Theorem eulerpartlems 34504

Description: Lemma for eulerpart 34526. (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 34503	. . . . 5 ⊢ 𝑆:((ℕ0 ↑m ℕ) ∩ 𝑅)⟶ℕ0
4	3	ffvelcdmi 7035	. . . 4 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) ∈ ℕ0)
5		nndiffz1 32859	. . . . 5 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆‘𝐴))) = (ℤ≥‘((𝑆‘𝐴) + 1)))
6	5	eleq2d 2822	. . . 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 574	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) ↔ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
9		simpr 484	. . . . . 6 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))))
10		eldif 3899	. . . . . 6 ⊢ (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
11	9, 10	sylib 218	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
12	11	simpld 494	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℕ)
13	1, 2	eulerpartlemelr 34501	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))
14	13	simpld 494	. . . . 5 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
15	14	ffvelcdmda 7036	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
16	12, 15	syldan 592	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) ∈ ℕ0)
17		simpl 482	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
18	4	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑆‘𝐴) ∈ ℕ0)
19	11	simprd 495	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ¬ 𝑡 ∈ (1...(𝑆‘𝐴)))
20		simpl 482	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ ℕ)
21		nnuz 12827	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
22	20, 21	eleqtrdi 2846	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ (ℤ≥‘1))
23		simpr 484	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℕ0)
24	23	nn0zd 12549	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℤ)
25		elfz5 13470	. . . . . . . . 9 ⊢ ((𝑡 ∈ (ℤ≥‘1) ∧ (𝑆‘𝐴) ∈ ℤ) → (𝑡 ∈ (1...(𝑆‘𝐴)) ↔ 𝑡 ≤ (𝑆‘𝐴)))
26	22, 24, 25	syl2anc 585	. . . . . . . 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 12499	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℝ)
29	20	nnred 12189	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ ℝ)
30	28, 29	ltnled 11293	. . . . . . 7 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → ((𝑆‘𝐴) < 𝑡 ↔ ¬ 𝑡 ≤ (𝑆‘𝐴)))
31	27, 30	bitr4d 282	. . . . . 6 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (¬ 𝑡 ∈ (1...(𝑆‘𝐴)) ↔ (𝑆‘𝐴) < 𝑡))
32	31	biimpa 476	. . . . 5 ⊢ (((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))) → (𝑆‘𝐴) < 𝑡)
33	12, 18, 19, 32	syl21anc 838	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑆‘𝐴) < 𝑡)
34	1, 2	eulerpartlemsv1 34500	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘))
35		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
36		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
37	35, 36	oveq12d 7385	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡))
38	37	cbvsumv 15658	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡)
39	34, 38	eqtr2di 2788	. . . . . . . . 9 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) = (𝑆‘𝐴))
40		breq2 5089	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑙 → ((𝑆‘𝐴) < 𝑡 ↔ (𝑆‘𝐴) < 𝑙))
41		fveq2 6840	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑙 → (𝐴‘𝑡) = (𝐴‘𝑙))
42	41	breq2d 5097	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑙 → (0 < (𝐴‘𝑡) ↔ 0 < (𝐴‘𝑙)))
43	40, 42	anbi12d 633	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑙 → (((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))))
44	43	cbvrexvw 3216	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙)))
45	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℕ0)
46	45	nn0red 12499	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℝ)
47	4	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℕ0)
48	47	nn0red 12499	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℝ)
49		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
50	49	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℕ)
51	50	nnred 12189	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℝ)
52		1zzd 12558	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 1 ∈ ℤ)
53	14	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝐴:ℕ⟶ℕ0)
54		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ)
55		eqidd 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) = (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)))
56		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → 𝑚 = 𝑡)
57	56	fveq2d 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → (𝐴‘𝑚) = (𝐴‘𝑡))
58	57, 56	oveq12d 7385	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → ((𝐴‘𝑚) · 𝑚) = ((𝐴‘𝑡) · 𝑡))
