plydiveu - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > plydiveu		Structured version Visualization version GIF version

Theorem plydiveu 23857

Description: Lemma for plydivalg 23858. (Contributed by Mario Carneiro, 24-Jul-2014.)

**Hypotheses**
Ref	Expression
plydiv.pl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
plydiv.tm	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
plydiv.rc	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)
plydiv.m1	⊢ (𝜑 → -1 ∈ 𝑆)
plydiv.f	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
plydiv.g	⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
plydiv.z	⊢ (𝜑 → 𝐺 ≠ 0_𝑝)
plydiv.r	⊢ 𝑅 = (𝐹 ∘_𝑓 − (𝐺 ∘_𝑓 · 𝑞))
plydiveu.q	⊢ (𝜑 → 𝑞 ∈ (Poly‘𝑆))
plydiveu.qd	⊢ (𝜑 → (𝑅 = 0_𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
plydiveu.t	⊢ 𝑇 = (𝐹 ∘_𝑓 − (𝐺 ∘_𝑓 · 𝑝))
plydiveu.p	⊢ (𝜑 → 𝑝 ∈ (Poly‘𝑆))
plydiveu.pd	⊢ (𝜑 → (𝑇 = 0_𝑝 ∨ (deg‘𝑇) < (deg‘𝐺)))

**Assertion**
Ref	Expression
plydiveu	⊢ (𝜑 → 𝑝 = 𝑞)

Distinct variable groups: 𝑥,𝑦 𝑞,𝑝,𝑥,𝑦,𝐹 𝜑,𝑥,𝑦 𝑥,𝑇,𝑦 𝐺,𝑝,𝑞,𝑥,𝑦 𝑅,𝑝,𝑥,𝑦 𝑆,𝑝,𝑞,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑞,𝑝) 𝑅(𝑞) 𝑇(𝑞,𝑝)

Proof of Theorem plydiveu Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idd 24	. . . 4 ⊢ (𝜑 → ((𝑝 ∘_𝑓 − 𝑞) = 0_𝑝 → (𝑝 ∘_𝑓 − 𝑞) = 0_𝑝))
2		plydiveu.q	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑞 ∈ (Poly‘𝑆))
3		plydiv.pl	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
4		plydiv.tm	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
5		plydiv.rc	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)
6		plydiv.m1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → -1 ∈ 𝑆)
7		plydiv.f	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
8		plydiv.g	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
9		plydiv.z	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ≠ 0_𝑝)
10		plydiv.r	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅 = (𝐹 ∘_𝑓 − (𝐺 ∘_𝑓 · 𝑞))
11	3, 4, 5, 6, 7, 8, 9, 10	plydivlem2 23853	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → 𝑅 ∈ (Poly‘𝑆))
12	2, 11	mpdan 699	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ (Poly‘𝑆))
13		plydiveu.p	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑝 ∈ (Poly‘𝑆))
14		plydiveu.t	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 = (𝐹 ∘_𝑓 − (𝐺 ∘_𝑓 · 𝑝))
15	3, 4, 5, 6, 7, 8, 9, 14	plydivlem2 23853	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Poly‘𝑆)) → 𝑇 ∈ (Poly‘𝑆))
16	13, 15	mpdan 699	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 ∈ (Poly‘𝑆))
