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Theorem dgrco 24322
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 24247 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 dgrco.3 . . 3 (𝜑𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 3759 . 2 (𝜑𝐹 ∈ (Poly‘ℂ))
4 dgrco.1 . . . 4 𝑀 = (deg‘𝐹)
5 dgrcl 24280 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
62, 5syl 17 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℕ0)
74, 6syl5eqel 2848 . . 3 (𝜑𝑀 ∈ ℕ0)
8 breq2 4813 . . . . . . 7 (𝑥 = 0 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 0))
98imbi1d 332 . . . . . 6 (𝑥 = 0 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
109ralbidv 3133 . . . . 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 331 . . . 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 4813 . . . . . . 7 (𝑥 = 𝑑 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑑))
1312imbi1d 332 . . . . . 6 (𝑥 = 𝑑 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1413ralbidv 3133 . . . . 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 331 . . . 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 4813 . . . . . . 7 (𝑥 = (𝑑 + 1) → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ (𝑑 + 1)))
1716imbi1d 332 . . . . . 6 (𝑥 = (𝑑 + 1) → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1817ralbidv 3133 . . . . 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 331 . . . 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 4813 . . . . . . 7 (𝑥 = 𝑀 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑀))
2120imbi1d 332 . . . . . 6 (𝑥 = 𝑀 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
2221ralbidv 3133 . . . . 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 331 . . . 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 24280 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑 → (deg‘𝐺) ∈ ℕ0)
2824, 27syl5eqel 2848 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
2928nn0cnd 11600 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
3029adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑁 ∈ ℂ)
3130mul02d 10488 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (0 · 𝑁) = 0)
32 simprr 789 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ≤ 0)
33 dgrcl 24280 . . . . . . . . . . . 12 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
3433ad2antrl 719 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℕ0)
3534nn0ge0d 11601 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 0 ≤ (deg‘𝑓))
3634nn0red 11599 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℝ)
37 0re 10295 . . . . . . . . . . 11 0 ∈ ℝ
38 letri3 10377 . . . . . . . . . . 11 (((deg‘𝑓) ∈ ℝ ∧ 0 ∈ ℝ) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
3936, 37, 38sylancl 580 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
4032, 35, 39mpbir2and 704 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = 0)
4140oveq1d 6857 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (0 · 𝑁))
4231, 41, 403eqtr4d 2809 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (deg‘𝑓))
43 fconstmpt 5333 . . . . . . . . 9 (ℂ × {(𝑓‘0)}) = (𝑦 ∈ ℂ ↦ (𝑓‘0))
44 0dgrb 24293 . . . . . . . . . . 11 (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
4544ad2antrl 719 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
4640, 45mpbid 223 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (ℂ × {(𝑓‘0)}))
4725adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 ∈ (Poly‘𝑆))
48 plyf 24245 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
4947, 48syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺:ℂ⟶ℂ)
5049ffvelrnda 6549 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) ∧ 𝑦 ∈ ℂ) → (𝐺𝑦) ∈ ℂ)
5149feqmptd 6438 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 = (𝑦 ∈ ℂ ↦ (𝐺𝑦)))
52 fconstmpt 5333 . . . . . . . . . . 11 (ℂ × {(𝑓‘0)}) = (𝑥 ∈ ℂ ↦ (𝑓‘0))
5346, 52syl6eq 2815 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑥 ∈ ℂ ↦ (𝑓‘0)))
54 eqidd 2766 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (𝑓‘0) = (𝑓‘0))
5550, 51, 53, 54fmptco 6587 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (𝑓𝐺) = (𝑦 ∈ ℂ ↦ (𝑓‘0)))
5643, 46, 553eqtr4a 2825 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑓𝐺))
5756fveq2d 6379 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = (deg‘(𝑓𝐺)))
5842, 57eqtr2d 2800 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))
5958expr 448 . . . . 5 ((𝜑𝑓 ∈ (Poly‘ℂ)) → ((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
6059ralrimiva 3113 . . . 4 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
61 fveq2 6375 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6261breq1d 4819 . . . . . . . . 9 (𝑓 = 𝑔 → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑔) ≤ 𝑑))
63 coeq1 5448 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑓𝐺) = (𝑔𝐺))
6463fveq2d 6379 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘(𝑓𝐺)) = (deg‘(𝑔𝐺)))
6561oveq1d 6857 . . . . . . . . . 10 (𝑓 = 𝑔 → ((deg‘𝑓) · 𝑁) = ((deg‘𝑔) · 𝑁))
6664, 65eqeq12d 2780 . . . . . . . . 9 (𝑓 = 𝑔 → ((deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))
6762, 66imbi12d 335 . . . . . . . 8 (𝑓 = 𝑔 → (((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))))
6867cbvralv 3319 . . . . . . 7 (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))
