Theorem eulerpartlems 32327

Description: Lemma for eulerpart 32349. (Contributed by Thierry Arnoux, 6-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)

Hypotheses

Ref	Expression
eulerpartlems.r	⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
eulerpartlems.s	⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))

Assertion

Ref	Expression
eulerpartlems	⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)

Distinct variable groups:   𝑓,𝑘,𝐴   𝑅,𝑓,𝑘   𝑡,𝑘,𝐴   𝑡,𝑅   𝑡,𝑆

Allowed substitution hints:   𝑆(𝑓,𝑘)

Proof of Theorem eulerpartlems

Dummy variables 𝑙 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eulerpartlems.r	. . . . . 6 ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}
2		eulerpartlems.s	. . . . . 6 ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))
3	1, 2	eulerpartlemsf 32326	. . . . 5 ⊢ 𝑆:((ℕ0 ↑m ℕ) ∩ 𝑅)⟶ℕ0
4	3	ffvelrni 6960	. . . 4 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) ∈ ℕ0)
5		nndiffz1 31107	. . . . 5 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆‘𝐴))) = (ℤ≥‘((𝑆‘𝐴) + 1)))
6	5	eleq2d 2824	. . . 4 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
7	4, 6	syl 17	. . 3 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
8	7	pm5.32i 575	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) ↔ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
9		simpr 485	. . . . . 6 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))))
10		eldif 3897	. . . . . 6 ⊢ (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
11	9, 10	sylib 217	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
12	11	simpld 495	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℕ)
13	1, 2	eulerpartlemelr 32324	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))
14	13	simpld 495	. . . . 5 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
15	14	ffvelrnda 6961	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
16	12, 15	syldan 591	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) ∈ ℕ0)
17		simpl 483	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
18	4	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑆‘𝐴) ∈ ℕ0)
19	11	simprd 496	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ¬ 𝑡 ∈ (1...(𝑆‘𝐴)))
20		simpl 483	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ ℕ)
21		nnuz 12621	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
22	20, 21	eleqtrdi 2849	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ (ℤ≥‘1))
23		simpr 485	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℕ0)
24	23	nn0zd 12424	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℤ)
25		elfz5 13248	. . . . . . . . 9 ⊢ ((𝑡 ∈ (ℤ≥‘1) ∧ (𝑆‘𝐴) ∈ ℤ) → (𝑡 ∈ (1...(𝑆‘𝐴)) ↔ 𝑡 ≤ (𝑆‘𝐴)))
26	22, 24, 25	syl2anc 584	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑡 ∈ (1...(𝑆‘𝐴)) ↔ 𝑡 ≤ (𝑆‘𝐴)))
27	26	notbid 318	. . . . . . 7 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (¬ 𝑡 ∈ (1...(𝑆‘𝐴)) ↔ ¬ 𝑡 ≤ (𝑆‘𝐴)))
28	23	nn0red 12294	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℝ)
29	20	nnred 11988	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ ℝ)
30	28, 29	ltnled 11122	. . . . . . 7 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → ((𝑆‘𝐴) < 𝑡 ↔ ¬ 𝑡 ≤ (𝑆‘𝐴)))
31	27, 30	bitr4d 281	. . . . . 6 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (¬ 𝑡 ∈ (1...(𝑆‘𝐴)) ↔ (𝑆‘𝐴) < 𝑡))
32	31	biimpa 477	. . . . 5 ⊢ (((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))) → (𝑆‘𝐴) < 𝑡)
33	12, 18, 19, 32	syl21anc 835	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑆‘𝐴) < 𝑡)
34	1, 2	eulerpartlemsv1 32323	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘))
35		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
36		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
37	35, 36	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡))
38	37	cbvsumv 15408	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡)
39	34, 38	eqtr2di 2795	. . . . . . . . 9 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) = (𝑆‘𝐴))
40		breq2 5078	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑙 → ((𝑆‘𝐴) < 𝑡 ↔ (𝑆‘𝐴) < 𝑙))
41		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑙 → (𝐴‘𝑡) = (𝐴‘𝑙))
42	41	breq2d 5086	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑙 → (0 < (𝐴‘𝑡) ↔ 0 < (𝐴‘𝑙)))
43	40, 42	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑙 → (((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))))
44	43	cbvrexvw 3384	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙)))
45	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℕ0)
46	45	nn0red 12294	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℝ)
47	4	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℕ0)
48	47	nn0red 12294	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℝ)
49		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
50	49	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℕ)
51	50	nnred 11988	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℝ)
52		1zzd 12351	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 1 ∈ ℤ)
53	14	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝐴:ℕ⟶ℕ0)
54		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ)
55		eqidd 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) = (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)))