17	12, 16, 3, 4, 6	plysub 23779	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑅 ∘_𝑓 − 𝑇) ∈ (Poly‘𝑆))
18		dgrcl 23793	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∘_𝑓 − 𝑇) ∈ (Poly‘𝑆) → (deg‘(𝑅 ∘_𝑓 − 𝑇)) ∈ ℕ₀)
19	17, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (deg‘(𝑅 ∘_𝑓 − 𝑇)) ∈ ℕ₀)
20	19	nn0red 11229	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘(𝑅 ∘_𝑓 − 𝑇)) ∈ ℝ)
21		dgrcl 23793	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (Poly‘𝑆) → (deg‘𝑇) ∈ ℕ₀)
22	16, 21	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (deg‘𝑇) ∈ ℕ₀)
23	22	nn0red 11229	. . . . . . . . . . . . 13 ⊢ (𝜑 → (deg‘𝑇) ∈ ℝ)
24		dgrcl 23793	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (Poly‘𝑆) → (deg‘𝑅) ∈ ℕ₀)
25	12, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (deg‘𝑅) ∈ ℕ₀)
26	25	nn0red 11229	. . . . . . . . . . . . 13 ⊢ (𝜑 → (deg‘𝑅) ∈ ℝ)
27	23, 26	ifcld 4081	. . . . . . . . . . . 12 ⊢ (𝜑 → if((deg‘𝑅) ≤ (deg‘𝑇), (deg‘𝑇), (deg‘𝑅)) ∈ ℝ)
28		dgrcl 23793	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
29	8, 28	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ₀)
30	29	nn0red 11229	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐺) ∈ ℝ)
31		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (deg‘𝑅) = (deg‘𝑅)
32		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (deg‘𝑇) = (deg‘𝑇)
33	31, 32	dgrsub 23832	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (Poly‘𝑆) ∧ 𝑇 ∈ (Poly‘𝑆)) → (deg‘(𝑅 ∘_𝑓 − 𝑇)) ≤ if((deg‘𝑅) ≤ (deg‘𝑇), (deg‘𝑇), (deg‘𝑅)))
34	12, 16, 33	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘(𝑅 ∘_𝑓 − 𝑇)) ≤ if((deg‘𝑅) ≤ (deg‘𝑇), (deg‘𝑇), (deg‘𝑅)))
35		plydiveu.pd	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑇 = 0_𝑝 ∨ (deg‘𝑇) < (deg‘𝐺)))
36		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (coeff‘𝑇) = (coeff‘𝑇)
37	32, 36	dgrlt 23826	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ (Poly‘𝑆) ∧ (deg‘𝐺) ∈ ℕ₀) → ((𝑇 = 0_𝑝 ∨ (deg‘𝑇) < (deg‘𝐺)) ↔ ((deg‘𝑇) ≤ (deg‘𝐺) ∧ ((coeff‘𝑇)‘(deg‘𝐺)) = 0)))
38	16, 29, 37	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑇 = 0_𝑝 ∨ (deg‘𝑇) < (deg‘𝐺)) ↔ ((deg‘𝑇) ≤ (deg‘𝐺) ∧ ((coeff‘𝑇)‘(deg‘𝐺)) = 0)))
39	35, 38	mpbid 221	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((deg‘𝑇) ≤ (deg‘𝐺) ∧ ((coeff‘𝑇)‘(deg‘𝐺)) = 0))
40	39	simpld 474	. . . . . . . . . . . . 13 ⊢ (𝜑 → (deg‘𝑇) ≤ (deg‘𝐺))
41		plydiveu.qd	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅 = 0_𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
42		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (coeff‘𝑅) = (coeff‘𝑅)
