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Theorem dgrco 24780
 Description: The degree of a composition of two polynomials is the product of the degrees. (Contributed by Mario Carneiro, 15-Sep-2014.)
Hypotheses
Ref Expression
dgrco.1 𝑀 = (deg‘𝐹)
dgrco.2 𝑁 = (deg‘𝐺)
dgrco.3 (𝜑𝐹 ∈ (Poly‘𝑆))
dgrco.4 (𝜑𝐺 ∈ (Poly‘𝑆))
Assertion
Ref Expression
dgrco (𝜑 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))

Proof of Theorem dgrco
Dummy variables 𝑓 𝑥 𝑦 𝑑 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 24705 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 dgrco.3 . . 3 (𝜑𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 3969 . 2 (𝜑𝐹 ∈ (Poly‘ℂ))
4 dgrco.1 . . . 4 𝑀 = (deg‘𝐹)
5 dgrcl 24738 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
62, 5syl 17 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℕ0)
74, 6eqeltrid 2922 . . 3 (𝜑𝑀 ∈ ℕ0)
8 breq2 5067 . . . . . . 7 (𝑥 = 0 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 0))
98imbi1d 343 . . . . . 6 (𝑥 = 0 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
109ralbidv 3202 . . . . 5 (𝑥 = 0 → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1110imbi2d 342 . . . 4 (𝑥 = 0 → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))) ↔ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))))
12 breq2 5067 . . . . . . 7 (𝑥 = 𝑑 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑑))
1312imbi1d 343 . . . . . 6 (𝑥 = 𝑑 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1413ralbidv 3202 . . . . 5 (𝑥 = 𝑑 → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1514imbi2d 342 . . . 4 (𝑥 = 𝑑 → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))) ↔ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))))
16 breq2 5067 . . . . . . 7 (𝑥 = (𝑑 + 1) → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ (𝑑 + 1)))
1716imbi1d 343 . . . . . 6 (𝑥 = (𝑑 + 1) → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1817ralbidv 3202 . . . . 5 (𝑥 = (𝑑 + 1) → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1918imbi2d 342 . . . 4 (𝑥 = (𝑑 + 1) → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))) ↔ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))))
20 breq2 5067 . . . . . . 7 (𝑥 = 𝑀 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑀))
2120imbi1d 343 . . . . . 6 (𝑥 = 𝑀 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
2221ralbidv 3202 . . . . 5 (𝑥 = 𝑀 → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
2322imbi2d 342 . . . 4 (𝑥 = 𝑀 → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))) ↔ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))))
24 dgrco.2 . . . . . . . . . . . 12 𝑁 = (deg‘𝐺)
25 dgrco.4 . . . . . . . . . . . . 13 (𝜑𝐺 ∈ (Poly‘𝑆))
26 dgrcl 24738 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑 → (deg‘𝐺) ∈ ℕ0)
2824, 27eqeltrid 2922 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
2928nn0cnd 11946 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
3029adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑁 ∈ ℂ)
3130mul02d 10827 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (0 · 𝑁) = 0)
32 simprr 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ≤ 0)
33 dgrcl 24738 . . . . . . . . . . . 12 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
3433ad2antrl 724 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℕ0)
3534nn0ge0d 11947 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 0 ≤ (deg‘𝑓))
3634nn0red 11945 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℝ)
37 0re 10632 . . . . . . . . . . 11 0 ∈ ℝ
38 letri3 10715 . . . . . . . . . . 11 (((deg‘𝑓) ∈ ℝ ∧ 0 ∈ ℝ) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
3936, 37, 38sylancl 586 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
4032, 35, 39mpbir2and 709 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = 0)
4140oveq1d 7163 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (0 · 𝑁))
4231, 41, 403eqtr4d 2871 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (deg‘𝑓))
43 fconstmpt 5613 . . . . . . . . 9 (ℂ × {(𝑓‘0)}) = (𝑦 ∈ ℂ ↦ (𝑓‘0))
44 0dgrb 24751 . . . . . . . . . . 11 (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
4544ad2antrl 724 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
4640, 45mpbid 233 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (ℂ × {(𝑓‘0)}))
