elaa2lem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > elaa2lem		Structured version Visualization version GIF version

Theorem elaa2lem 46393

Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 46394. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by AV, 1-Oct-2020.)

**Hypotheses**
Ref	Expression
elaa2lem.a	⊢ (𝜑 → 𝐴 ∈ 𝔸)
elaa2lem.an0	⊢ (𝜑 → 𝐴 ≠ 0)
elaa2lem.g	⊢ (𝜑 → 𝐺 ∈ (Poly‘ℤ))
elaa2lem.gn0	⊢ (𝜑 → 𝐺 ≠ 0_𝑝)
elaa2lem.ga	⊢ (𝜑 → (𝐺‘𝐴) = 0)
elaa2lem.m	⊢ 𝑀 = inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < )
elaa2lem.i	⊢ 𝐼 = (𝑘 ∈ ℕ₀ ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀)))
elaa2lem.f	⊢ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼‘𝑘) · (𝑧↑𝑘)))

**Assertion**
Ref	Expression
elaa2lem	⊢ (𝜑 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))

Distinct variable groups: 𝐴,𝑓 𝐴,𝑘,𝑧 𝑓,𝐹 𝑘,𝐺 𝑛,𝐺 𝑧,𝐺 𝑘,𝐼,𝑧 𝑘,𝑀 𝑛,𝑀 𝑧,𝑀 𝜑,𝑘,𝑧 Allowed substitution hints: 𝜑(𝑓,𝑛) 𝐴(𝑛) 𝐹(𝑧,𝑘,𝑛) 𝐺(𝑓) 𝐼(𝑓,𝑛) 𝑀(𝑓)

Proof of Theorem elaa2lem Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elaa2lem.f	. . . 4 ⊢ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼‘𝑘) · (𝑧↑𝑘)))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼‘𝑘) · (𝑧↑𝑘))))
3		zsscn 12487	. . . . 5 ⊢ ℤ ⊆ ℂ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → ℤ ⊆ ℂ)
5		elaa2lem.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℤ))
6		dgrcl 26185	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘ℤ) → (deg‘𝐺) ∈ ℕ₀)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ₀)
8	7	nn0zd 12504	. . . . . . 7 ⊢ (𝜑 → (deg‘𝐺) ∈ ℤ)
9		elaa2lem.m	. . . . . . . . 9 ⊢ 𝑀 = inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < )
10		ssrab2 4029	. . . . . . . . . 10 ⊢ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ ℕ₀
11		nn0uz 12780	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
12	10, 11	sseqtri 3979	. . . . . . . . . . . 12 ⊢ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ_≥‘0)
13	12	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ_≥‘0))
14		elaa2lem.gn0	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 ≠ 0_𝑝)
15	14	neneqd 2934	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝐺 = 0_𝑝)
16		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (deg‘𝐺) = (deg‘𝐺)
17		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
18	16, 17	dgreq0 26218	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ (Poly‘ℤ) → (𝐺 = 0_𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
19	5, 18	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺 = 0_𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
20	15, 19	mtbid 324	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)
21	20	neqned 2936	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0)
22	7, 21	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((deg‘𝐺) ∈ ℕ₀ ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
23		fveq2 6831	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (deg‘𝐺) → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘(deg‘𝐺)))
24	23	neeq1d 2988	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (deg‘𝐺) → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
25	24	elrab 3643	. . . . . . . . . . . . 13 ⊢ ((deg‘𝐺) ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ ((deg‘𝐺) ∈ ℕ₀ ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
26	22, 25	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐺) ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
27	26	ne0d 4291	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ≠ ∅)
28		infssuzcl 12836	. . . . . . . . . . 11 ⊢ (({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ_≥‘0) ∧ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ≠ ∅) → inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
29	13, 27, 28	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
30	10, 29	sselid 3928	. . . . . . . . 9 ⊢ (𝜑 → inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ ℕ₀)
31	9, 30	eqeltrid 2837	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
32	31	nn0zd 12504	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
33	8, 32	zsubcld 12592	. . . . . 6 ⊢ (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℤ)
34	9	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑀 = inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