56		simpr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → 𝑚 = 𝑡)
57	56	fveq2d 6778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → (𝐴‘𝑚) = (𝐴‘𝑡))
58	57, 56	oveq12d 7293	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → ((𝐴‘𝑚) · 𝑚) = ((𝐴‘𝑡) · 𝑡))
59		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ)
60		ffvelrn 6959	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
61	59	nnnn0d 12293	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ0)
62	60, 61	nn0mulcld 12298	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℕ0)
63	55, 58, 59, 62	fvmptd 6882	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))‘𝑡) = ((𝐴‘𝑡) · 𝑡))
64	53, 54, 63	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))‘𝑡) = ((𝐴‘𝑡) · 𝑡))
65	14	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝐴:ℕ⟶ℕ0)
66	65	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
67	54	nnnn0d 12293	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ0)
68	66, 67	nn0mulcld 12298	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℕ0)
69	68	nn0red 12294	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
70		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑡 → (𝐴‘𝑚) = (𝐴‘𝑡))
71		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑡 → 𝑚 = 𝑡)
72	70, 71	oveq12d 7293	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑡 → ((𝐴‘𝑚) · 𝑚) = ((𝐴‘𝑡) · 𝑡))
73	72	cbvmptv 5187	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) = (𝑡 ∈ ℕ ↦ ((𝐴‘𝑡) · 𝑡))
74	68, 73	fmptd 6988	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℕ0)
75		nn0sscn 12238	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 ⊆ ℂ
76		fss 6617	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ)
77	74, 75, 76	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ)
78		nnex 11979	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ ∈ V
79		0nn0 12248	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℕ0
80		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℂ ∖ {0}) = (ℂ ∖ {0})
81	80	ffs2 31063	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℕ ∈ V ∧ 0 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) = (◡(𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) “ (ℂ ∖ {0})))
82	78, 79, 81	mp3an12 1450	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) = (◡(𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) “ (ℂ ∖ {0})))
83	77, 82	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) = (◡(𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) “ (ℂ ∖ {0})))
84		frnnn0supp 12289	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℕ ∈ V ∧ 𝐴:ℕ⟶ℕ0) → (𝐴 supp 0) = (◡𝐴 “ ℕ))
85	78, 65, 84	sylancr 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴 supp 0) = (◡𝐴 “ ℕ))
86	13	simprd 496	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (◡𝐴 “ ℕ) ∈ Fin)
87	86	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (◡𝐴 “ ℕ) ∈ Fin)
88	85, 87	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴 supp 0) ∈ Fin)
89	78	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴:ℕ⟶ℕ0 → ℕ ∈ V)
90	79	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴:ℕ⟶ℕ0 → 0 ∈ ℕ0)
91		ffn 6600	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴:ℕ⟶ℕ0 → 𝐴 Fn ℕ)
92		simp3 1137	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → (𝐴‘𝑡) = 0)
93	92	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → ((𝐴‘𝑡) · 𝑡) = (0 · 𝑡))
94		simp2 1136	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → 𝑡 ∈ ℕ)
95	94	nncnd 11989	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → 𝑡 ∈ ℂ)
96	95	mul02d 11173	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → (0 · 𝑡) = 0)
97	93, 96	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → ((𝐴‘𝑡) · 𝑡) = 0)
98	73, 89, 90, 91, 97	suppss3 31059	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴:ℕ⟶ℕ0 → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ⊆ (𝐴 supp 0))
99	65, 98	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ⊆ (𝐴 supp 0))
100		ssfi 8956	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 supp 0) ∈ Fin ∧ ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ⊆ (𝐴 supp 0)) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ∈ Fin)
101	88, 99, 100	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ∈ Fin)
102	83, 101	eqeltrrd 2840	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (◡(𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) “ (ℂ ∖ {0})) ∈ Fin)
103	21, 52, 77, 102	fsumcvg4 31900	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → seq1( + , (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))) ∈ dom ⇝ )
104	21, 52, 64, 69, 103	isumrecl 15477	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
105	104	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
106		simprl 768	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) < 𝑙)
107	14	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴‘𝑙) ∈ ℕ0)
108	107	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝐴‘𝑙) ∈ ℕ0)
109	108	nn0red 12294	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝐴‘𝑙) ∈ ℝ)
110	109, 51	remulcld 11005	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → ((𝐴‘𝑙) · 𝑙) ∈ ℝ)
111	50	nnnn0d 12293	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℕ0)
112	111	nn0ge0d 12296	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 0 ≤ 𝑙)
