Theorem elaa2lem 47212

Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 47213. (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 ≠ 0𝑝)
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 12694	. . . . 5 ⊢ ℤ ⊆ ℂ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → ℤ ⊆ ℂ)
5		elaa2lem.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (Poly‘ℤ))
6		dgrcl 26545	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘ℤ) → (deg‘𝐺) ∈ ℕ0)
7	5, 6	syl 18	. . . . . . . 8 ⊢ (𝜑 → (deg‘𝐺) ∈ ℕ0)
8	7	nn0zd 12711	. . . . . . 7 ⊢ (𝜑 → (deg‘𝐺) ∈ ℤ)
9		elaa2lem.m	. . . . . . . . 9 ⊢ 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < )
10		ssrab2 4028	. . . . . . . . . 10 ⊢ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ⊆ ℕ0
11		nn0uz 12996	. . . . . . . . . . . . 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 ≠ 0𝑝)
15	14	neneqd 2961	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝐺 = 0𝑝)
16		eqid 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (deg‘𝐺) = (deg‘𝐺)
17		eqid 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
18	16, 17	dgreq0 26577	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ (Poly‘ℤ) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
19	5, 18	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
20	15, 19	mtbid 327	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)
21	20	neqned 2963	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0)
22	7, 21	jca 521	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
23		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (deg‘𝐺) → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘(deg‘𝐺)))
24	23	neeq1d 3015	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (deg‘𝐺) → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
25	24	elrab 3645	. . . . . . . . . . . . 13 ⊢ ((deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ ((deg‘𝐺) ∈ ℕ0 ∧ ((coeff‘𝐺)‘(deg‘𝐺)) ≠ 0))
26	22, 25	sylibr 237	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝐺) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
27	26	ne0d 4288	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ≠ ∅)
28		infssuzcl 13052	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
30	10, 29	sselid 3929	. . . . . . . . 9 ⊢ (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ∈ ℕ0)
31	9, 30	eqeltrid 2865	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
32	31	nn0zd 12711	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
33	8, 32	zsubcld 12801	. . . . . 6 ⊢ (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℤ)
34	9	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑀 = inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ))
35		infssuzle 13051	. . . . . . . . 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 596	. . . . . . . 8 ⊢ (𝜑 → inf({𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0}, ℝ, < ) ≤ (deg‘𝐺))
37	34, 36	eqbrtrd 5127	. . . . . . 7 ⊢ (𝜑 → 𝑀 ≤ (deg‘𝐺))
38	7	nn0red 12661	. . . . . . . 8 ⊢ (𝜑 → (deg‘𝐺) ∈ ℝ)
39	31	nn0red 12661	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
40	38, 39	subge0d 11899	. . . . . . 7 ⊢ (𝜑 → (0 ≤ ((deg‘𝐺) − 𝑀) ↔ 𝑀 ≤ (deg‘𝐺)))
41	37, 40	mpbird 260	. . . . . 6 ⊢ (𝜑 → 0 ≤ ((deg‘𝐺) − 𝑀))
42	33, 41	jca 521	. . . . 5 ⊢ (𝜑 → (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
43		elnn0z 12699	. . . . 5 ⊢ (((deg‘𝐺) − 𝑀) ∈ ℕ0 ↔ (((deg‘𝐺) − 𝑀) ∈ ℤ ∧ 0 ≤ ((deg‘𝐺) − 𝑀)))
44	42, 43	sylibr 237	. . . 4 ⊢ (𝜑 → ((deg‘𝐺) − 𝑀) ∈ ℕ0)
45		0zd 12698	. . . . . . . 8 ⊢ (𝐺 ∈ (Poly‘ℤ) → 0 ∈ ℤ)
46	17	coef2 26543	. . . . . . . 8 ⊢ ((𝐺 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝐺):ℕ0⟶ℤ)
47	5, 45, 46	syl2anc2 597	. . . . . . 7 ⊢ (𝜑 → (coeff‘𝐺):ℕ0⟶ℤ)
48	47	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (coeff‘𝐺):ℕ0⟶ℤ)
49		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
50	31	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
51	49, 50	nn0addcld 12664	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 𝑀) ∈ ℕ0)
