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Theorem dgrco 24076
 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 24001 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 dgrco.3 . . 3 (𝜑𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 3634 . 2 (𝜑𝐹 ∈ (Poly‘ℂ))
4 dgrco.1 . . . 4 𝑀 = (deg‘𝐹)
5 dgrcl 24034 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
62, 5syl 17 . . . 4 (𝜑 → (deg‘𝐹) ∈ ℕ0)
74, 6syl5eqel 2734 . . 3 (𝜑𝑀 ∈ ℕ0)
8 breq2 4689 . . . . . . 7 (𝑥 = 0 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 0))
98imbi1d 330 . . . . . 6 (𝑥 = 0 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
109ralbidv 3015 . . . . 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 329 . . . 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 4689 . . . . . . 7 (𝑥 = 𝑑 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑑))
1312imbi1d 330 . . . . . 6 (𝑥 = 𝑑 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1413ralbidv 3015 . . . . 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 329 . . . 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 4689 . . . . . . 7 (𝑥 = (𝑑 + 1) → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ (𝑑 + 1)))
1716imbi1d 330 . . . . . 6 (𝑥 = (𝑑 + 1) → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ (𝑑 + 1) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
1817ralbidv 3015 . . . . 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 329 . . . 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 4689 . . . . . . 7 (𝑥 = 𝑀 → ((deg‘𝑓) ≤ 𝑥 ↔ (deg‘𝑓) ≤ 𝑀))
2120imbi1d 330 . . . . . 6 (𝑥 = 𝑀 → (((deg‘𝑓) ≤ 𝑥 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
2221ralbidv 3015 . . . . 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 329 . . . 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 24034 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑 → (deg‘𝐺) ∈ ℕ0)
2824, 27syl5eqel 2734 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
2928nn0cnd 11391 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
3029adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑁 ∈ ℂ)
3130mul02d 10272 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (0 · 𝑁) = 0)
32 simprr 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ≤ 0)
33 dgrcl 24034 . . . . . . . . . . . 12 (𝑓 ∈ (Poly‘ℂ) → (deg‘𝑓) ∈ ℕ0)
3433ad2antrl 764 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℕ0)
3534nn0ge0d 11392 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 0 ≤ (deg‘𝑓))
3634nn0red 11390 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) ∈ ℝ)
37 0re 10078 . . . . . . . . . . 11 0 ∈ ℝ
38 letri3 10161 . . . . . . . . . . 11 (((deg‘𝑓) ∈ ℝ ∧ 0 ∈ ℝ) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
3936, 37, 38sylancl 695 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ ((deg‘𝑓) ≤ 0 ∧ 0 ≤ (deg‘𝑓))))
4032, 35, 39mpbir2and 977 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = 0)
4140oveq1d 6705 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (0 · 𝑁))
4231, 41, 403eqtr4d 2695 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) · 𝑁) = (deg‘𝑓))
43 fconstmpt 5197 . . . . . . . . 9 (ℂ × {(𝑓‘0)}) = (𝑦 ∈ ℂ ↦ (𝑓‘0))
44 0dgrb 24047 . . . . . . . . . . 11 (𝑓 ∈ (Poly‘ℂ) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
4544ad2antrl 764 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → ((deg‘𝑓) = 0 ↔ 𝑓 = (ℂ × {(𝑓‘0)})))
4640, 45mpbid 222 . . . . . . . . 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 23999 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
4947, 48syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺:ℂ⟶ℂ)
5049ffvelrnda 6399 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) ∧ 𝑦 ∈ ℂ) → (𝐺𝑦) ∈ ℂ)
5149feqmptd 6288 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝐺 = (𝑦 ∈ ℂ ↦ (𝐺𝑦)))
52 fconstmpt 5197 . . . . . . . . . . 11 (ℂ × {(𝑓‘0)}) = (𝑥 ∈ ℂ ↦ (𝑓‘0))
5346, 52syl6eq 2701 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑥 ∈ ℂ ↦ (𝑓‘0)))
54 eqidd 2652 . . . . . . . . . 10 (𝑥 = (𝐺𝑦) → (𝑓‘0) = (𝑓‘0))
5550, 51, 53, 54fmptco 6436 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (𝑓𝐺) = (𝑦 ∈ ℂ ↦ (𝑓‘0)))
5643, 46, 553eqtr4a 2711 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → 𝑓 = (𝑓𝐺))
5756fveq2d 6233 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘𝑓) = (deg‘(𝑓𝐺)))
5842, 57eqtr2d 2686 . . . . . 6 ((𝜑 ∧ (𝑓 ∈ (Poly‘ℂ) ∧ (deg‘𝑓) ≤ 0)) → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))
5958expr 642 . . . . 5 ((𝜑𝑓 ∈ (Poly‘ℂ)) → ((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
6059ralrimiva 2995 . . . 4 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 0 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
61 fveq2 6229 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘𝑓) = (deg‘𝑔))
6261breq1d 4695 . . . . . . . . 9 (𝑓 = 𝑔 → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑔) ≤ 𝑑))
63 coeq1 5312 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑓𝐺) = (𝑔𝐺))
6463fveq2d 6233 . . . . . . . . . 10 (𝑓 = 𝑔 → (deg‘(𝑓𝐺)) = (deg‘(𝑔𝐺)))
6561oveq1d 6705 . . . . . . . . . 10 (𝑓 = 𝑔 → ((deg‘𝑓) · 𝑁) = ((deg‘𝑔) · 𝑁))
6664, 65eqeq12d 2666 . . . . . . . . 9 (𝑓 = 𝑔 → ((deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))
