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Theorem dgrco 26329
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 26253 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 dgrco.3 . . 3 (𝜑𝐹 ∈ (Poly‘𝑆))
31, 2sselid 3992 . 2 (𝜑𝐹 ∈ (Poly‘ℂ))
4 dgrco.1 . . . 4 𝑀 = (deg‘𝐹)
5 dgrcl 26286 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
62, 5syl 17 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℕ0)
74, 6eqeltrid 2842 . . 3 (𝜑𝑀 ∈ ℕ0)
8 breq2 5151 . . . . . . 7 (𝑥 = 0 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 0))
98imbi1d 341 . . . . . 6 (𝑥 = 0 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
109ralbidv 3175 . . . . 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 340 . . . 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 5151 . . . . . . 7 (𝑥 = 𝑑 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑑))
1312imbi1d 341 . . . . . 6 (𝑥 = 𝑑 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1413ralbidv 3175 . . . . 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 340 . . . 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 5151 . . . . . . 7 (𝑥 = (𝑑 + 1) → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ (𝑑 + 1)))
1716imbi1d 341 . . . . . 6 (𝑥 = (𝑑 + 1) → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1817ralbidv 3175 . . . . 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 340 . . . 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 5151 . . . . . . 7 (𝑥 = 𝑀 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑀))
2120imbi1d 341 . . . . . 6 (𝑥 = 𝑀 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
2221ralbidv 3175 . . . . 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 340 . . . 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 26286 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑 → (deg‘𝐺) ∈ ℕ0)
2824, 27eqeltrid 2842 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
2928nn0cnd 12586 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
3029adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑁 ∈ ℂ)
3130mul02d 11456 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (0 · 𝑁) = 0)
32 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ≤ 0)
33 dgrcl 26286 . . . . . . . . . . . 12 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
3433ad2antrl 728 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℕ0)
3534nn0ge0d 12587 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 0 ≤ (deg‘𝑓))
3634nn0red 12585 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℝ)
37 0re 11260 . . . . . . . . . . 11 0 ∈ ℝ
38 letri3 11343 . . . . . . . . . . 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 713 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = 0)
4140oveq1d 7445 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (0 · 𝑁))
4231, 41, 403eqtr4d 2784 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (deg‘𝑓))
43 fconstmpt 5750 . . . . . . . . 9 (ℂ × {(𝑓‘0)}) = (𝑦 ∈ ℂ ↦ (𝑓‘0))
44 0dgrb 26299 . . . . . . . . . . 11 (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
4544ad2antrl 728 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
4640, 45mpbid 232 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (ℂ × {(𝑓‘0)}))
4725adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 ∈ (Poly‘𝑆))
48 plyf 26251 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
4947, 48syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺:ℂ⟶ℂ)
5049ffvelcdmda 7103 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) ∧ 𝑦 ∈ ℂ) → (𝐺𝑦) ∈ ℂ)
5149feqmptd 6976 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 = (𝑦 ∈ ℂ ↦ (𝐺𝑦)))
52 fconstmpt 5750 . . . . . . . . . . 11 (ℂ × {(𝑓‘0)}) = (𝑥 ∈ ℂ ↦ (𝑓‘0))
5346, 52eqtrdi 2790 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑥 ∈ ℂ ↦ (𝑓‘0)))
54 eqidd 2735 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (𝑓‘0) = (𝑓‘0))
5550, 51, 53, 54fmptco 7148 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (𝑓𝐺) = (𝑦 ∈ ℂ ↦ (𝑓‘0)))
5643, 46, 553eqtr4a 2800 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑓𝐺))
5756fveq2d 6910 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = (deg‘(𝑓𝐺)))
5842, 57eqtr2d 2775 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))
5958expr 456 . . . . 5 ((𝜑𝑓 ∈ (Poly‘ℂ)) → ((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
6059ralrimiva 3143 . . . 4 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
61 fveq2 6906 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6261breq1d 5157 . . . . . . . . 9 (𝑓 = 𝑔 → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑔) ≤ 𝑑))
63 coeq1 5870 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑓𝐺) = (𝑔𝐺))
6463fveq2d 6910 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘(𝑓𝐺)) = (deg‘(𝑔𝐺)))
6561oveq1d 7445 . . . . . . . . . 10 (𝑓 = 𝑔 → ((deg‘𝑓) · 𝑁) = ((deg‘𝑔) · 𝑁))
6664, 65eqeq12d 2750 . . . . . . . . 9 (𝑓 = 𝑔 → ((deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))
6762, 66imbi12d 344 . . . . . . . 8 (𝑓 = 𝑔 → (((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))))