52	48, 51	ffvelcdmd 7083	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
53		elaa2lem.i	. . . . 5 ⊢ 𝐼 = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀)))
54	52, 53	fmptd 7112	. . . 4 ⊢ (𝜑 → 𝐼:ℕ0⟶ℤ)
55		elplyr 26512	. . . 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 1398	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))((𝐼‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘ℤ))
57	2, 56	eqeltrd 2861	. 2 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))
58		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ≤ ((deg‘𝐺) − 𝑀))
59	58	iftrued 4490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
60		iffalse 4491	. . . . . . . . . . 11 ⊢ (¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
61	60	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = 0)
62		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀))
63	38	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) ∈ ℝ)
64	39	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑀 ∈ ℝ)
65	63, 64	resubcld 11737	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) ∈ ℝ)
66		nn0re 12608	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
67	66	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℝ)
68	65, 67	ltnled 11450	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)))
69	62, 68	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) − 𝑀) < 𝑘)
70	63, 64, 67	ltsubaddd 11905	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (((deg‘𝐺) − 𝑀) < 𝑘 ↔ (deg‘𝐺) < (𝑘 + 𝑀)))
71	69, 70	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (deg‘𝐺) < (𝑘 + 𝑀))
72		olc 882	. . . . . . . . . . . . 13 ⊢ ((deg‘𝐺) < (𝑘 + 𝑀) → (𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)))
73	71, 72	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)))
74	5	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → 𝐺 ∈ (Poly‘ℤ))
75	51	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → (𝑘 + 𝑀) ∈ ℕ0)
76	16, 17	dgrlt 26578	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (Poly‘ℤ) ∧ (𝑘 + 𝑀) ∈ ℕ0) → ((𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)) ↔ ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)))
77	74, 75, 76	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((𝐺 = 0𝑝 ∨ (deg‘𝐺) < (𝑘 + 𝑀)) ↔ ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)))
78	73, 77	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((deg‘𝐺) ≤ (𝑘 + 𝑀) ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0))
79	78	simprd 501	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ≤ ((deg‘𝐺) − 𝑀)) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = 0)
80	61, 79	eqtr4d 2799	. . . . . . . . 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 825	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
82	81	mpteq2dva 5198	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)) = (𝑘 ∈ ℕ0 ↦ ((coeff‘𝐺)‘(𝑘 + 𝑀))))
83	47, 4	fssd 6725	. . . . . . . . . 10 ⊢ (𝜑 → (coeff‘𝐺):ℕ0⟶ℂ)
84	83	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (coeff‘𝐺):ℕ0⟶ℂ)
85		elfznn0 13747	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...((deg‘𝐺) − 𝑀)) → 𝑘 ∈ ℕ0)
86	85	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑘 ∈ ℕ0)
87	31	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝑀 ∈ ℕ0)
88	86, 87	nn0addcld 12664	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑘 + 𝑀) ∈ ℕ0)
89	84, 88	ffvelcdmd 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℂ)
90		eqidd 2762	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...((deg‘𝐺) − 𝑀)) = (0...((deg‘𝐺) − 𝑀)))
91		simpl 488	. . . . . . . . . . . . . 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 7005	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
94	91, 86, 93	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
95	94	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
96	95	oveq1d 7433	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼‘𝑘) · (𝑧↑𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧↑𝑘)))
