Theorem eulerpartlems 29749

Description: Lemma for eulerpart 29771. (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 29748	. . . . 5 ⊢ 𝑆:((ℕ0 ↑𝑚 ℕ) ∩ 𝑅)⟶ℕ0
4	3	ffvelrni 6266	. . . 4 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘𝐴) ∈ ℕ0)
5		nndiffz1 28936	. . . . 5 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆‘𝐴))) = (ℤ≥‘((𝑆‘𝐴) + 1)))
6	5	eleq2d 2673	. . . 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 667	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) ↔ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
9		simpr 476	. . . . . 6 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))))
10		eldif 3550	. . . . . 6 ⊢ (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
11	9, 10	sylib 207	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
12	11	simpld 474	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℕ)
13	1, 2	eulerpartlemelr 29746	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))
14	13	simpld 474	. . . . 5 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
15	14	ffvelrnda 6267	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
16	12, 15	syldan 486	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) ∈ ℕ0)
17		simpl 472	. . . 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 478	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ¬ 𝑡 ∈ (1...(𝑆‘𝐴)))
20		simpl 472	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ ℕ)
21		nnuz 11599	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
22	20, 21	syl6eleq 2698	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ (ℤ≥‘1))
23		simpr 476	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℕ0)
24	23	nn0zd 11356	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℤ)
25		elfz5 12205	. . . . . . . . 9 ⊢ ((𝑡 ∈ (ℤ≥‘1) ∧ (𝑆‘𝐴) ∈ ℤ) → (𝑡 ∈ (1...(𝑆‘𝐴)) ↔ 𝑡 ≤ (𝑆‘𝐴)))
26	22, 24, 25	syl2anc 691	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑡 ∈ (1...(𝑆‘𝐴)) ↔ 𝑡 ≤ (𝑆‘𝐴)))
27	26	notbid 307	. . . . . . 7 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (¬ 𝑡 ∈ (1...(𝑆‘𝐴)) ↔ ¬ 𝑡 ≤ (𝑆‘𝐴)))
28	23	nn0red 11229	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℝ)
29	20	nnred 10912	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ ℝ)
30	28, 29	ltnled 10063	. . . . . . 7 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → ((𝑆‘𝐴) < 𝑡 ↔ ¬ 𝑡 ≤ (𝑆‘𝐴)))
31	27, 30	bitr4d 270	. . . . . 6 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (¬ 𝑡 ∈ (1...(𝑆‘𝐴)) ↔ (𝑆‘𝐴) < 𝑡))
32	31	biimpa 500	. . . . 5 ⊢ (((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))) → (𝑆‘𝐴) < 𝑡)
33	12, 18, 19, 32	syl21anc 1317	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑆‘𝐴) < 𝑡)
34	1, 2	eulerpartlemsv1 29745	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘))
35		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
36		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
37	35, 36	oveq12d 6567	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡))
38	37	cbvsumv 14274	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡)
39	34, 38	syl6req 2661	. . . . . . . . 9 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) = (𝑆‘𝐴))
40		breq2 4587	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑙 → ((𝑆‘𝐴) < 𝑡 ↔ (𝑆‘𝐴) < 𝑙))
41		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑙 → (𝐴‘𝑡) = (𝐴‘𝑙))
42	41	breq2d 4595	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑙 → (0 < (𝐴‘𝑡) ↔ 0 < (𝐴‘𝑙)))
43	40, 42	anbi12d 743	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑙 → (((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))))
44	43	cbvrexv 3148	. . . . . . . . . . 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 11229	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℝ)
47	4	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℕ0)
48	47	nn0red 11229	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℝ)
49		simpr 476	. . . . . . . . . . . . . . . . 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 10912	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℝ)
52		1zzd 11285	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 1 ∈ ℤ)
53	14	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝐴:ℕ⟶ℕ0)
54		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ)
55		eqidd 2611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) = (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)))
56		simpr 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → 𝑚 = 𝑡)
57	56	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → (𝐴‘𝑚) = (𝐴‘𝑡))
58	57, 56	oveq12d 6567	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → ((𝐴‘𝑚) · 𝑚) = ((𝐴‘𝑡) · 𝑡))
59		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ)
60		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
61	59	nnnn0d 11228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ0)
62	60, 61	nn0mulcld 11233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℕ0)
63	55, 58, 59, 62	fvmptd 6197	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))‘𝑡) = ((𝐴‘𝑡) · 𝑡))
64	53, 54, 63	syl2anc 691	. . . . . . . . . . . . . . . . 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	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
67	54	nnnn0d 11228	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ0)
68	66, 67	nn0mulcld 11233	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℕ0)
69	68	nn0red 11229	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
70		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑡 → (𝐴‘𝑚) = (𝐴‘𝑡))
71		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑡 → 𝑚 = 𝑡)
72	70, 71	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑡 → ((𝐴‘𝑚) · 𝑚) = ((𝐴‘𝑡) · 𝑡))
73	72	cbvmptv 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) = (𝑡 ∈ ℕ ↦ ((𝐴‘𝑡) · 𝑡))
74	68, 73	fmptd 6292	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℕ0)
75		nn0sscn 11174	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 ⊆ ℂ
76		fss 5969	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ)
77	74, 75, 76	sylancl 693	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ)
78		nnex 10903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ ∈ V
79		0nn0 11184	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℕ0
80		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℂ ∖ {0}) = (ℂ ∖ {0})
81	80	ffs2 28891	. . . . . . . . . . . . . . . . . . . . 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 1406	. . . . . . . . . . . . . . . . . . . 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 11226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℕ ∈ V ∧ 𝐴:ℕ⟶ℕ0) → (𝐴 supp 0) = (◡𝐴 “ ℕ))