59		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ)
60		ffvelcdm 7033	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
61	59	nnnn0d 12498	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ0)
62	60, 61	nn0mulcld 12503	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℕ0)
63	55, 58, 59, 62	fvmptd 6955	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))‘𝑡) = ((𝐴‘𝑡) · 𝑡))
64	53, 54, 63	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))‘𝑡) = ((𝐴‘𝑡) · 𝑡))
65	14	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝐴:ℕ⟶ℕ0)
66	65	ffvelcdmda 7036	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
67	54	nnnn0d 12498	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ0)
68	66, 67	nn0mulcld 12503	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℕ0)
69	68	nn0red 12499	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
70		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑡 → (𝐴‘𝑚) = (𝐴‘𝑡))
71		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑡 → 𝑚 = 𝑡)
72	70, 71	oveq12d 7385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑡 → ((𝐴‘𝑚) · 𝑚) = ((𝐴‘𝑡) · 𝑡))
73	72	cbvmptv 5189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) = (𝑡 ∈ ℕ ↦ ((𝐴‘𝑡) · 𝑡))
74	68, 73	fmptd 7066	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℕ0)
75		nn0sscn 12442	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 ⊆ ℂ
76		fss 6684	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ)
77	74, 75, 76	sylancl 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ)
78		nnex 12180	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ ∈ V
79		0nn0 12452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℕ0
80		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℂ ∖ {0}) = (ℂ ∖ {0})
81	80	ffs2 32800	. . . . . . . . . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . . . . . . . . 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		fcdmnn0supp 12494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℕ ∈ V ∧ 𝐴:ℕ⟶ℕ0) → (𝐴 supp 0) = (◡𝐴 “ ℕ))
85	78, 65, 84	sylancr 588	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴 supp 0) = (◡𝐴 “ ℕ))
86	13	simprd 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (◡𝐴 “ ℕ) ∈ Fin)
87	86	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (◡𝐴 “ ℕ) ∈ Fin)
88	85, 87	eqeltrd 2836	. . . . . . . . . . . . . . . . . . . 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 6668	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴:ℕ⟶ℕ0 → 𝐴 Fn ℕ)
92		simp3 1139	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → (𝐴‘𝑡) = 0)
93	92	oveq1d 7382	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → ((𝐴‘𝑡) · 𝑡) = (0 · 𝑡))
94		simp2 1138	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → 𝑡 ∈ ℕ)
95	94	nncnd 12190	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → 𝑡 ∈ ℂ)
96	95	mul02d 11344	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → (0 · 𝑡) = 0)
97	93, 96	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → ((𝐴‘𝑡) · 𝑡) = 0)
98	73, 89, 90, 91, 97	suppss3 32796	. . . . . . . . . . . . . . . . . . . . 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 9107	. . . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ∈ Fin)
102	83, 101	eqeltrrd 2837	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (◡(𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) “ (ℂ ∖ {0})) ∈ Fin)
103	21, 52, 77, 102	fsumcvg4 34094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → seq1( + , (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))) ∈ dom ⇝ )
104	21, 52, 64, 69, 103	isumrecl 15727	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
105	104	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
106		simprl 771	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) < 𝑙)
107	14	ffvelcdmda 7036	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴‘𝑙) ∈ ℕ0)
108	107	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝐴‘𝑙) ∈ ℕ0)
109	108	nn0red 12499	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝐴‘𝑙) ∈ ℝ)
110	109, 51	remulcld 11175	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → ((𝐴‘𝑙) · 𝑙) ∈ ℝ)
111	50	nnnn0d 12498	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℕ0)
112	111	nn0ge0d 12501	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 0 ≤ 𝑙)
113		simprr 773	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 0 < (𝐴‘𝑙))
114		elnnnn0b 12481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴‘𝑙) ∈ ℕ ↔ ((𝐴‘𝑙) ∈ ℕ0 ∧ 0 < (𝐴‘𝑙)))
115		nnge1 12205	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴‘𝑙) ∈ ℕ → 1 ≤ (𝐴‘𝑙))
116	114, 115	sylbir 235	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑙) ∈ ℕ0 ∧ 0 < (𝐴‘𝑙)) → 1 ≤ (𝐴‘𝑙))