97	90, 96	sumeq12rdv 15866	. . . . . . . . . 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 5198	. . . . . . . . 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 2796	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝑧↑𝑘))))
100	57, 44, 89, 99	coeeq2 26554	. . . . . . 7 ⊢ (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ ((deg‘𝐺) − 𝑀), ((coeff‘𝐺)‘(𝑘 + 𝑀)), 0)))
101	82, 100, 92	3eqtr4d 2806	. . . . . 6 ⊢ (𝜑 → (coeff‘𝐹) = 𝐼)
102	101	fveq1d 6885	. . . . 5 ⊢ (𝜑 → ((coeff‘𝐹)‘0) = (𝐼‘0))
103		oveq1 7425	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 + 𝑀) = (0 + 𝑀))
104	103	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑘 + 𝑀) = (0 + 𝑀))
105	3, 32	sselid 3929	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
106	105	addlidd 11504	. . . . . . . . 9 ⊢ (𝜑 → (0 + 𝑀) = 𝑀)
107	106	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 0) → (0 + 𝑀) = 𝑀)
108	104, 107	eqtrd 2796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑘 + 𝑀) = 𝑀)
109	108	fveq2d 6887	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘𝑀))
110		0nn0 12614	. . . . . . 7 ⊢ 0 ∈ ℕ0
111	110	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℕ0)
112	47, 31	ffvelcdmd 7083	. . . . . 6 ⊢ (𝜑 → ((coeff‘𝐺)‘𝑀) ∈ ℤ)
113	92, 109, 111, 112	fvmptd 6999	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = ((coeff‘𝐺)‘𝑀))
114		eqidd 2762	. . . . 5 ⊢ (𝜑 → ((coeff‘𝐺)‘𝑀) = ((coeff‘𝐺)‘𝑀))
115	102, 113, 114	3eqtrd 2800	. . . 4 ⊢ (𝜑 → ((coeff‘𝐹)‘0) = ((coeff‘𝐺)‘𝑀))
116	34, 29	eqeltrd 2861	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
117		fveq2 6883	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑀))
118	117	neeq1d 3015	. . . . . . 7 ⊢ (𝑛 = 𝑀 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑀) ≠ 0))
119	118	elrab 3645	. . . . . 6 ⊢ (𝑀 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
120	116, 119	sylib 221	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑀) ≠ 0))
121	120	simprd 501	. . . 4 ⊢ (𝜑 → ((coeff‘𝐺)‘𝑀) ≠ 0)
122	115, 121	eqnetrd 3023	. . 3 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ≠ 0)
123	5, 45	syl 18	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
124		aasscn 26634	. . . . . . . . . . 11 ⊢ 𝔸 ⊆ ℂ
125		elaa2lem.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝔸)
126	124, 125	sselid 3929	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℂ)
127	91, 126	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → 𝐴 ∈ ℂ)
128	127, 86	expcld 14282	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐴↑𝑘) ∈ ℂ)
129	89, 128	mulcld 11322	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)) ∈ ℂ)
130		fvoveq1 7441	. . . . . . . 8 ⊢ (𝑘 = (𝑗 − 𝑀) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) = ((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)))
131		oveq2 7426	. . . . . . . 8 ⊢ (𝑘 = (𝑗 − 𝑀) → (𝐴↑𝑘) = (𝐴↑(𝑗 − 𝑀)))
132	130, 131	oveq12d 7436	. . . . . . 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 15939	. . . . . 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 3929	. . . . . . . . . 10 ⊢ (𝜑 → (deg‘𝐺) ∈ ℂ)
135	134, 105	npcand 11666	. . . . . . . . 9 ⊢ (𝜑 → (((deg‘𝐺) − 𝑀) + 𝑀) = (deg‘𝐺))
136	106, 135	oveq12d 7436	. . . . . . . 8 ⊢ (𝜑 → ((0 + 𝑀)...(((deg‘𝐺) − 𝑀) + 𝑀)) = (𝑀...(deg‘𝐺)))
137	136	sumeq1d 15860	. . . . . . 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 13649	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑗 ∈ ℤ)
139	138	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
140	3, 139	sselid 3929	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℂ)
141	105	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℂ)
142	140, 141	npcand 11666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((𝑗 − 𝑀) + 𝑀) = 𝑗)
143	142	fveq2d 6887	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) = ((coeff‘𝐺)‘𝑗))
144	143	oveq1d 7433	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗 − 𝑀))))
145	126	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ∈ ℂ)