35		infssuzle 12835	. . . . . . . . 9 ⊢ (({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ_≥‘0) ∧ (deg‘𝐺) ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}) → inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ (deg‘𝐺))
36	13, 26, 35	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ (deg‘𝐺))
37	34, 36	eqbrtrd 5117	. . . . . . 7 ⊢ (𝜑 → 𝑀 ≤ (deg‘𝐺))
38	7	nn0red 12454	. . . . . . . 8 ⊢ (𝜑 → (deg‘𝐺) ∈ ℝ)
39	31	nn0red 12454	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
40	38, 39	subge0d 11718	. . . . . . 7 ⊢ (𝜑 → (0 ≤ ((deg‘𝐺) − 𝑀) ↔ 𝑀 ≤ (deg‘𝐺)))
41	37, 40	mpbird 257	. . . . . 6 ⊢ (𝜑 → 0 ≤ ((deg‘𝐺) − 𝑀))
42	33, 41	jca 511	. . . . 5 ⊢ (𝜑 → (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
43		elnn0z 12492	. . . . 5 ⊢ (((deg‘𝐺) − 𝑀) ∈ ℕ₀ ↔ (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
44	42, 43	sylibr 234	. . . 4 ⊢ (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℕ₀)
45		0zd 12491	. . . . . . . 8 ⊢ (𝐺 ∈ (Poly‘ℤ) → 0 ∈ ℤ)
46	17	coef2 26183	. . . . . . . 8 ⊢ ((𝐺 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝐺):ℕ₀⟶ℤ)
47	5, 45, 46	syl2anc2 585	. . . . . . 7 ⊢ (𝜑 → (coeff‘𝐺):ℕ₀⟶ℤ)
48	47	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (coeff‘𝐺):ℕ₀⟶ℤ)
49		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
50	31	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝑀 ∈ ℕ₀)
51	49, 50	nn0addcld 12457	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝑘 + 𝑀) ∈ ℕ₀)
52	48, 51	ffvelcdmd 7027	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
53		elaa2lem.i	. . . . 5 ⊢ 𝐼 = (𝑘 ∈ ℕ₀ ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀)))
54	52, 53	fmptd 7056	. . . 4 ⊢ (𝜑 → 𝐼:ℕ₀⟶ℤ)
55		elplyr 26153	. . . 4 ⊢ ((ℤ ⊆ ℂ ∧ ((deg‘𝐺) − 𝑀) ∈ ℕ₀ ∧ 𝐼:ℕ₀⟶ℤ) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘ℤ))
56	4, 44, 54, 55	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘ℤ))
57	2, 56	eqeltrd 2833	. 2 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))
58		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ≤ ((deg‘𝐺) − 𝑀))
59	58	iftrued 4484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
60		iffalse 4485	. . . . . . . . . . 11 ⊢ (¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
61	60	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
62		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀))
63	38	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) ∈ ℝ)
64	39	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑀 ∈ ℝ)
65	63, 64	resubcld 11556	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) ∈ ℝ)
66		nn0re 12401	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
67	66	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℝ)
68	65, 67	ltnled 11271	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)))
69	62, 68	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) < 𝑘)
70	63, 64, 67	ltsubaddd 11724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ (deg‘𝐺) < (𝑘 + 𝑀)))
71	69, 70	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) < (𝑘 + 𝑀))
72		olc 868	. . . . . . . . . . . . 13 ⊢ ((deg‘𝐺) < (𝑘 + 𝑀) → (𝐺 = 0_𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)))
73	71, 72	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝐺 = 0_𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)))
74	5	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝐺 ∈ (Poly‘ℤ))
75	51	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝑘 + 𝑀) ∈ ℕ₀)
76	16, 17	dgrlt 26219	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (Poly‘ℤ) ∧ (𝑘 + 𝑀) ∈ ℕ₀) → ((𝐺 = 0_𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)) ↔ ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)))
77	74, 75, 76	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((𝐺 = 0_𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)) ↔ ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)))
78	73, 77	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0))
79	78	simprd 495	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)
80	61, 79	eqtr4d 2771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
81	59, 80	pm2.61dan 812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
82	81	mpteq2dva 5188	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ ℕ₀ ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)) = (𝑘 ∈ ℕ₀ ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
83	47, 4	fssd 6676	. . . . . . . . . 10 ⊢ (𝜑 → (coeff‘𝐺):ℕ₀⟶ℂ)