4725adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 ∈ (Poly‘𝑆))
48 plyf 24703 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
4947, 48syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺:ℂ⟶ℂ)
5049ffvelrnda 6847 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) ∧ 𝑦 ∈ ℂ) → (𝐺𝑦) ∈ ℂ)
5149feqmptd 6730 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 = (𝑦 ∈ ℂ ↦ (𝐺𝑦)))
52 fconstmpt 5613 . . . . . . . . . . 11 (ℂ × {(𝑓‘0)}) = (𝑥 ∈ ℂ ↦ (𝑓‘0))
5346, 52syl6eq 2877 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑥 ∈ ℂ ↦ (𝑓‘0)))
54 eqidd 2827 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (𝑓‘0) = (𝑓‘0))
5550, 51, 53, 54fmptco 6887 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (𝑓𝐺) = (𝑦 ∈ ℂ ↦ (𝑓‘0)))
5643, 46, 553eqtr4a 2887 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑓𝐺))
5756fveq2d 6671 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = (deg‘(𝑓𝐺)))
5842, 57eqtr2d 2862 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))
5958expr 457 . . . . 5 ((𝜑𝑓 ∈ (Poly‘ℂ)) → ((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
6059ralrimiva 3187 . . . 4 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
61 fveq2 6667 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6261breq1d 5073 . . . . . . . . 9 (𝑓 = 𝑔 → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑔) ≤ 𝑑))
63 coeq1 5727 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑓𝐺) = (𝑔𝐺))
6463fveq2d 6671 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘(𝑓𝐺)) = (deg‘(𝑔𝐺)))
6561oveq1d 7163 . . . . . . . . . 10 (𝑓 = 𝑔 → ((deg‘𝑓) · 𝑁) = ((deg‘𝑔) · 𝑁))
6664, 65eqeq12d 2842 . . . . . . . . 9 (𝑓 = 𝑔 → ((deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))
6762, 66imbi12d 346 . . . . . . . 8 (𝑓 = 𝑔 → (((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))))
6867cbvralvw 3455 . . . . . . 7 (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))
6933ad2antrl 724 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℕ0)
7069nn0red 11945 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℝ)
71 nn0p1nn 11925 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ)
7271ad2antlr 723 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℕ)
7372nnred 11642 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℝ)
7470, 73leloed 10772 . . . . . . . . . 10 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) ≤ (𝑑 + 1) ↔ ((deg‘𝑓) < (𝑑 + 1) ∨ (deg‘𝑓) = (𝑑 + 1))))
75 simplr 765 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → 𝑑 ∈ ℕ0)
76 nn0leltp1 12030 . . . . . . . . . . . . 13 (((deg‘𝑓) ∈ ℕ0𝑑 ∈ ℕ0) → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑓) < (𝑑 + 1)))
7769, 75, 76syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑓) < (𝑑 + 1)))
78 fveq2 6667 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
7978breq1d 5073 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘𝑓) ≤ 𝑑))
80 coeq1 5727 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑓 → (𝑔𝐺) = (𝑓𝐺))
8180fveq2d 6671 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘(𝑔𝐺)) = (deg‘(𝑓𝐺)))
8278oveq1d 7163 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → ((deg‘𝑔) · 𝑁) = ((deg‘𝑓) · 𝑁))
8381, 82eqeq12d 2842 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
8479, 83imbi12d 346 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
8584rspcva 3625 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) → ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
8685adantl 482 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
8777, 86sylbird 261 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) < (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
88 eqid 2826 . . . . . . . . . . . . 13 (deg‘𝑓) = (deg‘𝑓)
89 simprll 775 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → 𝑓 ∈ (Poly‘ℂ))
901, 25sseldi 3969 . . . . . . . . . . . . . 14 (𝜑𝐺 ∈ (Poly‘ℂ))
9190ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → 𝐺 ∈ (Poly‘ℂ))
92 eqid 2826 . . . . . . . . . . . . 13 (coeff‘𝑓) = (coeff‘𝑓)
93 simplr 765 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → 𝑑 ∈ ℕ0)
94 simprr 769 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → (deg‘𝑓) = (𝑑 + 1))
95 simprlr 776 . . . . . . . . . . . . . 14 (((𝜑𝑑 ∈ ℕ0) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))