146		elaa2lem.an0	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ≠ 0)
147	146	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝐴 ≠ 0)
148	32	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℤ)
149	145, 147, 148, 139	expsubd 14293	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑(𝑗 − 𝑀)) = ((𝐴↑𝑗) / (𝐴↑𝑀)))
150	149	oveq2d 7434	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗 − 𝑀))) = (((coeff‘𝐺)‘𝑗) · ((𝐴↑𝑗) / (𝐴↑𝑀))))
151	83	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
152		0red 11304	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ∈ ℝ)
153	39	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ∈ ℝ)
154	139	zred 12796	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
155	31	nn0ge0d 12663	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ 𝑀)
156	155	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑀)
157		elfzle1 13653	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑀...(deg‘𝐺)) → 𝑀 ≤ 𝑗)
158	157	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑀 ≤ 𝑗)
159	152, 153, 154, 156, 158	letrd 11460	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 0 ≤ 𝑗)
160	139, 159	jca 521	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
161		elnn0z 12699	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ0 ↔ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
162	160, 161	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
163	151, 162	ffvelcdmd 7083	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
164	145, 162	expcld 14282	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑𝑗) ∈ ℂ)
165	126, 31	expcld 14282	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℂ)
166	165	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑𝑀) ∈ ℂ)
167	145, 147, 148	expne0d 14288	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (𝐴↑𝑀) ≠ 0)
168	163, 164, 166, 167	divassd 12121	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = (((coeff‘𝐺)‘𝑗) · ((𝐴↑𝑗) / (𝐴↑𝑀))))
169	168	eqcomd 2767	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · ((𝐴↑𝑗) / (𝐴↑𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
170	150, 169	eqtr2d 2797	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = (((coeff‘𝐺)‘𝑗) · (𝐴↑(𝑗 − 𝑀))))
171	144, 170	eqtr4d 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘((𝑗 − 𝑀) + 𝑀)) · (𝐴↑(𝑗 − 𝑀))) = ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
172	171	sumeq2dv 15862	. . . . . . 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 2796	. . . . . 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 2871	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
175		fzss1 13690	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘0) → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
176	174, 175	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑀...(deg‘𝐺)) ⊆ (0...(deg‘𝐺)))
177	163, 164	mulcld 11322	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) ∈ ℂ)
178	177, 166, 167	divcld 12086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(deg‘𝐺))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) ∈ ℂ)
179	32	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℤ)
180	8	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (deg‘𝐺) ∈ ℤ)
181		eldifi 4078	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ (0...(deg‘𝐺)))
182	181	elfzelzd 13650	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℤ)
183	182	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℤ)
184		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 < 𝑀)
185	39	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ∈ ℝ)
186	183	zred 12796	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ ℝ)
187	185, 186	lenltd 11449	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → (𝑀 ≤ 𝑗 ↔ ¬ 𝑗 < 𝑀))
188	184, 187	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑀 ≤ 𝑗)
189		elfzle2 13654	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ≤ (deg‘𝐺))
