Theorem eulerpartlems 34817

Description: Lemma for eulerpart 34839. (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 34816	. . . . 5 ⊢ 𝑆:((ℕ0 ↑m ℕ) ∩ 𝑅)⟶ℕ0
4	3	ffvelcdmi 7082	. . . 4 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) ∈ ℕ0)
5		nndiffz1 33203	. . . . 5 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (ℕ ∖ (1...(𝑆‘𝐴))) = (ℤ≥‘((𝑆‘𝐴) + 1)))
6	5	eleq2d 2851	. . . 4 ⊢ ((𝑆‘𝐴) ∈ ℕ0 → (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
7	4, 6	syl 18	. . 3 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
8	7	pm5.32i 585	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) ↔ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))))
9		eldif 3916	. . . . . 6 ⊢ (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
10	9	bilani 510	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))))
11	10	simpld 500	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℕ)
12	1, 2	eulerpartlemelr 34814	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))
13	12	simpld 500	. . . . 5 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0)
14	13	ffvelcdmda 7083	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
15	11, 14	syldan 603	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) ∈ ℕ0)
16		simpl 488	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅))
17	4	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑆‘𝐴) ∈ ℕ0)
18	10	simprd 501	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ¬ 𝑡 ∈ (1...(𝑆‘𝐴)))
19		simpl 488	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ ℕ)
20		nnuz 12919	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
21	19, 20	eleqtrdi 2875	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ (ℤ≥‘1))
22		simpr 490	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℕ0)
23	22	nn0zd 12633	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℤ)
24		elfz5 13562	. . . . . . . . 9 ⊢ ((𝑡 ∈ (ℤ≥‘1) ∧ (𝑆‘𝐴) ∈ ℤ) → (𝑡 ∈ (1...(𝑆‘𝐴)) ↔ 𝑡 ≤ (𝑆‘𝐴)))
25	21, 23, 24	syl2anc 596	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑡 ∈ (1...(𝑆‘𝐴)) ↔ 𝑡 ≤ (𝑆‘𝐴)))
26	25	notbid 321	. . . . . . 7 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (¬ 𝑡 ∈ (1...(𝑆‘𝐴)) ↔ ¬ 𝑡 ≤ (𝑆‘𝐴)))
27	22	nn0red 12583	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (𝑆‘𝐴) ∈ ℝ)
28	19	nnred 12265	. . . . . . . 8 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → 𝑡 ∈ ℝ)
29	27, 28	ltnled 11374	. . . . . . 7 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → ((𝑆‘𝐴) < 𝑡 ↔ ¬ 𝑡 ≤ (𝑆‘𝐴)))
30	26, 29	bitr4d 285	. . . . . 6 ⊢ ((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) → (¬ 𝑡 ∈ (1...(𝑆‘𝐴)) ↔ (𝑆‘𝐴) < 𝑡))
31	30	biimpa 482	. . . . 5 ⊢ (((𝑡 ∈ ℕ ∧ (𝑆‘𝐴) ∈ ℕ0) ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))) → (𝑆‘𝐴) < 𝑡)
32	11, 17, 18, 31	syl21anc 851	. . . 4 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝑆‘𝐴) < 𝑡)
33	1, 2	eulerpartlemsv1 34813	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘))
34		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡))
35		id 23	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡)
36	34, 35	oveq12d 7437	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡))
37	36	cbvsumv 15773	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡)
38	33, 37	eqtr2di 2817	. . . . . . . . 9 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) = (𝑆‘𝐴))
39		breq2 5115	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑙 → ((𝑆‘𝐴) < 𝑡 ↔ (𝑆‘𝐴) < 𝑙))
40		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑙 → (𝐴‘𝑡) = (𝐴‘𝑙))
41	40	breq2d 5123	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑙 → (0 < (𝐴‘𝑡) ↔ 0 < (𝐴‘𝑙)))
42	39, 41	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑙 → (((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))))
43	42	cbvrexvw 3246	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙)))
44	4	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℕ0)
45	44	nn0red 12583	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℝ)
46	4	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℕ0)
47	46	nn0red 12583	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) ∈ ℝ)
48		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℕ)
49	48	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℕ)
50	49	nnred 12265	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℝ)
51		1zzd 12642	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 1 ∈ ℤ)
52	13	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝐴:ℕ⟶ℕ0)
53		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ)
54		eqidd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) = (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)))
55		simpr 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → 𝑚 = 𝑡)
56	55	fveq2d 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → (𝐴‘𝑚) = (𝐴‘𝑡))
57	56, 55	oveq12d 7437	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) ∧ 𝑚 = 𝑡) → ((𝐴‘𝑚) · 𝑚) = ((𝐴‘𝑡) · 𝑡))
58		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ)
59		ffvelcdm 7080	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
60	58	nnnn0d 12582	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ0)
61	59, 60	nn0mulcld 12587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℕ0)
62	54, 57, 58, 61	fvmptd 7001	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))‘𝑡) = ((𝐴‘𝑡) · 𝑡))
63	52, 53, 62	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))‘𝑡) = ((𝐴‘𝑡) · 𝑡))
