . . . . . . . . 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 867 . . . . . . . . 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 2807 . . . . . . . 8 (𝜑 → (𝐷 + 1) = ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
8132oveq1d 6859 . . . . . . . 8 (𝜑 → ((deg‘(Xp𝑓 − (ℂ × {𝐴}))) + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))) = (1 + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))))
8260, 80, 813eqtrrd 2804 . . . . . . 7 (𝜑 → (1 + (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))) = (1 + 𝐷))
8353, 56, 58, 82addcanad 10497 . . . . . 6 (𝜑 → (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐷)
84 fveqeq2 6386 . . . . . . . 8 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → ((deg‘𝑔) = 𝐷 ↔ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))) = 𝐷))
85 cnveq 5466 . . . . . . . . . . 11 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → 𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))))
8685imaeq1d 5649 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (𝑔 “ {0}) = ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}))
8786eleq1d 2829 . . . . . . . . 9 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → ((𝑔 “ {0}) ∈ Fin ↔ ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin))
8886fveq2d 6381 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (♯‘(𝑔 “ {0})) = (♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})))
89 fveq2 6377 . . . . . . . . . 10 (𝑔 = (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) → (deg‘𝑔) = (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
9088, 89breq12d 4824 . . . . . . . . 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 624 . . . . . . . 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 335 . . . . . . 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 4474 . . . . . . . 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 578 . . . . . . 7 (𝜑 → (𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) ∈ ((Poly‘ℂ) ∖ {0𝑝}))
9692, 93, 95rspcdva 3468 . . . . . 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 488 . . . 4 (𝜑 → ((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin)
99 snfi 8247 . . . 4 {𝐴} ∈ Fin
100 unfi 8436 . . . 4 ((((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∈ Fin ∧ {𝐴} ∈ Fin) → (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10198, 99, 100sylancl 580 . . 3 (𝜑 → (((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴}) ∈ Fin)
10252, 101eqeltrd 2844 . 2 (𝜑𝑅 ∈ Fin)
10352fveq2d 6381 . . 3 (𝜑 → (♯‘𝑅) = (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})))
104 hashcl 13352 . . . . . 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 11601 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ∈ ℝ)
107 hashcl 13352 . . . . . . 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 11601 . . . . 5 (𝜑 → (♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ∈ ℝ)
110 peano2re 10465 . . . . 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 24283 . . . . . 6 (𝐹 ∈ (Poly‘ℂ) → (deg‘𝐹) ∈ ℕ0)
1134, 112syl 17 . . . . 5 (𝜑 → (deg‘𝐹) ∈ ℕ0)
114113nn0red 11601 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℝ)
115 hashun2 13377 . . . . . 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 580 . . . . 5 (𝜑 → (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})))
117 hashsng 13364 . . . . . . 7 (𝐴 ∈ ℂ → (♯‘{𝐴}) = 1)
11811, 117syl 17 . . . . . 6 (𝜑 → (♯‘{𝐴}) = 1)
119118oveq2d 6860 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + (♯‘{𝐴})) = ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
120116, 119breqtrd 4837 . . . 4 (𝜑 → (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1))
12157nn0red 11601 . . . . . 6 (𝜑𝐷 ∈ ℝ)
122 1red 10296 . . . . . 6 (𝜑 → 1 ∈ ℝ)
12397simprd 489 . . . . . . 7 (𝜑 → (♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ (deg‘(𝐹 quot (Xp𝑓 − (ℂ × {𝐴})))))
124123, 83breqtrd 4837 . . . . . 6 (𝜑 → (♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) ≤ 𝐷)
125109, 121, 122, 124leadd1dd 10897 . . . . 5 (𝜑 → ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (𝐷 + 1))
126125, 62breqtrrd 4839 . . . 4 (𝜑 → ((♯‘((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0})) + 1) ≤ (deg‘𝐹))
127106, 111, 114, 120, 126letrd 10450 . . 3 (𝜑 → (♯‘(((𝐹 quot (Xp𝑓 − (ℂ × {𝐴}))) “ {0}) ∪ {𝐴})) ≤ (deg‘𝐹))
128103, 127eqbrtrd 4833 . 2 (𝜑 → (♯‘𝑅) ≤ (deg‘𝐹))
129102, 128jca 507 1 (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cdif 3731  cun 3732  wss 3734  {csn 4336   class class class wbr 4811   × cxp 5277  ccnv 5278  cima 5282   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6844  𝑓 cof 7095  Fincfn 8162  cc 10189  cr 10190  0cc0 10191  1c1 10192   + caddc 10194   · cmul 10196  cle 10331  cmin 10522  0cn0 11540  chash 13324  0𝑝c0p 23730  Polycply 24234  Xpcidp 24235  degcdgr 24237   quot cquot 24339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269  ax-addf 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-of 7097  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-map 8064  df-pm 8065  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-sup 8557  df-inf 8558  df-oi 8624  df-card 9018  df-cda 9245  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-xnn0 11613  df-z 11627  df-uz 11890  df-rp 12032  df-fz 12537  df-fzo 12677  df-fl 12804  df-seq 13012  df-exp 13071  df-hash 13325  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-clim 14507  df-rlim 14508  df-sum 14705  df-0p 23731  df-ply 24238  df-idp 24239  df-coe 24240  df-dgr 24241  df-quot 24340
This theorem is referenced by:  fta1  24357
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