43	31, 42	dgrlt 23826	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ (Poly‘𝑆) ∧ (deg‘𝐺) ∈ ℕ₀) → ((𝑅 = 0_𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)) ↔ ((deg‘𝑅) ≤ (deg‘𝐺) ∧ ((coeff‘𝑅)‘(deg‘𝐺)) = 0)))
44	12, 29, 43	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑅 = 0_𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)) ↔ ((deg‘𝑅) ≤ (deg‘𝐺) ∧ ((coeff‘𝑅)‘(deg‘𝐺)) = 0)))
45	41, 44	mpbid 221	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((deg‘𝑅) ≤ (deg‘𝐺) ∧ ((coeff‘𝑅)‘(deg‘𝐺)) = 0))
46	45	simpld 474	. . . . . . . . . . . . 13 ⊢ (𝜑 → (deg‘𝑅) ≤ (deg‘𝐺))
47		breq1 4586	. . . . . . . . . . . . . 14 ⊢ ((deg‘𝑇) = if((deg‘𝑅) ≤ (deg‘𝑇), (deg‘𝑇), (deg‘𝑅)) → ((deg‘𝑇) ≤ (deg‘𝐺) ↔ if((deg‘𝑅) ≤ (deg‘𝑇), (deg‘𝑇), (deg‘𝑅)) ≤ (deg‘𝐺)))
48		breq1 4586	. . . . . . . . . . . . . 14 ⊢ ((deg‘𝑅) = if((deg‘𝑅) ≤ (deg‘𝑇), (deg‘𝑇), (deg‘𝑅)) → ((deg‘𝑅) ≤ (deg‘𝐺) ↔ if((deg‘𝑅) ≤ (deg‘𝑇), (deg‘𝑇), (deg‘𝑅)) ≤ (deg‘𝐺)))
49	47, 48	ifboth 4074	. . . . . . . . . . . . 13 ⊢ (((deg‘𝑇) ≤ (deg‘𝐺) ∧ (deg‘𝑅) ≤ (deg‘𝐺)) → if((deg‘𝑅) ≤ (deg‘𝑇), (deg‘𝑇), (deg‘𝑅)) ≤ (deg‘𝐺))
50	40, 46, 49	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → if((deg‘𝑅) ≤ (deg‘𝑇), (deg‘𝑇), (deg‘𝑅)) ≤ (deg‘𝐺))
51	20, 27, 30, 34, 50	letrd 10073	. . . . . . . . . . 11 ⊢ (𝜑 → (deg‘(𝑅 ∘_𝑓 − 𝑇)) ≤ (deg‘𝐺))
52	51	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (deg‘(𝑅 ∘_𝑓 − 𝑇)) ≤ (deg‘𝐺))
53	13, 2, 3, 4, 6	plysub 23779	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑝 ∘_𝑓 − 𝑞) ∈ (Poly‘𝑆))
54		dgrcl 23793	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∘_𝑓 − 𝑞) ∈ (Poly‘𝑆) → (deg‘(𝑝 ∘_𝑓 − 𝑞)) ∈ ℕ₀)
55	53, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (deg‘(𝑝 ∘_𝑓 − 𝑞)) ∈ ℕ₀)
56		nn0addge1 11216	. . . . . . . . . . . . 13 ⊢ (((deg‘𝐺) ∈ ℝ ∧ (deg‘(𝑝 ∘_𝑓 − 𝑞)) ∈ ℕ₀) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝑝 ∘_𝑓 − 𝑞))))
57	30, 55, 56	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝑝 ∘_𝑓 − 𝑞))))
58	57	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝑝 ∘_𝑓 − 𝑞))))
59		plyf 23758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
60	7, 59	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
61	60	ffvelrnda 6267	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐹‘𝑧) ∈ ℂ)
62	8, 2, 3, 4	plymul 23778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 ∘_𝑓 · 𝑞) ∈ (Poly‘𝑆))
63		plyf 23758	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∘_𝑓 · 𝑞) ∈ (Poly‘𝑆) → (𝐺 ∘_𝑓 · 𝑞):ℂ⟶ℂ)
64	62, 63	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺 ∘_𝑓 · 𝑞):ℂ⟶ℂ)