64	13	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝐴:ℕ⟶ℕ0)
65	64	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℕ0)
66	53	nnnn0d 12582	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℕ0)
67	65, 66	nn0mulcld 12587	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℕ0)
68	67	nn0red 12583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
69		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑡 → (𝐴‘𝑚) = (𝐴‘𝑡))
70		id 23	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑡 → 𝑚 = 𝑡)
71	69, 70	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑡 → ((𝐴‘𝑚) · 𝑚) = ((𝐴‘𝑡) · 𝑡))
72	71	cbvmptv 5217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) = (𝑡 ∈ ℕ ↦ ((𝐴‘𝑡) · 𝑡))
73	67, 72	fmptd 7113	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℕ0)
74		nn0sscn 12526	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 ⊆ ℂ
75		fss 6726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ)
76	73, 74, 75	sylancl 598	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ)
77		nnex 12256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ ∈ V
78		0nn0 12536	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℕ0
79		eqid 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℂ ∖ {0}) = (ℂ ∖ {0})
80	79	ffs2 33144	. . . . . . . . . . . . . . . . . . . . 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})))
81	77, 78, 80	mp3an12 1480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)):ℕ⟶ℂ → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) = (◡(𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) “ (ℂ ∖ {0})))
82	76, 81	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) = (◡(𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) “ (ℂ ∖ {0})))
83		fcdmnn0supp 12578	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℕ ∈ V ∧ 𝐴:ℕ⟶ℕ0) → (𝐴 supp 0) = (◡𝐴 “ ℕ))
84	77, 64, 83	sylancr 599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴 supp 0) = (◡𝐴 “ ℕ))
85	12	simprd 501	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (◡𝐴 “ ℕ) ∈ Fin)
86	85	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (◡𝐴 “ ℕ) ∈ Fin)
87	84, 86	eqeltrd 2865	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴 supp 0) ∈ Fin)
88	77	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴:ℕ⟶ℕ0 → ℕ ∈ V)
89	78	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴:ℕ⟶ℕ0 → 0 ∈ ℕ0)
90		ffn 6709	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴:ℕ⟶ℕ0 → 𝐴 Fn ℕ)
91		simp3 1156	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → (𝐴‘𝑡) = 0)
92	91	oveq1d 7434	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → ((𝐴‘𝑡) · 𝑡) = (0 · 𝑡))
93		simp2 1155	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → 𝑡 ∈ ℕ)
94	93	nncnd 12266	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → 𝑡 ∈ ℂ)
95	94	mul02d 11425	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → (0 · 𝑡) = 0)
96	92, 95	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴:ℕ⟶ℕ0 ∧ 𝑡 ∈ ℕ ∧ (𝐴‘𝑡) = 0) → ((𝐴‘𝑡) · 𝑡) = 0)
97	72, 88, 89, 90, 96	suppss3 33140	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴:ℕ⟶ℕ0 → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ⊆ (𝐴 supp 0))
98	64, 97	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ⊆ (𝐴 supp 0))
99		ssfi 9164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 supp 0) ∈ Fin ∧ ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ⊆ (𝐴 supp 0)) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ∈ Fin)
100	87, 98, 99	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) supp 0) ∈ Fin)
101	82, 100	eqeltrrd 2866	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (◡(𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚)) “ (ℂ ∖ {0})) ∈ Fin)
102	20, 51, 76, 101	fsumcvg4 34406	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → seq1( + , (𝑚 ∈ ℕ ↦ ((𝐴‘𝑚) · 𝑚))) ∈ dom ⇝ )
103	20, 51, 63, 68, 102	isumrecl 15841	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
104	103	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ∈ ℝ)
105		simprl 783	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) < 𝑙)
106	13	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴‘𝑙) ∈ ℕ0)
107	106	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝐴‘𝑙) ∈ ℕ0)
108	107	nn0red 12583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝐴‘𝑙) ∈ ℝ)
109	108, 50	remulcld 11256	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → ((𝐴‘𝑙) · 𝑙) ∈ ℝ)
110	49	nnnn0d 12582	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ∈ ℕ0)
111	110	nn0ge0d 12585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 0 ≤ 𝑙)
112		simprr 785	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 0 < (𝐴‘𝑙))
113		elnnnn0b 12565	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴‘𝑙) ∈ ℕ ↔ ((𝐴‘𝑙) ∈ ℕ0 ∧ 0 < (𝐴‘𝑙)))
114		nnge1 12281	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴‘𝑙) ∈ ℕ → 1 ≤ (𝐴‘𝑙))
115	113, 114	sylbir 238	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑙) ∈ ℕ0 ∧ 0 < (𝐴‘𝑙)) → 1 ≤ (𝐴‘𝑙))
116	107, 112, 115	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 1 ≤ (𝐴‘𝑙))
117	50, 108, 111, 116	lemulge12d 12170	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ≤ ((𝐴‘𝑙) · 𝑙))
118	106	nn0cnd 12584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → (𝐴‘𝑙) ∈ ℂ)
119	48	nncnd 12266	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → 𝑙 ∈ ℂ)
120	118, 119	mulcld 11246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝐴‘𝑙) · 𝑙) ∈ ℂ)