84	83	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (coeff‘𝐺):ℕ₀⟶ℂ)
85		elfznn0 13527	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℕ₀)
86	85	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑘 ∈ ℕ₀)
87	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑀 ∈ ℕ₀)
88	86, 87	nn0addcld 12457	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ₀)
89	84, 88	ffvelcdmd 7027	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℂ)
90		eqidd 2734	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...((deg‘𝐺) − 𝑀)) = (0...((deg‘𝐺) − 𝑀)))
91		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝜑)
92	53	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 = (𝑘 ∈ ℕ₀ ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
93	92, 52	fvmpt2d 6951	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
94	91, 86, 93	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
95	94	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
96	95	oveq1d 7370	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼‘𝑘) · (𝑧↑𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧↑𝑘)))
97	90, 96	sumeq12rdv 15621	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧↑𝑘)))
98	97	mpteq2dva 5188	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧↑𝑘))))
99	2, 98	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧↑𝑘))))
100	57, 44, 89, 99	coeeq2 26194	. . . . . . 7 ⊢ (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ₀ ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)))
101	82, 100, 92	3eqtr4d 2778	. . . . . 6 ⊢ (𝜑 → (coeff‘𝐹) = 𝐼)
102	101	fveq1d 6833	. . . . 5 ⊢ (𝜑 → ((coeff‘𝐹)‘0) = (𝐼‘0))
103		oveq1 7362	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 + 𝑀) = (0 + 𝑀))
104	103	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑘 + 𝑀) = (0 + 𝑀))
105	3, 32	sselid 3928	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
106	105	addlidd 11325	. . . . . . . . 9 ⊢ (𝜑 → (0 + 𝑀) = 𝑀)
107	106	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 0) → (0 + 𝑀) = 𝑀)
108	104, 107	eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑘 + 𝑀) = 𝑀)
109	108	fveq2d 6835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘𝑀))
110		0nn0 12407	. . . . . . 7 ⊢ 0 ∈ ℕ₀
111	110	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ₀)
112	47, 31	ffvelcdmd 7027	. . . . . 6 ⊢ (𝜑 → ((coeff‘𝐺)‘𝑀) ∈ ℤ)
113	92, 109, 111, 112	fvmptd 6945	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = ((coeff‘𝐺)‘𝑀))
114		eqidd 2734	. . . . 5 ⊢ (𝜑 → ((coeff‘𝐺)‘𝑀) = ((coeff‘𝐺)‘𝑀))
115	102, 113, 114	3eqtrd 2772	. . . 4 ⊢ (𝜑 → ((coeff‘𝐹)‘0) = ((coeff‘𝐺)‘𝑀))
116	34, 29	eqeltrd 2833	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
117		fveq2 6831	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑀))
118	117	neeq1d 2988	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑀) ≠ 0))
119	118	elrab 3643	. . . . . 6 ⊢ (𝑀 ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑀 ∈ ℕ₀ ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
120	116, 119	sylib 218	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ₀ ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
121	120	simprd 495	. . . 4 ⊢ (𝜑 → ((coeff‘𝐺)‘𝑀) ≠ 0)
122	115, 121	eqnetrd 2996	. . 3 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ≠ 0)
123	5, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
124		aasscn 26273	. . . . . . . . . . 11 ⊢ 𝔸 ⊆ ℂ
125		elaa2lem.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝔸)
126	124, 125	sselid 3928	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℂ)
127	91, 126	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝐴 ∈ ℂ)
128	127, 86	expcld 14060	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐴↑𝑘) ∈ ℂ)
129	89, 128	mulcld 11143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)) ∈ ℂ)
130		fvoveq1 7378	. . . . . . . 8 ⊢ (𝑘 = (𝑗 − 𝑀) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)))
131		oveq2 7363	. . . . . . . 8 ⊢ (𝑘 = (𝑗 − 𝑀) → (𝐴↑𝑘) = (𝐴↑(𝑗 − 𝑀)))
132	130, 131	oveq12d 7373	. . . . . . 7 ⊢ (𝑘 = (𝑗 − 𝑀) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)) = (((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))))
133	32, 123, 33, 129, 132	fsumshft 15694	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)) = Σ𝑗 ∈ ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀))(((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))))