65	64	ffvelrnda 6267	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺 ∘_𝑓 · 𝑞)‘𝑧) ∈ ℂ)
66	8, 13, 3, 4	plymul 23778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 ∘_𝑓 · 𝑝) ∈ (Poly‘𝑆))
67		plyf 23758	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∘_𝑓 · 𝑝) ∈ (Poly‘𝑆) → (𝐺 ∘_𝑓 · 𝑝):ℂ⟶ℂ)
68	66, 67	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺 ∘_𝑓 · 𝑝):ℂ⟶ℂ)
69	68	ffvelrnda 6267	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺 ∘_𝑓 · 𝑝)‘𝑧) ∈ ℂ)
70	61, 65, 69	nnncan1d 10305	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑞)‘𝑧)) − ((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑝)‘𝑧))) = (((𝐺 ∘_𝑓 · 𝑝)‘𝑧) − ((𝐺 ∘_𝑓 · 𝑞)‘𝑧)))
71	70	mpteq2dva 4672	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ (((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑞)‘𝑧)) − ((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑝)‘𝑧)))) = (𝑧 ∈ ℂ ↦ (((𝐺 ∘_𝑓 · 𝑝)‘𝑧) − ((𝐺 ∘_𝑓 · 𝑞)‘𝑧))))
72		cnex 9896	. . . . . . . . . . . . . . . . . 18 ⊢ ℂ ∈ V
73	72	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℂ ∈ V)
74	61, 65	subcld 10271	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑞)‘𝑧)) ∈ ℂ)
75	61, 69	subcld 10271	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑝)‘𝑧)) ∈ ℂ)
76	60	feqmptd 6159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ (𝐹‘𝑧)))
77	64	feqmptd 6159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺 ∘_𝑓 · 𝑞) = (𝑧 ∈ ℂ ↦ ((𝐺 ∘_𝑓 · 𝑞)‘𝑧)))
78	73, 61, 65, 76, 77	offval2 6812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 ∘_𝑓 − (𝐺 ∘_𝑓 · 𝑞)) = (𝑧 ∈ ℂ ↦ ((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑞)‘𝑧))))
79	10, 78	syl5eq 2656	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 = (𝑧 ∈ ℂ ↦ ((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑞)‘𝑧))))
80	68	feqmptd 6159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺 ∘_𝑓 · 𝑝) = (𝑧 ∈ ℂ ↦ ((𝐺 ∘_𝑓 · 𝑝)‘𝑧)))
81	73, 61, 69, 76, 80	offval2 6812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 ∘_𝑓 − (𝐺 ∘_𝑓 · 𝑝)) = (𝑧 ∈ ℂ ↦ ((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑝)‘𝑧))))
82	14, 81	syl5eq 2656	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 = (𝑧 ∈ ℂ ↦ ((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑝)‘𝑧))))
83	73, 74, 75, 79, 82	offval2 6812	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑅 ∘_𝑓 − 𝑇) = (𝑧 ∈ ℂ ↦ (((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑞)‘𝑧)) − ((𝐹‘𝑧) − ((𝐺 ∘_𝑓 · 𝑝)‘𝑧)))))
84	73, 69, 65, 80, 77	offval2 6812	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐺 ∘_𝑓 · 𝑝) ∘_𝑓 − (𝐺 ∘_𝑓 · 𝑞)) = (𝑧 ∈ ℂ ↦ (((𝐺 ∘_𝑓 · 𝑝)‘𝑧) − ((𝐺 ∘_𝑓 · 𝑞)‘𝑧))))