96 fveq2 6667 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘𝑔) = (deg‘))
9796breq1d 5073 . . . . . . . . . . . . . . . 16 (𝑔 = → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘) ≤ 𝑑))
98 coeq1 5727 . . . . . . . . . . . . . . . . . 18 (𝑔 = → (𝑔𝐺) = (𝐺))
9998fveq2d 6671 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘(𝑔𝐺)) = (deg‘(𝐺)))
10096oveq1d 7163 . . . . . . . . . . . . . . . . 17 (𝑔 = → ((deg‘𝑔) · 𝑁) = ((deg‘) · 𝑁))
10199, 100eqeq12d 2842 . . . . . . . . . . . . . . . 16 (𝑔 = → ((deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝐺)) = ((deg‘) · 𝑁)))
10297, 101imbi12d 346 . . . . . . . . . . . . . . 15 (𝑔 = → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘) ≤ 𝑑 → (deg‘(𝐺)) = ((deg‘) · 𝑁))))
103102cbvralvw 3455 . . . . . . . . . . . . . 14 (∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ∀ ∈ (Poly‘ℂ)((deg‘) ≤ 𝑑 → (deg‘(𝐺)) = ((deg‘) · 𝑁)))
10495, 103sylib 219 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → ∀ ∈ (Poly‘ℂ)((deg‘) ≤ 𝑑 → (deg‘(𝐺)) = ((deg‘) · 𝑁)))
10588, 24, 89, 91, 92, 93, 94, 104dgrcolem2 24779 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) ∧ (deg‘𝑓) = (𝑑 + 1))) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))
106105expr 457 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) = (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
10787, 106jaod 855 . . . . . . . . . 10 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (((deg‘𝑓) < (𝑑 + 1) ∨ (deg‘𝑓) = (𝑑 + 1)) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
10874, 107sylbid 241 . . . . . . . . 9 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → ((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
109108expr 457 . . . . . . . 8 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑓 ∈ (Poly‘ℂ)) → (∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) → ((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
110109ralrimdva 3194 . . . . . . 7 ((𝜑𝑑 ∈ ℕ0) → (∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
11168, 110syl5bi 243 . . . . . 6 ((𝜑𝑑 ∈ ℕ0) → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
112111expcom 414 . . . . 5 (𝑑 ∈ ℕ0 → (𝜑 → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))))
113112a2d 29 . . . 4 (𝑑 ∈ ℕ0 → ((𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))) → (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))))
11411, 15, 19, 23, 60, 113nn0ind 12066 . . 3 (𝑀 ∈ ℕ0 → (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1157, 114mpcom 38 . 2 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
1167nn0red 11945 . . 3 (𝜑𝑀 ∈ ℝ)
117116leidd 11195 . 2 (𝜑𝑀𝑀)
118 fveq2 6667 . . . . . 6 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
119118, 4syl6eqr 2879 . . . . 5 (𝑓 = 𝐹 → (deg‘𝑓) = 𝑀)
120119breq1d 5073 . . . 4 (𝑓 = 𝐹 → ((deg‘𝑓) ≤ 𝑀𝑀𝑀))
121 coeq1 5727 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝐺) = (𝐹𝐺))
122121fveq2d 6671 . . . . 5 (𝑓 = 𝐹 → (deg‘(𝑓𝐺)) = (deg‘(𝐹𝐺)))
123119oveq1d 7163 . . . . 5 (𝑓 = 𝐹 → ((deg‘𝑓) · 𝑁) = (𝑀 · 𝑁))
124122, 123eqeq12d 2842 . . . 4 (𝑓 = 𝐹 → ((deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝐹𝐺)) = (𝑀 · 𝑁)))
125120, 124imbi12d 346 . . 3 (𝑓 = 𝐹 → (((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ (𝑀𝑀 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))))
126125rspcv 3622 . 2 (𝐹 ∈ (Poly‘ℂ) → (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) → (𝑀𝑀 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))))
1273, 115, 117, 126syl3c 66 1 (𝜑 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 843   = wceq 1530   ∈ wcel 2107  ∀wral 3143  {csn 4564   class class class wbr 5063   ↦ cmpt 5143   × cxp 5552   ∘ ccom 5558  ⟶wf 6348  ‘cfv 6352  (class class class)co 7148  ℂcc 10524  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531   < clt 10664   ≤ cle 10665  ℕcn 11627  ℕ0cn0 11886  Polycply 24689  coeffccoe 24691  degcdgr 24692 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7399  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-fl 13152  df-seq 13360  df-exp 13420  df-hash 13681  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-clim 14835  df-rlim 14836  df-sum 15033  df-0p 24186  df-ply 24693  df-coe 24695  df-dgr 24696 This theorem is referenced by:  taylply2  24871  ftalem7  25570
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