134	3, 8	sselid 3928	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘𝐺) ∈ ℂ)
135	134, 105	npcand 11487	. . . . . . . . 9 ⊢ (𝜑 → (((deg‘𝐺) − 𝑀) + 𝑀) = (deg‘𝐺))
136	106, 135	oveq12d 7373	. . . . . . . 8 ⊢ (𝜑 → ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀)) = (𝑀...(deg‘𝐺)))
137	136	sumeq1d 15614	. . . . . . 7 ⊢ (𝜑 → Σ𝑗 ∈ ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀))(((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))(((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))))
138		elfzelz 13431	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑗 ∈ ℤ)
139	138	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
140	3, 139	sselid 3928	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℂ)
141	105	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℂ)
142	140, 141	npcand 11487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((𝑗 − 𝑀) + 𝑀) = 𝑗)
143	142	fveq2d 6835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) = ((coeff‘𝐺)‘𝑗))
144	143	oveq1d 7370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗 − 𝑀))))
145	126	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ∈ ℂ)
146		elaa2lem.an0	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ≠ 0)
147	146	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ≠ 0)
148	32	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℤ)
149	145, 147, 148, 139	expsubd 14071	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑(𝑗 − 𝑀)) = ((𝐴↑𝑗) / (𝐴↑𝑀)))
150	149	oveq2d 7371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗 − 𝑀))) = (((coeff‘𝐺)‘𝑗) · ((𝐴↑𝑗) / (𝐴↑𝑀))))
151	83	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (coeff‘𝐺):ℕ₀⟶ℂ)
152		0red 11126	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ∈ ℝ)
153	39	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℝ)
154	139	zred 12587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
155	31	nn0ge0d 12456	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ 𝑀)
156	155	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑀)
157		elfzle1 13434	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑀 ≤ 𝑗)
158	157	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ≤ 𝑗)
159	152, 153, 154, 156, 158	letrd 11281	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑗)
160	139, 159	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
161		elnn0z 12492	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ₀ ↔ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
162	160, 161	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ₀)
163	151, 162	ffvelcdmd 7027	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
164	145, 162	expcld 14060	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑𝑗) ∈ ℂ)
165	126, 31	expcld 14060	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℂ)
166	165	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑𝑀) ∈ ℂ)
167	145, 147, 148	expne0d 14066	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑𝑀) ≠ 0)
168	163, 164, 166, 167	divassd 11943	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = (((coeff‘𝐺)‘𝑗) · ((𝐴↑𝑗) / (𝐴↑𝑀))))
169	168	eqcomd 2739	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · ((𝐴↑𝑗) / (𝐴↑𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
170	150, 169	eqtr2d 2769	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗 − 𝑀))))
171	144, 170	eqtr4d 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
172	171	sumeq2dv 15616	. . . . . . 7 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))(((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
173	137, 172	eqtrd 2768	. . . . . 6 ⊢ (𝜑 → Σ𝑗 ∈ ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀))(((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))) = Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
174	31, 11	eleqtrdi 2843	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
175		fzss1 13470	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ_≥‘0) → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
176	174, 175	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
177	163, 164	mulcld 11143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) ∈ ℂ)
178	177, 166, 167	divcld 11908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) ∈ ℂ)
179	32	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℤ)