85	71, 83, 84	3eqtr4d 2654	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅 ∘_𝑓 − 𝑇) = ((𝐺 ∘_𝑓 · 𝑝) ∘_𝑓 − (𝐺 ∘_𝑓 · 𝑞)))
86		plyf 23758	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
87	8, 86	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
88		plyf 23758	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ (Poly‘𝑆) → 𝑝:ℂ⟶ℂ)
89	13, 88	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑝:ℂ⟶ℂ)
90		plyf 23758	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 ∈ (Poly‘𝑆) → 𝑞:ℂ⟶ℂ)
91	2, 90	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑞:ℂ⟶ℂ)
92		subdi 10342	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
93	92	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
94	73, 87, 89, 91, 93	caofdi 6831	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺 ∘_𝑓 · (𝑝 ∘_𝑓 − 𝑞)) = ((𝐺 ∘_𝑓 · 𝑝) ∘_𝑓 − (𝐺 ∘_𝑓 · 𝑞)))
95	85, 94	eqtr4d 2647	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑅 ∘_𝑓 − 𝑇) = (𝐺 ∘_𝑓 · (𝑝 ∘_𝑓 − 𝑞)))
96	95	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝜑 → (deg‘(𝑅 ∘_𝑓 − 𝑇)) = (deg‘(𝐺 ∘_𝑓 · (𝑝 ∘_𝑓 − 𝑞))))
97	96	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (deg‘(𝑅 ∘_𝑓 − 𝑇)) = (deg‘(𝐺 ∘_𝑓 · (𝑝 ∘_𝑓 − 𝑞))))
98	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → 𝐺 ∈ (Poly‘𝑆))
99	9	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → 𝐺 ≠ 0_𝑝)
100	53	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (𝑝 ∘_𝑓 − 𝑞) ∈ (Poly‘𝑆))
101		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝)
102		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (deg‘𝐺) = (deg‘𝐺)
103		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (deg‘(𝑝 ∘_𝑓 − 𝑞)) = (deg‘(𝑝 ∘_𝑓 − 𝑞))
104	102, 103	dgrmul 23830	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0_𝑝) ∧ ((𝑝 ∘_𝑓 − 𝑞) ∈ (Poly‘𝑆) ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝)) → (deg‘(𝐺 ∘_𝑓 · (𝑝 ∘_𝑓 − 𝑞))) = ((deg‘𝐺) + (deg‘(𝑝 ∘_𝑓 − 𝑞))))
105	98, 99, 100, 101, 104	syl22anc 1319	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (deg‘(𝐺 ∘_𝑓 · (𝑝 ∘_𝑓 − 𝑞))) = ((deg‘𝐺) + (deg‘(𝑝 ∘_𝑓 − 𝑞))))
106	97, 105	eqtrd 2644	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (deg‘(𝑅 ∘_𝑓 − 𝑇)) = ((deg‘𝐺) + (deg‘(𝑝 ∘_𝑓 − 𝑞))))
107	58, 106	breqtrrd 4611	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (deg‘𝐺) ≤ (deg‘(𝑅 ∘_𝑓 − 𝑇)))
108	20, 30	letri3d 10058	. . . . . . . . . . 11 ⊢ (𝜑 → ((deg‘(𝑅 ∘_𝑓 − 𝑇)) = (deg‘𝐺) ↔ ((deg‘(𝑅 ∘_𝑓 − 𝑇)) ≤ (deg‘𝐺) ∧ (deg‘𝐺) ≤ (deg‘(𝑅 ∘_𝑓 − 𝑇)))))