190	181, 189	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ≤ (deg‘𝐺))
191	190	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ≤ (deg‘𝐺))
192	179, 180, 183, 188, 191	elfzd 13640	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → 𝑗 ∈ (𝑀...(deg‘𝐺)))
193		eldifn 4079	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
194	193	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ 𝑗 < 𝑀) → ¬ 𝑗 ∈ (𝑀...(deg‘𝐺)))
195	192, 194	condan 830	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 < 𝑀)
196	195	adantr 486	. . . . . . . . . . . 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 13747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...(deg‘𝐺)) → 𝑗 ∈ ℕ0)
200	181, 199	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
201	200	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℕ0)
202		neqne 2964	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ((coeff‘𝐺)‘𝑗) = 0 → ((coeff‘𝐺)‘𝑗) ≠ 0)
203	202	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ((coeff‘𝐺)‘𝑗) ≠ 0)
204	201, 203	jca 521	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
205		fveq2 6883	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((coeff‘𝐺)‘𝑛) = ((coeff‘𝐺)‘𝑗))
206	205	neeq1d 3015	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → (((coeff‘𝐺)‘𝑛) ≠ 0 ↔ ((coeff‘𝐺)‘𝑗) ≠ 0))
207	206	elrab 3645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0} ↔ (𝑗 ∈ ℕ0 ∧ ((coeff‘𝐺)‘𝑗) ≠ 0))
208	204, 207	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
209	208	adantll 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ {𝑛 ∈ ℕ0 ∣ ((coeff‘𝐺)‘𝑛) ≠ 0})
210		infssuzle 13051	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 5127	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 ≤ 𝑗)
213	39	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑀 ∈ ℝ)
214	182	zred 12796	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺))) → 𝑗 ∈ ℝ)
215	214	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → 𝑗 ∈ ℝ)
216	213, 215	lenltd 11449	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → (𝑀 ≤ 𝑗 ↔ ¬ 𝑗 < 𝑀))
217	212, 216	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) ∧ ¬ ((coeff‘𝐺)‘𝑗) = 0) → ¬ 𝑗 < 𝑀)
218	196, 217	condan 830	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((coeff‘𝐺)‘𝑗) = 0)
219	218	oveq1d 7433	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) = (0 · (𝐴↑𝑗)))
220	126	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝐴 ∈ ℂ)
221	200	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → 𝑗 ∈ ℕ0)
222	220, 221	expcld 14282	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (𝐴↑𝑗) ∈ ℂ)
223	222	mul02d 11501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 · (𝐴↑𝑗)) = 0)
224	219, 223	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) = 0)
225	224	oveq1d 7433	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = (0 / (𝐴↑𝑀)))
226	126, 146, 32	expne0d 14288	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴↑𝑀) ≠ 0)
227	165, 226	div0d 12085	. . . . . . . . 9 ⊢ (𝜑 → (0 / (𝐴↑𝑀)) = 0)
228	227	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → (0 / (𝐴↑𝑀)) = 0)
229	225, 228	eqtrd 2796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((0...(deg‘𝐺)) ∖ (𝑀...(deg‘𝐺)))) → ((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)) = 0)
230		fzfid 14109	. . . . . . 7 ⊢ (𝜑 → (0...(deg‘𝐺)) ∈ Fin)
231	176, 178, 229, 230	fsumss 15884	. . . . . 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 2800	. . . . 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 603	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ)
234	53	fvmpt2 7003	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ0 ∧ ((coeff‘𝐺)‘(𝑘 + 𝑀)) ∈ ℤ) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
235	86, 233, 234	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
236	235	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝐼‘𝑘) = ((coeff‘𝐺)‘(𝑘 + 𝑀)))
237		oveq1 7425	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑧↑𝑘) = (𝐴↑𝑘))