6933ad2antrl 719 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℕ0)
7069nn0red 11599 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℝ)
71 nn0p1nn 11579 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ)
7271ad2antlr 718 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℕ)
7372nnred 11291 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℝ)
7470, 73leloed 10434 . . . . . . . . . 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 785 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → 𝑑 ∈ ℕ0)
76 nn0leltp1 11683 . . . . . . . . . . . . 13 (((deg‘𝑓) ∈ ℕ0𝑑 ∈ ℕ0) → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑓) < (𝑑 + 1)))
7769, 75, 76syl2anc 579 . . . . . . . . . . . 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 6375 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
7978breq1d 4819 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘𝑓) ≤ 𝑑))
80 coeq1 5448 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑓 → (𝑔𝐺) = (𝑓𝐺))
8180fveq2d 6379 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘(𝑔𝐺)) = (deg‘(𝑓𝐺)))
8278oveq1d 6857 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → ((deg‘𝑔) · 𝑁) = ((deg‘𝑓) · 𝑁))
8381, 82eqeq12d 2780 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
8479, 83imbi12d 335 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
8584rspcva 3459 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))) → ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
8685adantl 473 . . . . . . . . . . . 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 251 . . . . . . . . . . 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 2765 . . . . . . . . . . . . 13 (deg‘𝑓) = (deg‘𝑓)
89 simprll 797 . . . . . . . . . . . . 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 3759 . . . . . . . . . . . . . 14 (𝜑𝐺 ∈ (Poly‘ℂ))
9190ad2antrr 717 . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . 13 (coeff‘𝑓) = (coeff‘𝑓)
93 simplr 785 . . . . . . . . . . . . 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 789 . . . . . . . . . . . . 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 798 . . . . . . . . . . . . . 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 6375 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘𝑔) = (deg‘))
9796breq1d 4819 . . . . . . . . . . . . . . . 16 (𝑔 = → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘) ≤ 𝑑))
98 coeq1 5448 . . . . . . . . . . . . . . . . . 18 (𝑔 = → (𝑔𝐺) = (𝐺))
9998fveq2d 6379 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘(𝑔𝐺)) = (deg‘(𝐺)))
10096oveq1d 6857 . . . . . . . . . . . . . . . . 17 (𝑔 = → ((deg‘𝑔) · 𝑁) = ((deg‘) · 𝑁))
10199, 100eqeq12d 2780 . . . . . . . . . . . . . . . 16 (𝑔 = → ((deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝐺)) = ((deg‘) · 𝑁)))
10297, 101imbi12d 335 . . . . . . . . . . . . . . 15 (𝑔 = → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘) ≤ 𝑑 → (deg‘(𝐺)) = ((deg‘) · 𝑁))))
103102cbvralv 3319 . . . . . . . . . . . . . 14 (∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ∀ ∈ (Poly‘ℂ)((deg‘) ≤ 𝑑 → (deg‘(𝐺)) = ((deg‘) · 𝑁)))
10495, 103sylib 209 . . . . . . . . . . . . 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 24321 . . . . . . . . . . . 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 448 . . . . . . . . . . 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 885 . . . . . . . . . 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 231 . . . . . . . . 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 448 . . . . . . . 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 3116 . . . . . . 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 233 . . . . . 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 402 . . . . 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 11719 . . 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 11599 . . 3 (𝜑𝑀 ∈ ℝ)
117116leidd 10848 . 2 (𝜑𝑀𝑀)
118 fveq2 6375 . . . . . 6 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
119118, 4syl6eqr 2817 . . . . 5 (𝑓 = 𝐹 → (deg‘𝑓) = 𝑀)
120119breq1d 4819 . . . 4 (𝑓 = 𝐹 → ((deg‘𝑓) ≤ 𝑀𝑀𝑀))
121 coeq1 5448 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝐺) = (𝐹𝐺))
122121fveq2d 6379 . . . . 5 (𝑓 = 𝐹 → (deg‘(𝑓𝐺)) = (deg‘(𝐹𝐺)))
123119oveq1d 6857 . . . . 5 (𝑓 = 𝐹 → ((deg‘𝑓) · 𝑁) = (𝑀 · 𝑁))
124122, 123eqeq12d 2780 . . . 4 (𝑓 = 𝐹 → ((deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝐹𝐺)) = (𝑀 · 𝑁)))
125120, 124imbi12d 335 . . 3 (𝑓 = 𝐹 → (((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ (𝑀𝑀 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))))
126125rspcv 3457 . 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 197  wa 384  wo 873   = wceq 1652  wcel 2155  wral 3055  {csn 4334   class class class wbr 4809  cmpt 4888   × cxp 5275  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194   < clt 10328  cle 10329  cn 11274  0cn0 11538  Polycply 24231  coeffccoe 24233  degcdgr 24234
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 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268
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 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-rlim 14505  df-sum 14702  df-0p 23728  df-ply 24235  df-coe 24237  df-dgr 24238
This theorem is referenced by:  taylply2  24413  ftalem7  25096
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