109	108	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → ((deg‘(𝑅 ∘_𝑓 − 𝑇)) = (deg‘𝐺) ↔ ((deg‘(𝑅 ∘_𝑓 − 𝑇)) ≤ (deg‘𝐺) ∧ (deg‘𝐺) ≤ (deg‘(𝑅 ∘_𝑓 − 𝑇)))))
110	52, 107, 109	mpbir2and 959	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (deg‘(𝑅 ∘_𝑓 − 𝑇)) = (deg‘𝐺))
111	110	fveq2d 6107	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘(𝑅 ∘_𝑓 − 𝑇))) = ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘𝐺)))
112	42, 36	coesub 23817	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (Poly‘𝑆) ∧ 𝑇 ∈ (Poly‘𝑆)) → (coeff‘(𝑅 ∘_𝑓 − 𝑇)) = ((coeff‘𝑅) ∘_𝑓 − (coeff‘𝑇)))
113	12, 16, 112	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (coeff‘(𝑅 ∘_𝑓 − 𝑇)) = ((coeff‘𝑅) ∘_𝑓 − (coeff‘𝑇)))
114	113	fveq1d 6105	. . . . . . . . . . 11 ⊢ (𝜑 → ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘𝐺)) = (((coeff‘𝑅) ∘_𝑓 − (coeff‘𝑇))‘(deg‘𝐺)))
115	42	coef3 23792	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (Poly‘𝑆) → (coeff‘𝑅):ℕ₀⟶ℂ)
116		ffn 5958	. . . . . . . . . . . . . 14 ⊢ ((coeff‘𝑅):ℕ₀⟶ℂ → (coeff‘𝑅) Fn ℕ₀)
117	12, 115, 116	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (coeff‘𝑅) Fn ℕ₀)
118	36	coef3 23792	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ (Poly‘𝑆) → (coeff‘𝑇):ℕ₀⟶ℂ)
119		ffn 5958	. . . . . . . . . . . . . 14 ⊢ ((coeff‘𝑇):ℕ₀⟶ℂ → (coeff‘𝑇) Fn ℕ₀)
120	16, 118, 119	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (coeff‘𝑇) Fn ℕ₀)
121		nn0ex 11175	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
122	121	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ₀ ∈ V)
123		inidm 3784	. . . . . . . . . . . . 13 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
124	45	simprd 478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘𝑅)‘(deg‘𝐺)) = 0)
125	124	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (deg‘𝐺) ∈ ℕ₀) → ((coeff‘𝑅)‘(deg‘𝐺)) = 0)
126	39	simprd 478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘𝑇)‘(deg‘𝐺)) = 0)
127	126	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (deg‘𝐺) ∈ ℕ₀) → ((coeff‘𝑇)‘(deg‘𝐺)) = 0)
128	117, 120, 122, 122, 123, 125, 127	ofval 6804	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (deg‘𝐺) ∈ ℕ₀) → (((coeff‘𝑅) ∘_𝑓 − (coeff‘𝑇))‘(deg‘𝐺)) = (0 − 0))
129	29, 128	mpdan 699	. . . . . . . . . . 11 ⊢ (𝜑 → (((coeff‘𝑅) ∘_𝑓 − (coeff‘𝑇))‘(deg‘𝐺)) = (0 − 0))
130	114, 129	eqtrd 2644	. . . . . . . . . 10 ⊢ (𝜑 → ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘𝐺)) = (0 − 0))
131		0m0e0 11007	. . . . . . . . . 10 ⊢ (0 − 0) = 0
132	130, 131	syl6eq 2660	. . . . . . . . 9 ⊢ (𝜑 → ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘𝐺)) = 0)