121		id 23	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑙 → 𝑡 = 𝑙)
122	40, 121	oveq12d 7437	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑙 → ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
123	122	sumsn 15822	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑙 ∈ ℕ ∧ ((𝐴‘𝑙) · 𝑙) ∈ ℂ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
124	48, 120, 123	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) = ((𝐴‘𝑙) · 𝑙))
125		snfi 9047	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑙} ∈ Fin
126	125	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → {𝑙} ∈ Fin)
127	48	snssd 4754	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → {𝑙} ⊆ ℕ)
128	67	nn0ge0d 12585	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 0 ≤ ((𝐴‘𝑡) · 𝑡))
129	20, 51, 126, 127, 63, 68, 128, 102	isumless 15924	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → Σ𝑡 ∈ {𝑙} ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
130	124, 129	eqbrtrrd 5137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) → ((𝐴‘𝑙) · 𝑙) ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
131	130	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → ((𝐴‘𝑙) · 𝑙) ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
132	50, 109, 104, 117, 131	letrd 11384	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → 𝑙 ≤ Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
133	47, 50, 104, 105, 132	ltletrd 11387	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑙 ∈ ℕ) ∧ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) < Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
134	133	r19.29an 3171	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → (𝑆‘𝐴) < Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡))
135	45, 134	gtned 11362	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ ∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙))) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴))
136	135	ex 418	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (∃𝑙 ∈ ℕ ((𝑆‘𝐴) < 𝑙 ∧ 0 < (𝐴‘𝑙)) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴)))
137	43, 136	biimtrid 245	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) → Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) ≠ (𝑆‘𝐴)))
138	137	necon2bd 2976	. . . . . . . . 9 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (Σ𝑡 ∈ ℕ ((𝐴‘𝑡) · 𝑡) = (𝑆‘𝐴) → ¬ ∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡))))
139	38, 138	mpd 16	. . . . . . . 8 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → ¬ ∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
140		ralnex 3093	. . . . . . . 8 ⊢ (∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)) ↔ ¬ ∃𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
141	139, 140	sylibr 237	. . . . . . 7 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → ∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
142		imnan 405	. . . . . . . 8 ⊢ (((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)) ↔ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
143	142	ralbii 3113	. . . . . . 7 ⊢ (∀𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)) ↔ ∀𝑡 ∈ ℕ ¬ ((𝑆‘𝐴) < 𝑡 ∧ 0 < (𝐴‘𝑡)))
144	141, 143	sylibr 237	. . . . . 6 ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → ∀𝑡 ∈ ℕ ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)))
145	144	r19.21bi 3259	. . . . 5 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝑆‘𝐴) < 𝑡 → ¬ 0 < (𝐴‘𝑡)))
146	145	imp 412	. . . 4 ⊢ (((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ (𝑆‘𝐴) < 𝑡) → ¬ 0 < (𝐴‘𝑡))
147	16, 11, 32, 146	syl21anc 851	. . 3 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ¬ 0 < (𝐴‘𝑡))
148		nn0re 12530	. . . . . 6 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (𝐴‘𝑡) ∈ ℝ)
149		0red 11228	. . . . . 6 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → 0 ∈ ℝ)
150	148, 149	lenltd 11373	. . . . 5 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → ((𝐴‘𝑡) ≤ 0 ↔ ¬ 0 < (𝐴‘𝑡)))
151		nn0le0eq0 12549	. . . . 5 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → ((𝐴‘𝑡) ≤ 0 ↔ (𝐴‘𝑡) = 0))
152	150, 151	bitr3d 284	. . . 4 ⊢ ((𝐴‘𝑡) ∈ ℕ0 → (¬ 0 < (𝐴‘𝑡) ↔ (𝐴‘𝑡) = 0))
153	152	biimpa 482	. . 3 ⊢ (((𝐴‘𝑡) ∈ ℕ0 ∧ ¬ 0 < (𝐴‘𝑡)) → (𝐴‘𝑡) = 0)
154	15, 147, 153	syl2anc 596	. 2 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) = 0)
155	8, 154	sylbir 238	1 ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  {csn 4591   class class class wbr 5111   ↦ cmpt 5194  ^◡ccnv 5662   “ cima 5666  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   supp csupp 8162   ↑_m cmap 8830  Fincfn 8949  ℂcc 11115  ℝcr 11116  0cc0 11117  1c1 11118   + caddc 11120   · cmul 11122   < clt 11260   ≤ cle 11261  ℕcn 12250  ℕ₀cn0 12521  ℤcz 12608  ℤ_≥cuz 12880  ...cfz 13553  Σcsu 15763

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-pm 8833  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-inf 9410  df-oi 9479  df-card 9941  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-div 11889  df-nn 12251  df-2 12320  df-3 12321  df-n0 12522  df-z 12609  df-uz 12881  df-rp 13035  df-fz 13554  df-fzo 13702  df-fl 13845  df-seq 14058  df-exp 14118  df-hash 14387  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-clim 15565  df-rlim 15566  df-sum 15764

This theorem is used by:  eulerpartlemsv3  34818  eulerpartlemgc  34819