117	108, 113, 116	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 1 ≤ (𝐴‘𝑙))
118	51, 109, 112, 117	lemulge12d 12094	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ≤ ((𝐴‘𝑙) · 𝑙))
119	107	nn0cnd 12500	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴‘𝑙) ∈ ℂ)
120	49	nncnd 12190	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℂ)
121	119, 120	mulcld 11165	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝐴‘𝑙) · 𝑙) ∈ ℂ)
122		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑙 → 𝑡 = 𝑙)
123	41, 122	oveq12d 7385	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑙 → ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
124	123	sumsn 15708	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑙 ∈ ℕ ∧ ((𝐴‘𝑙) · 𝑙) ∈ ℂ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
125	49, 121, 124	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
126		snfi 8990	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑙} ∈ Fin
127	126	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → {𝑙} ∈ Fin)
128	49	snssd 4730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → {𝑙} ⊆ ℕ)
129	68	nn0ge0d 12501	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 0 ≤ ((𝐴‘𝑡) · 𝑡))
130	21, 52, 127, 128, 64, 69, 129, 103	isumless 15810	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
131	125, 130	eqbrtrrd 5109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝐴‘𝑙) · 𝑙) ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
132	131	adantr 480	. . . . . . . . . . . . . . . 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 11303	. . . . . . . . . . . . . . 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 11306	. . . . . . . . . . . . . 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 3141	. . . . . . . . . . . . 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 11281	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴))
137	136	ex 412	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙)) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴)))
138	44, 137	biimtrid 242	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴)))
139	138	necon2bd 2948	. . . . . . . . 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 3063	. . . . . . . 8 ⊢ (∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ¬ ∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
142	140, 141	sylibr 234	. . . . . . 7 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → ∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
143		imnan 399	. . . . . . . 8 ⊢ (((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)) ↔ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
144	143	ralbii 3083	. . . . . . 7 ⊢ (∀𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)) ↔ ∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
145	142, 144	sylibr 234	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → ∀𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)))
146	145	r19.21bi 3229	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)))
147	146	imp 406	. . . 4 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ (𝑆‘𝐴) < 𝑡) → ¬ 0 < (𝐴‘𝑡))
148	17, 12, 33, 147	syl21anc 838	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ¬ 0 < (𝐴‘𝑡))
149		nn0re 12446	. . . . . 6 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (𝐴‘𝑡) ∈ ℝ)
150		0red 11147	. . . . . 6 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → 0 ∈ ℝ)
151	149, 150	lenltd 11292	. . . . 5 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → ((𝐴‘𝑡) ≤ 0 ↔ ¬ 0 < (𝐴‘𝑡)))
152		nn0le0eq0 12465	. . . . 5 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → ((𝐴‘𝑡) ≤ 0 ↔ (𝐴‘𝑡) = 0))
153	151, 152	bitr3d 281	. . . 4 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (¬ 0 < (𝐴‘𝑡) ↔ (𝐴‘𝑡) = 0))
154	153	biimpa 476	. . 3 ⊢ (((𝐴‘𝑡) ∈ ℕ0 ∧ ¬ 0 < (𝐴‘𝑡)) → (𝐴‘𝑡) = 0)
155	16, 148, 154	syl2anc 585	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) = 0)
156	8, 155	sylbir 235	1 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ∖ cdif 3886   ∩ cin 3888   ⊆ wss 3889  {csn 4567   class class class wbr 5085   ↦ cmpt 5166  ^◡ccnv 5630   “ cima 5634  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   supp csupp 8110   ↑_m cmap 8773  Fincfn 8893  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043   < clt 11179   ≤ cle 11180  ℕcn 12174  ℕ₀cn0 12437  ℤcz 12524  ℤ_≥cuz 12788  ...cfz 13461  Σcsu 15648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-fl 13751  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-rlim 15451  df-sum 15649

This theorem is referenced by:  eulerpartlemsv3  34505  eulerpartlemgc  34506