133	132	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘𝐺)) = 0)
134	111, 133	eqtrd 2644	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘(𝑅 ∘_𝑓 − 𝑇))) = 0)
135		eqid 2610	. . . . . . . . . 10 ⊢ (deg‘(𝑅 ∘_𝑓 − 𝑇)) = (deg‘(𝑅 ∘_𝑓 − 𝑇))
136		eqid 2610	. . . . . . . . . 10 ⊢ (coeff‘(𝑅 ∘_𝑓 − 𝑇)) = (coeff‘(𝑅 ∘_𝑓 − 𝑇))
137	135, 136	dgreq0 23825	. . . . . . . . 9 ⊢ ((𝑅 ∘_𝑓 − 𝑇) ∈ (Poly‘𝑆) → ((𝑅 ∘_𝑓 − 𝑇) = 0_𝑝 ↔ ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘(𝑅 ∘_𝑓 − 𝑇))) = 0))
138	17, 137	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑅 ∘_𝑓 − 𝑇) = 0_𝑝 ↔ ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘(𝑅 ∘_𝑓 − 𝑇))) = 0))
139	138	biimpar 501	. . . . . . 7 ⊢ ((𝜑 ∧ ((coeff‘(𝑅 ∘_𝑓 − 𝑇))‘(deg‘(𝑅 ∘_𝑓 − 𝑇))) = 0) → (𝑅 ∘_𝑓 − 𝑇) = 0_𝑝)
140	134, 139	syldan 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝) → (𝑅 ∘_𝑓 − 𝑇) = 0_𝑝)
141	140	ex 449	. . . . 5 ⊢ (𝜑 → ((𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝 → (𝑅 ∘_𝑓 − 𝑇) = 0_𝑝))
142		plymul0or 23840	. . . . . . 7 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑝 ∘_𝑓 − 𝑞) ∈ (Poly‘𝑆)) → ((𝐺 ∘_𝑓 · (𝑝 ∘_𝑓 − 𝑞)) = 0_𝑝 ↔ (𝐺 = 0_𝑝 ∨ (𝑝 ∘_𝑓 − 𝑞) = 0_𝑝)))
143	8, 53, 142	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ((𝐺 ∘_𝑓 · (𝑝 ∘_𝑓 − 𝑞)) = 0_𝑝 ↔ (𝐺 = 0_𝑝 ∨ (𝑝 ∘_𝑓 − 𝑞) = 0_𝑝)))
144	95	eqeq1d 2612	. . . . . 6 ⊢ (𝜑 → ((𝑅 ∘_𝑓 − 𝑇) = 0_𝑝 ↔ (𝐺 ∘_𝑓 · (𝑝 ∘_𝑓 − 𝑞)) = 0_𝑝))
145	9	neneqd 2787	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐺 = 0_𝑝)
146		biorf 419	. . . . . . 7 ⊢ (¬ 𝐺 = 0_𝑝 → ((𝑝 ∘_𝑓 − 𝑞) = 0_𝑝 ↔ (𝐺 = 0_𝑝 ∨ (𝑝 ∘_𝑓 − 𝑞) = 0_𝑝)))
147	145, 146	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑝 ∘_𝑓 − 𝑞) = 0_𝑝 ↔ (𝐺 = 0_𝑝 ∨ (𝑝 ∘_𝑓 − 𝑞) = 0_𝑝)))
148	143, 144, 147	3bitr4d 299	. . . . 5 ⊢ (𝜑 → ((𝑅 ∘_𝑓 − 𝑇) = 0_𝑝 ↔ (𝑝 ∘_𝑓 − 𝑞) = 0_𝑝))
149	141, 148	sylibd 228	. . . 4 ⊢ (𝜑 → ((𝑝 ∘_𝑓 − 𝑞) ≠ 0_𝑝 → (𝑝 ∘_𝑓 − 𝑞) = 0_𝑝))
150	1, 149	pm2.61dne 2868	. . 3 ⊢ (𝜑 → (𝑝 ∘_𝑓 − 𝑞) = 0_𝑝)
151		df-0p 23243	. . 3 ⊢ 0_𝑝 = (ℂ × {0})
152	150, 151	syl6eq 2660	. 2 ⊢ (𝜑 → (𝑝 ∘_𝑓 − 𝑞) = (ℂ × {0}))
153		ofsubeq0 10894	. . 3 ⊢ ((ℂ ∈ V ∧ 𝑝:ℂ⟶ℂ ∧ 𝑞:ℂ⟶ℂ) → ((𝑝 ∘_𝑓 − 𝑞) = (ℂ × {0}) ↔ 𝑝 = 𝑞))
154	73, 89, 91, 153	syl3anc 1318	. 2 ⊢ (𝜑 → ((𝑝 ∘_𝑓 − 𝑞) = (ℂ × {0}) ↔ 𝑝 = 𝑞))
155	152, 154	mpbid 221	1 ⊢ (𝜑 → 𝑝 = 𝑞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ifcif 4036 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕ₀cn0 11169 0_𝑝c0p 23242 Polycply 23744 coeffccoe 23746 degcdgr 23747

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-0p 23243 df-ply 23748 df-coe 23750 df-dgr 23751

This theorem is referenced by: plydivalg 23858

W3C validator