85	78, 65, 84	sylancr 694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴 supp 0) = (◡𝐴 “ ℕ))
86	13	simprd 478	. . . . . . . . . . . . . . . . . . . . . 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 2688	. . . . . . . . . . . . . . . . . . . 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 5958	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴:ℕ⟶ℕ0 → 𝐴 Fn ℕ)
92		simp3 1056	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → (𝐴‘𝑡) = 0)
93	92	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → ((𝐴‘𝑡) · 𝑡) = (0 · 𝑡))
94		simp2 1055	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → 𝑡 ∈ ℕ)
95	94	nncnd 10913	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → 𝑡 ∈ ℂ)
96	95	mul02d 10113	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → (0 · 𝑡) = 0)
97	93, 96	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → ((𝐴‘𝑡) · 𝑡) = 0)
98	73, 89, 90, 91, 97	suppss3 28890	. . . . . . . . . . . . . . . . . . . . 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 8065	. . . . . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ∈ Fin)
102	83, 101	eqeltrrd 2689	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (◡(𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) “ (ℂ ∖ {0})) ∈ Fin)
103	21, 52, 77, 102	fsumcvg4 29324	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → seq1( + , (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))) ∈ dom ⇝ )
104	21, 52, 64, 69, 103	isumrecl 14338	. . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) < 𝑙)
107	14	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . 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 11229	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝐴‘𝑙) ∈ ℝ)
110	109, 51	remulcld 9949	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → ((𝐴‘𝑙) · 𝑙) ∈ ℝ)
111	50	nnnn0d 11228	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℕ0)
112	111	nn0ge0d 11231	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 0 ≤ 𝑙)
113		simprr 792	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 0 < (𝐴‘𝑙))
114		elnnnn0b 11214	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴‘𝑙) ∈ ℕ ↔ ((𝐴‘𝑙) ∈ ℕ0 ∧ 0 < (𝐴‘𝑙)))
115		nnge1 10923	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴‘𝑙) ∈ ℕ → 1 ≤ (𝐴‘𝑙))
116	114, 115	sylbir 224	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑙) ∈ ℕ0 ∧ 0 < (𝐴‘𝑙)) → 1 ≤ (𝐴‘𝑙))
117	108, 113, 116	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 1 ≤ (𝐴‘𝑙))
118	51, 109, 112, 117	lemulge12d 10841	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ≤ ((𝐴‘𝑙) · 𝑙))
119	107	nn0cnd 11230	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴‘𝑙) ∈ ℂ)
120	49	nncnd 10913	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℂ)
121	119, 120	mulcld 9939	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝐴‘𝑙) · 𝑙) ∈ ℂ)
122		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑙 → 𝑡 = 𝑙)
123	41, 122	oveq12d 6567	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑙 → ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
124	123	sumsn 14319	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑙 ∈ ℕ ∧ ((𝐴‘𝑙) · 𝑙) ∈ ℂ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
125	49, 121, 124	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
126		snfi 7923	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑙} ∈ Fin
127	126	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → {𝑙} ∈ Fin)
128	49	snssd 4281	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → {𝑙} ⊆ ℕ)
129	68	nn0ge0d 11231	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 0 ≤ ((𝐴‘𝑡) · 𝑡))
130	21, 52, 127, 128, 64, 69, 129, 103	isumless 14416	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
131	125, 130	eqbrtrrd 4607	. . . . . . . . . . . . . . . . 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 10073	. . . . . . . . . . . . . . 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 10076	. . . . . . . . . . . . . 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 3059	. . . . . . . . . . . . 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 10051	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴))
137	136	ex 449	. . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → (∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴)))
139	138	necon2bd 2798	. . . . . . . . 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 2975	. . . . . . . 8 ⊢ (∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ¬ ∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
142	140, 141	sylibr 223	. . . . . . 7 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → ∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
143		imnan 437	. . . . . . . 8 ⊢ (((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)) ↔ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
144	143	ralbii 2963	. . . . . . 7 ⊢ (∀𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)) ↔ ∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
145	142, 144	sylibr 223	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) → ∀𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)))
146	145	r19.21bi 2916	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)))
147	146	imp 444	. . . 4 ⊢ (((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ (𝑆‘𝐴) < 𝑡) → ¬ 0 < (𝐴‘𝑡))
148	17, 12, 33, 147	syl21anc 1317	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ¬ 0 < (𝐴‘𝑡))
149		nn0re 11178	. . . . . 6 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (𝐴‘𝑡) ∈ ℝ)
150		0red 9920	. . . . . 6 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → 0 ∈ ℝ)
151	149, 150	lenltd 10062	. . . . 5 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → ((𝐴‘𝑡) ≤ 0 ↔ ¬ 0 < (𝐴‘𝑡)))
152		nn0le0eq0 11198	. . . . 5 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → ((𝐴‘𝑡) ≤ 0 ↔ (𝐴‘𝑡) = 0))
153	151, 152	bitr3d 269	. . . 4 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (¬ 0 < (𝐴‘𝑡) ↔ (𝐴‘𝑡) = 0))
154	153	biimpa 500	. . 3 ⊢ (((𝐴‘𝑡) ∈ ℕ0 ∧ ¬ 0 < (𝐴‘𝑡)) → (𝐴‘𝑡) = 0)
155	16, 148, 154	syl2anc 691	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) = 0)
156	8, 155	sylbir 224	1 ⊢ ((𝐴 ∈ ((ℕ0 ↑𝑚 ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   supp csupp 7182   ↑_𝑚 cmap 7744  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  Σcsu 14264

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

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265

This theorem is referenced by:  eulerpartlemsv3  29750  eulerpartlemgc  29751