6762, 66imbi12d 333 . . . . . . . 8 (𝑓 = 𝑔 → (((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁))))
6867cbvralv 3201 . . . . . . 7 (∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))
6933ad2antrl 764 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℕ0)
7069nn0red 11390 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (deg‘𝑓) ∈ ℝ)
71 nn0p1nn 11370 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ)
7271ad2antlr 763 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℕ)
7372nnred 11073 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → (𝑑 + 1) ∈ ℝ)
7470, 73leloed 10218 . . . . . . . . . 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 807 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑓 ∈ (Poly‘ℂ) ∧ ∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)))) → 𝑑 ∈ ℕ0)
76 nn0leltp1 11474 . . . . . . . . . . . . 13 (((deg‘𝑓) ∈ ℕ0𝑑 ∈ ℕ0) → ((deg‘𝑓) ≤ 𝑑 ↔ (deg‘𝑓) < (𝑑 + 1)))
7769, 75, 76syl2anc 694 . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
7978breq1d 4695 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘𝑓) ≤ 𝑑))
80 coeq1 5312 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑓 → (𝑔𝐺) = (𝑓𝐺))
8180fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → (deg‘(𝑔𝐺)) = (deg‘(𝑓𝐺)))
8278oveq1d 6705 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑓 → ((deg‘𝑔) · 𝑁) = ((deg‘𝑓) · 𝑁))
8381, 82eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑔 = 𝑓 → ((deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))
8479, 83imbi12d 333 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘𝑓) ≤ 𝑑 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁))))
8584rspcva 3338 . . . . . . . . . . . . 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 250 . . . . . . . . . . 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 2651 . . . . . . . . . . . . 13 (deg‘𝑓) = (deg‘𝑓)
89 simprll 819 . . . . . . . . . . . . 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 3634 . . . . . . . . . . . . . 14 (𝜑𝐺 ∈ (Poly‘ℂ))
9190ad2antrr 762 . . . . . . . . . . . . 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 2651 . . . . . . . . . . . . 13 (coeff‘𝑓) = (coeff‘𝑓)
93 simplr 807 . . . . . . . . . . . . 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 811 . . . . . . . . . . . . 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 820 . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘𝑔) = (deg‘))
9796breq1d 4695 . . . . . . . . . . . . . . . 16 (𝑔 = → ((deg‘𝑔) ≤ 𝑑 ↔ (deg‘) ≤ 𝑑))
98 coeq1 5312 . . . . . . . . . . . . . . . . . 18 (𝑔 = → (𝑔𝐺) = (𝐺))
9998fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑔 = → (deg‘(𝑔𝐺)) = (deg‘(𝐺)))
10096oveq1d 6705 . . . . . . . . . . . . . . . . 17 (𝑔 = → ((deg‘𝑔) · 𝑁) = ((deg‘) · 𝑁))
10199, 100eqeq12d 2666 . . . . . . . . . . . . . . . 16 (𝑔 = → ((deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁) ↔ (deg‘(𝐺)) = ((deg‘) · 𝑁)))
10297, 101imbi12d 333 . . . . . . . . . . . . . . 15 (𝑔 = → (((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ((deg‘) ≤ 𝑑 → (deg‘(𝐺)) = ((deg‘) · 𝑁))))
103102cbvralv 3201 . . . . . . . . . . . . . 14 (∀𝑔 ∈ (Poly‘ℂ)((deg‘𝑔) ≤ 𝑑 → (deg‘(𝑔𝐺)) = ((deg‘𝑔) · 𝑁)) ↔ ∀ ∈ (Poly‘ℂ)((deg‘) ≤ 𝑑 → (deg‘(𝐺)) = ((deg‘) · 𝑁)))
10495, 103sylib 208 . . . . . . . . . . . . 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 24075 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 394 . . . . . . . . . 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 230 . . . . . . . . 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 642 . . . . . . . 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 2998 . . . . . . 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 232 . . . . . 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 450 . . . . 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 11510 . . 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 11390 . . 3 (𝜑𝑀 ∈ ℝ)
117116leidd 10632 . 2 (𝜑𝑀𝑀)
118 fveq2 6229 . . . . . 6 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
119118, 4syl6eqr 2703 . . . . 5 (𝑓 = 𝐹 → (deg‘𝑓) = 𝑀)
120119breq1d 4695 . . . 4 (𝑓 = 𝐹 → ((deg‘𝑓) ≤ 𝑀𝑀𝑀))
121 coeq1 5312 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝐺) = (𝐹𝐺))
122121fveq2d 6233 . . . . 5 (𝑓 = 𝐹 → (deg‘(𝑓𝐺)) = (deg‘(𝐹𝐺)))
123119oveq1d 6705 . . . . 5 (𝑓 = 𝐹 → ((deg‘𝑓) · 𝑁) = (𝑀 · 𝑁))
124122, 123eqeq12d 2666 . . . 4 (𝑓 = 𝐹 → ((deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁) ↔ (deg‘(𝐹𝐺)) = (𝑀 · 𝑁)))
125120, 124imbi12d 333 . . 3 (𝑓 = 𝐹 → (((deg‘𝑓) ≤ 𝑀 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)) ↔ (𝑀𝑀 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))))
126125rspcv 3336 . 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 196   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  {csn 4210   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141   ∘ ccom 5147  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112   ≤ cle 10113  ℕcn 11058  ℕ0cn0 11330  Polycply 23985  coeffccoe 23987  degcdgr 23988 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-0p 23482  df-ply 23989  df-coe 23991  df-dgr 23992 This theorem is referenced by:  taylply2  24167  ftalem7  24850
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