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Theorem dgrco 24792
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 24717 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 dgrco.3 . . 3 (𝜑𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 3962 . 2 (𝜑𝐹 ∈ (Poly‘ℂ))
4 dgrco.1 . . . 4 𝑀 = (deg‘𝐹)
5 dgrcl 24750 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
62, 5syl 17 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℕ0)
74, 6eqeltrid 2914 . . 3 (𝜑𝑀 ∈ ℕ0)
8 breq2 5061 . . . . . . 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 3194 . . . . 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 5061 . . . . . . 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 3194 . . . . 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 5061 . . . . . . 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 3194 . . . . 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 5061 . . . . . . 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 3194 . . . . 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 24750 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑 → (deg‘𝐺) ∈ ℕ0)
2824, 27eqeltrid 2914 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
2928nn0cnd 11945 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
3029adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑁 ∈ ℂ)
3130mul02d 10826 . . . . . . . 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 24750 . . . . . . . . . . . 12 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
3433ad2antrl 724 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℕ0)
3534nn0ge0d 11946 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 0 ≤ (deg‘𝑓))
3634nn0red 11944 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℝ)
37 0re 10631 . . . . . . . . . . 11 0 ∈ ℝ
38 letri3 10714 . . . . . . . . . . 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 7160 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (0 · 𝑁))
4231, 41, 403eqtr4d 2863 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (deg‘𝑓))
43 fconstmpt 5607 . . . . . . . . 9 (ℂ × {(𝑓‘0)}) = (𝑦 ∈ ℂ ↦ (𝑓‘0))
44 0dgrb 24763 . . . . . . . . . . 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 24715 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
4947, 48syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺:ℂ⟶ℂ)
5049ffvelrnda 6843 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) ∧ 𝑦 ∈ ℂ) → (𝐺𝑦) ∈ ℂ)
5149feqmptd 6726 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 = (𝑦 ∈ ℂ ↦ (𝐺𝑦)))
52 fconstmpt 5607 . . . . . . . . . . 11 (ℂ × {(𝑓‘0)}) = (𝑥 ∈ ℂ ↦ (𝑓‘0))
5346, 52syl6eq 2869 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑥 ∈ ℂ ↦ (𝑓‘0)))
54 eqidd 2819 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (𝑓‘0) = (𝑓‘0))
5550, 51, 53, 54fmptco 6883 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (𝑓𝐺) = (𝑦 ∈ ℂ ↦ (𝑓‘0)))
5643, 46, 553eqtr4a 2879 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑓𝐺))
5756fveq2d 6667 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = (deg‘(𝑓𝐺)))
5842, 57eqtr2d 2854 . . . . . 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 3179 . . . 4 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
61 fveq2 6663 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6261breq1d 5067 . . . . . . . . 9 (𝑓 = 𝑔 → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑔) ≤ 𝑑))
63 coeq1 5721 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑓𝐺) = (𝑔𝐺))
6463fveq2d 6667 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘(𝑓𝐺)) = (deg‘(𝑔𝐺)))
6561oveq1d 7160 . . . . . . . . . 10 (𝑓 = 𝑔 → ((deg‘𝑓) · 𝑁) = ((deg‘𝑔) · 𝑁))
6664, 65eqeq12d 2834 . . . . . . . . 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 3447 . . . . . . 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 11944 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℝ)
71 nn0p1nn 11924 . . . . . . . . . . . . 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 11641 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℝ)
7470, 73leloed 10771 . . . . . . . . . 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 12029 . . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
7978breq1d 5067 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘𝑓) ≤ 𝑑))
80 coeq1 5721 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑓 → (𝑔𝐺) = (𝑓𝐺))
8180fveq2d 6667 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘(𝑔𝐺)) = (deg‘(𝑓𝐺)))
8278oveq1d 7160 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → ((deg‘𝑔) · 𝑁) = ((deg‘𝑓) · 𝑁))
8381, 82eqeq12d 2834 . . . . . . . . . . . . . . 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 3618 . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . 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 3962 . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘𝑔) = (deg‘))
9796breq1d 5067 . . . . . . . . . . . . . . . 16 (𝑔 = → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘) ≤ 𝑑))
98 coeq1 5721 . . . . . . . . . . . . . . . . . 18 (𝑔 = → (𝑔𝐺) = (𝐺))
9998fveq2d 6667 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘(𝑔𝐺)) = (deg‘(𝐺)))
10096oveq1d 7160 . . . . . . . . . . . . . . . . 17 (𝑔 = → ((deg‘𝑔) · 𝑁) = ((deg‘) · 𝑁))
10199, 100eqeq12d 2834 . . . . . . . . . . . . . . . 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 3447 . . . . . . . . . . . . . 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 24791 . . . . . . . . . . . 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 853 . . . . . . . . . 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 3186 . . . . . . 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 12065 . . 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 11944 . . 3 (𝜑𝑀 ∈ ℝ)
117116leidd 11194 . 2 (𝜑𝑀𝑀)
118 fveq2 6663 . . . . . 6 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
119118, 4syl6eqr 2871 . . . . 5 (𝑓 = 𝐹 → (deg‘𝑓) = 𝑀)
120119breq1d 5067 . . . 4 (𝑓 = 𝐹 → ((deg‘𝑓) ≤ 𝑀𝑀𝑀))
121 coeq1 5721 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝐺) = (𝐹𝐺))
122121fveq2d 6667 . . . . 5 (𝑓 = 𝐹 → (deg‘(𝑓𝐺)) = (deg‘(𝐹𝐺)))
123119oveq1d 7160 . . . . 5 (𝑓 = 𝐹 → ((deg‘𝑓) · 𝑁) = (𝑀 · 𝑁))
124122, 123eqeq12d 2834 . . . 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 3615 . 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 841   = wceq 1528  wcel 2105  wral 3135  {csn 4557   class class class wbr 5057  cmpt 5137   × cxp 5546  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664  cn 11626  0cn0 11885  Polycply 24701  coeffccoe 24703  degcdgr 24704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-0p 24198  df-ply 24705  df-coe 24707  df-dgr 24708
This theorem is referenced by:  taylply2  24883  ftalem7  25583
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