113		simprr 770	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 0 < (𝐴‘𝑙))
114		elnnnn0b 12277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴‘𝑙) ∈ ℕ ↔ ((𝐴‘𝑙) ∈ ℕ0 ∧ 0 < (𝐴‘𝑙)))
115		nnge1 12001	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴‘𝑙) ∈ ℕ → 1 ≤ (𝐴‘𝑙))
116	114, 115	sylbir 234	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑙) ∈ ℕ0 ∧ 0 < (𝐴‘𝑙)) → 1 ≤ (𝐴‘𝑙))
117	108, 113, 116	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 1 ≤ (𝐴‘𝑙))
118	51, 109, 112, 117	lemulge12d 11913	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ≤ ((𝐴‘𝑙) · 𝑙))
119	107	nn0cnd 12295	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴‘𝑙) ∈ ℂ)
120	49	nncnd 11989	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℂ)
121	119, 120	mulcld 10995	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝐴‘𝑙) · 𝑙) ∈ ℂ)
122		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑙 → 𝑡 = 𝑙)
123	41, 122	oveq12d 7293	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑙 → ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
124	123	sumsn 15458	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑙 ∈ ℕ ∧ ((𝐴‘𝑙) · 𝑙) ∈ ℂ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
125	49, 121, 124	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
126		snfi 8834	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑙} ∈ Fin
127	126	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → {𝑙} ∈ Fin)
128	49	snssd 4742	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → {𝑙} ⊆ ℕ)
129	68	nn0ge0d 12296	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 0 ≤ ((𝐴‘𝑡) · 𝑡))
130	21, 52, 127, 128, 64, 69, 129, 103	isumless 15557	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
131	125, 130	eqbrtrrd 5098	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝐴‘𝑙) · 𝑙) ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
132	131	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → ((𝐴‘𝑙) · 𝑙) ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
133	51, 110, 105, 118, 132	letrd 11132	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
134	48, 51, 105, 106, 133	ltletrd 11135	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) < Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
135	134	r19.29an 3217	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) < Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
136	46, 135	gtned 11110	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴))
137	136	ex 413	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙)) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴)))
138	44, 137	syl5bi 241	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴)))
139	138	necon2bd 2959	. . . . . . . . 9 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) = (𝑆‘𝐴) → ¬ ∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡))))
140	39, 139	mpd 15	. . . . . . . 8 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → ¬ ∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
141		ralnex 3167	. . . . . . . 8 ⊢ (∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ¬ ∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
142	140, 141	sylibr 233	. . . . . . 7 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → ∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
143		imnan 400	. . . . . . . 8 ⊢ (((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)) ↔ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
144	143	ralbii 3092	. . . . . . 7 ⊢ (∀𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)) ↔ ∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
145	142, 144	sylibr 233	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → ∀𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)))
146	145	r19.21bi 3134	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)))
147	146	imp 407	. . . 4 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ (𝑆‘𝐴) < 𝑡) → ¬ 0 < (𝐴‘𝑡))
148	17, 12, 33, 147	syl21anc 835	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ¬ 0 < (𝐴‘𝑡))
149		nn0re 12242	. . . . . 6 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (𝐴‘𝑡) ∈ ℝ)
150		0red 10978	. . . . . 6 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → 0 ∈ ℝ)
151	149, 150	lenltd 11121	. . . . 5 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → ((𝐴‘𝑡) ≤ 0 ↔ ¬ 0 < (𝐴‘𝑡)))
152		nn0le0eq0 12261	. . . . 5 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → ((𝐴‘𝑡) ≤ 0 ↔ (𝐴‘𝑡) = 0))
153	151, 152	bitr3d 280	. . . 4 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (¬ 0 < (𝐴‘𝑡) ↔ (𝐴‘𝑡) = 0))
154	153	biimpa 477	. . 3 ⊢ (((𝐴‘𝑡) ∈ ℕ0 ∧ ¬ 0 < (𝐴‘𝑡)) → (𝐴‘𝑡) = 0)
155	16, 148, 154	syl2anc 584	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) = 0)
156	8, 155	sylbir 234	1 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  {csn 4561   class class class wbr 5074   ↦ cmpt 5157  ^◡ccnv 5588   “ cima 5592  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   supp csupp 7977   ↑_m cmap 8615  Fincfn 8733  ℂcc 10869  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876   < clt 11009   ≤ cle 11010  ℕcn 11973  ℕ₀cn0 12233  ℤcz 12319  ℤ_≥cuz 12582  ...cfz 13239  Σcsu 15397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398

This theorem is referenced by:  eulerpartlemsv3  32328  eulerpartlemgc  32329