180	8	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (deg‘𝐺) ∈ ℤ)
181		eldifi 4080	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ (0...(deg‘𝐺)))
182	181	elfzelzd 13432	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
183	182	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℤ)
184		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 < 𝑀)
185	39	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℝ)
186	183	zred 12587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℝ)
187	185, 186	lenltd 11270	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (𝑀 ≤ 𝑗 ↔ ¬ 𝑗 < 𝑀))
188	184, 187	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ≤ 𝑗)
189		elfzle2 13435	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ≤ (deg‘𝐺))
190	181, 189	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ≤ (deg‘𝐺))
191	190	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ≤ (deg‘𝐺))
192	179, 180, 183, 188, 191	elfzd 13422	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ (𝑀...(deg‘𝐺)))
193		eldifn 4081	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
194	193	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
195	192, 194	condan 817	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 < 𝑀)
196	195	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 < 𝑀)
197	9	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 = inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
198	12	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ_≥‘0))
199		elfznn0 13527	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ∈ ℕ₀)
200	181, 199	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ₀)
201	200	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℕ₀)
202		neqne 2937	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ((coeff‘𝐺)‘𝑗) = 0 → ((coeff‘𝐺)‘𝑗) ≠ 0)
203	202	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ((coeff‘𝐺)‘𝑗) ≠ 0)
204	201, 203	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑗 ∈ ℕ₀ ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
205		fveq2 6831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑗))
206	205	neeq1d 2988	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑗) ≠ 0))
207	206	elrab 3643	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑗 ∈ ℕ₀ ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
208	204, 207	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
209	208	adantll 714	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
210		infssuzle 12835	. . . . . . . . . . . . . . 15 ⊢ (({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ (ℤ_≥‘0) ∧ 𝑗 ∈ {𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}) → inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ 𝑗)
211	198, 209, 210	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → inf({𝑛 ∈ ℕ₀ ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ 𝑗)
212	197, 211	eqbrtrd 5117	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 ≤ 𝑗)
213	39	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 ∈ ℝ)
214	182	zred 12587	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
215	214	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℝ)
216	213, 215	lenltd 11270	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑀 ≤ 𝑗 ↔ ¬ 𝑗 < 𝑀))
217	212, 216	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ¬ 𝑗 < 𝑀)
218	196, 217	condan 817	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((coeff‘𝐺)‘𝑗) = 0)
219	218	oveq1d 7370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) = (0 · (𝐴↑𝑗)))
220	126	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝐴 ∈ ℂ)
221	200	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 ∈ ℕ₀)
222	220, 221	expcld 14060	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (𝐴↑𝑗) ∈ ℂ)
223	222	mul02d 11322	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 · (𝐴↑𝑗)) = 0)
224	219, 223	eqtrd 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) = 0)
225	224	oveq1d 7370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = (0 / (𝐴↑𝑀)))
226	126, 146, 32	expne0d 14066	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴↑𝑀) ≠ 0)
227	165, 226	div0d 11907	. . . . . . . . 9 ⊢ (𝜑 → (0 / (𝐴↑𝑀)) = 0)
228	227	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 / (𝐴↑𝑀)) = 0)
229	225, 228	eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = 0)