6867cbvralvw 3234 . . . . . . 7 (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))
6933ad2antrl 728 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℕ0)
7069nn0red 12585 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℝ)
71 nn0p1nn 12562 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ)
7271ad2antlr 727 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℕ)
7372nnred 12278 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℝ)
7470, 73leloed 11401 . . . . . . . . . 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 769 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → 𝑑 ∈ ℕ0)
76 nn0leltp1 12674 . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
7978breq1d 5157 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘𝑓) ≤ 𝑑))
80 coeq1 5870 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑓 → (𝑔𝐺) = (𝑓𝐺))
8180fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘(𝑔𝐺)) = (deg‘(𝑓𝐺)))
8278oveq1d 7445 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → ((deg‘𝑔) · 𝑁) = ((deg‘𝑓) · 𝑁))
8381, 82eqeq12d 2750 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
8479, 83imbi12d 344 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
8584rspcva 3619 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) → ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
8685adantl 481 . . . . . . . . . . . 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 260 . . . . . . . . . . 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 2734 . . . . . . . . . . . . 13 (deg‘𝑓) = (deg‘𝑓)
89 simprll 779 . . . . . . . . . . . . 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, 25sselid 3992 . . . . . . . . . . . . . 14 (𝜑𝐺 ∈ (Poly‘ℂ))
9190ad2antrr 726 . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . 13 (coeff‘𝑓) = (coeff‘𝑓)
93 simplr 769 . . . . . . . . . . . . 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 773 . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘𝑔) = (deg‘))
9796breq1d 5157 . . . . . . . . . . . . . . . 16 (𝑔 = → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘) ≤ 𝑑))
98 coeq1 5870 . . . . . . . . . . . . . . . . . 18 (𝑔 = → (𝑔𝐺) = (𝐺))
9998fveq2d 6910 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘(𝑔𝐺)) = (deg‘(𝐺)))
10096oveq1d 7445 . . . . . . . . . . . . . . . . 17 (𝑔 = → ((deg‘𝑔) · 𝑁) = ((deg‘) · 𝑁))
10199, 100eqeq12d 2750 . . . . . . . . . . . . . . . 16 (𝑔 = → ((deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝐺)) = ((deg‘) · 𝑁)))
10297, 101imbi12d 344 . . . . . . . . . . . . . . 15 (𝑔 = → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘) ≤ 𝑑 → (deg‘(𝐺)) = ((deg‘) · 𝑁))))
103102cbvralvw 3234 . . . . . . . . . . . . . 14 (∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ∀ ∈ (Poly‘ℂ)((deg‘) ≤ 𝑑 → (deg‘(𝐺)) = ((deg‘) · 𝑁)))
10495, 103sylib 218 . . . . . . . . . . . . 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 26328 . . . . . . . . . . . 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 456 . . . . . . . . . . 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 859 . . . . . . . . . 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 240 . . . . . . . . 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 456 . . . . . . . 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 3151 . . . . . . 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, 110biimtrid 242 . . . . . 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 413 . . . . 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 12710 . . 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 12585 . . 3 (𝜑𝑀 ∈ ℝ)
117116leidd 11826 . 2 (𝜑𝑀𝑀)
118 fveq2 6906 . . . . . 6 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
119118, 4eqtr4di 2792 . . . . 5 (𝑓 = 𝐹 → (deg‘𝑓) = 𝑀)
120119breq1d 5157 . . . 4 (𝑓 = 𝐹 → ((deg‘𝑓) ≤ 𝑀𝑀𝑀))
121 coeq1 5870 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝐺) = (𝐹𝐺))
122121fveq2d 6910 . . . . 5 (𝑓 = 𝐹 → (deg‘(𝑓𝐺)) = (deg‘(𝐹𝐺)))
123119oveq1d 7445 . . . . 5 (𝑓 = 𝐹 → ((deg‘𝑓) · 𝑁) = (𝑀 · 𝑁))
124122, 123eqeq12d 2750 . . . 4 (𝑓 = 𝐹 → ((deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝐹𝐺)) = (𝑀 · 𝑁)))
125120, 124imbi12d 344 . . 3 (𝑓 = 𝐹 → (((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ (𝑀𝑀 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))))
126125rspcv 3617 . 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 206  wa 395  wo 847   = wceq 1536  wcel 2105  wral 3058  {csn 4630   class class class wbr 5147  cmpt 5230   × cxp 5686  ccom 5692  wf 6558  cfv 6562  (class class class)co 7430  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157   < clt 11292  cle 11293  cn 12263  0cn0 12523  Polycply 26237  coeffccoe 26239  degcdgr 26240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-0p 25718  df-ply 26241  df-coe 26243  df-dgr 26244
This theorem is referenced by:  taylply2  26423  taylply2OLD  26424  ftalem7  27136
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