238	237	ad2antlr 740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → (𝑧↑𝑘) = (𝐴↑𝑘))
239	236, 238	oveq12d 7436	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 = 𝐴) ∧ 𝑘 ∈ (0...((deg‘𝐺) − 𝑀))) → ((𝐼‘𝑘) · (𝑧↑𝑘)) = (((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)))
240	239	sumeq2dv 15862	. . . . . 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 14109	. . . . . . 7 ⊢ (𝜑 → (0...((deg‘𝐺) − 𝑀)) ∈ Fin)
242	241, 129	fsumcl 15892	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)) ∈ ℂ)
243	2, 240, 126, 242	fvmptd 6999	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) = Σ𝑘 ∈ (0...((deg‘𝐺) − 𝑀))(((coeff‘𝐺)‘(𝑘 + 𝑀)) · (𝐴↑𝑘)))
244	17, 16	coeid2 26551	. . . . . . . 8 ⊢ ((𝐺 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝐺‘𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)))
245	5, 126, 244	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝐴) = Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)))
246	245	oveq1d 7433	. . . . . 6 ⊢ (𝜑 → ((𝐺‘𝐴) / (𝐴↑𝑀)) = (Σ𝑗 ∈ (0...(deg‘𝐺))(((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
247	83	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → (coeff‘𝐺):ℕ0⟶ℂ)
248	199	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → 𝑗 ∈ ℕ0)
249	247, 248	ffvelcdmd 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → ((coeff‘𝐺)‘𝑗) ∈ ℂ)
250	126	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → 𝐴 ∈ ℂ)
251	250, 248	expcld 14282	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → (𝐴↑𝑗) ∈ ℂ)
252	249, 251	mulcld 11322	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...(deg‘𝐺))) → (((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) ∈ ℂ)
253	230, 165, 252, 226	fsumdivc 15945	. . . . . 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 2796	. . . . 5 ⊢ (𝜑 → ((𝐺‘𝐴) / (𝐴↑𝑀)) = Σ𝑗 ∈ (0...(deg‘𝐺))((((coeff‘𝐺)‘𝑗) · (𝐴↑𝑗)) / (𝐴↑𝑀)))
255	232, 243, 254	3eqtr4d 2806	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝐺‘𝐴) / (𝐴↑𝑀)))
256		elaa2lem.ga	. . . . 5 ⊢ (𝜑 → (𝐺‘𝐴) = 0)
257	256	oveq1d 7433	. . . 4 ⊢ (𝜑 → ((𝐺‘𝐴) / (𝐴↑𝑀)) = (0 / (𝐴↑𝑀)))
258	255, 257, 227	3eqtrd 2800	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) = 0)
259	122, 258	jca 521	. 2 ⊢ (𝜑 → (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹‘𝐴) = 0))
260		fveq2 6883	. . . . . 6 ⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
261	260	fveq1d 6885	. . . . 5 ⊢ (𝑓 = 𝐹 → ((coeff‘𝑓)‘0) = ((coeff‘𝐹)‘0))
262	261	neeq1d 3015	. . . 4 ⊢ (𝑓 = 𝐹 → (((coeff‘𝑓)‘0) ≠ 0 ↔ ((coeff‘𝐹)‘0) ≠ 0))
263		fveq1 6882	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘𝐴) = (𝐹‘𝐴))
264	263	eqeq1d 2763	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝐴) = 0 ↔ (𝐹‘𝐴) = 0))
265	262, 264	anbi12d 644	. . 3 ⊢ (𝑓 = 𝐹 → ((((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0) ↔ (((coeff‘𝐹)‘0) ≠ 0 ∧ (𝐹‘𝐴) = 0)))
266	265	rspcev 3577	. 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 596	1 ⊢ (𝜑 → ∃𝑓 ∈ (Poly‘ℤ)(((coeff‘𝑓)‘0) ≠ 0 ∧ (𝑓‘𝐴) = 0))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  {crab 3413   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  infcinf 9426  ℂcc 11191  ℝcr 11192  0cc0 11193   + caddc 11196   · cmul 11198   < clt 11336   ≤ cle 11337   − cmin 11534   / cdiv 11966  ℕ₀cn0 12599  ℤcz 12686  ℤ_≥cuz 12958  ...cfz 13632  ↑cexp 14197  Σcsu 15846  0_𝑝c0p 25983  Polycply 26495  coeffccoe 26497  degcdgr 26498  𝔸caa 26630

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-map 8842  df-pm 8843  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9427  df-inf 9428  df-oi 9497  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-n0 12600  df-z 12687  df-uz 12959  df-rp 13114  df-fz 13633  df-fzo 13782  df-fl 13925  df-seq 14138  df-exp 14198  df-hash 14468  df-cj 15259  df-re 15260  df-im 15261  df-sqrt 15395  df-abs 15396  df-clim 15648  df-rlim 15649  df-sum 15847  df-0p 25984  df-ply 26499  df-coe 26501  df-dgr 26502  df-aa 26631

This theorem is used by:  elaa2  47213