230		fzfid 13887	. . . . . . 7 ⊢ (𝜑 → (0...(deg‘𝐺)) ∈ Fin)
231	176, 178, 229, 230	fsumss 15639	. . . . . 6 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
232	133, 173, 231	3eqtrd 2772	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
233	86, 52	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
234	53	fvmpt2 6949	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
235	86, 233, 234	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
236	235	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
237		oveq1 7362	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑧↑𝑘) = (𝐴↑𝑘))
238	237	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑧↑𝑘) = (𝐴↑𝑘))
239	236, 238	oveq12d 7373	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼‘𝑘) · (𝑧↑𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)))
240	239	sumeq2dv 15616	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝐴) → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)))
241		fzfid 13887	. . . . . . 7 ⊢ (𝜑 → (0...((deg‘𝐺) − 𝑀)) ∈ Fin)
242	241, 129	fsumcl 15647	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)) ∈ ℂ)
243	2, 240, 126, 242	fvmptd 6945	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)))
244	17, 16	coeid2 26191	. . . . . . . 8 ⊢ ((𝐺 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝐺‘𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)))
245	5, 126, 244	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)))
246	245	oveq1d 7370	. . . . . 6 ⊢ (𝜑 → ((𝐺‘𝐴) / (𝐴↑𝑀)) = (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
247	83	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → (coeff‘𝐺):ℕ₀⟶ℂ)
248	199	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → 𝑗 ∈ ℕ₀)
249	247, 248	ffvelcdmd 7027	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
250	126	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → 𝐴 ∈ ℂ)
251	250, 248	expcld 14060	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → (𝐴↑𝑗) ∈ ℂ)
252	249, 251	mulcld 11143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) ∈ ℂ)
253	230, 165, 252, 226	fsumdivc 15700	. . . . . 6 ⊢ (𝜑 → (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
254	246, 253	eqtrd 2768	. . . . 5 ⊢ (𝜑 → ((𝐺‘𝐴) / (𝐴↑𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
255	232, 243, 254	3eqtr4d 2778	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝐺‘𝐴) / (𝐴↑𝑀)))
256		elaa2lem.ga	. . . . 5 ⊢ (𝜑 → (𝐺‘𝐴) = 0)
257	256	oveq1d 7370	. . . 4 ⊢ (𝜑 → ((𝐺‘𝐴) / (𝐴↑𝑀)) = (0 / (𝐴↑𝑀)))
258	255, 257, 227	3eqtrd 2772	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
259	122, 258	jca 511	. 2 ⊢ (𝜑 → (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹‘𝐴) = 0))
260		fveq2 6831	. . . . . 6 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
261	260	fveq1d 6833	. . . . 5 ⊢ (𝑓 = 𝐹 → ((coeff‘𝑓)‘0) = ((coeff‘𝐹)‘0))
262	261	neeq1d 2988	. . . 4 ⊢ (𝑓 = 𝐹 → (((coeff‘𝑓)‘0) ≠ 0 ↔ ((coeff‘𝐹)‘0) ≠ 0))
263		fveq1 6830	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘𝐴) = (𝐹‘𝐴))
264	263	eqeq1d 2735	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝐴) = 0 ↔ (𝐹‘𝐴) = 0))
265	262, 264	anbi12d 632	. . 3 ⊢ (𝑓 = 𝐹 → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) ↔ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹‘𝐴) = 0)))
266	265	rspcev 3573	. 2 ⊢ ((𝐹 ∈ (Poly‘ℤ) ∧ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹‘𝐴) = 0)) → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))
267	57, 259, 266	syl2anc 584	1 ⊢ (𝜑 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∃wrex 3057 {crab 3396 ∖ cdif 3895 ⊆ wss 3898 ∅c0 4282 ifcif 4476 class class class wbr 5095 ↦ cmpt 5176 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 infcinf 9336 ℂcc 11015 ℝcr 11016 0cc0 11017 + caddc 11020 · cmul 11022 < clt 11157 ≤ cle 11158 − cmin 11355 / cdiv 11785 ℕ₀cn0 12392 ℤcz 12479 ℤ_≥cuz 12742 ...cfz 13414 ↑cexp 13975 Σcsu 15600 0_𝑝c0p 25617 Polycply 26136 coeffccoe 26138 degcdgr 26139 𝔸caa 26269

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-pm 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-n0 12393 df-z 12480 df-uz 12743 df-rp 12897 df-fz 13415 df-fzo 13562 df-fl 13703 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-clim 15402 df-rlim 15403 df-sum 15601 df-0p 25618 df-ply 26140 df-coe 26142 df-dgr 26143 df-aa 26270

This theorem is referenced by: elaa2 46394

W3C validator