Theorem eulerpartlemb 34545

Description: Lemma for eulerpart 34559. The set of all partitions of 𝑁 is finite. (Contributed by Mario Carneiro, 26-Jan-2015.)

Hypotheses

Ref	Expression
eulerpart.p	⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
eulerpart.o	⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d	⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}
eulerpart.j	⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h	⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m	⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})

Assertion

Ref	Expression
eulerpartlemb	⊢ 𝑃 ∈ Fin

Distinct variable groups:   𝑓,𝑔,𝑘,𝑥,𝑦   𝑓,𝑁,𝑔,𝑥   𝑃,𝑔

Allowed substitution hints:   𝐷(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑘,𝑛,𝑟)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝐽(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑀(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)   𝑁(𝑦,𝑧,𝑘,𝑛,𝑟)   𝑂(𝑥,𝑦,𝑧,𝑓,𝑔,𝑘,𝑛,𝑟)

Proof of Theorem eulerpartlemb

Step	Hyp	Ref	Expression
1		fzfid 13908	. . . 4 ⊢ (⊤ → (1...𝑁) ∈ Fin)
2		fzfi 13907	. . . . . 6 ⊢ (0...𝑁) ∈ Fin
3		snfi 8992	. . . . . 6 ⊢ {0} ∈ Fin
4	2, 3	ifcli 4529	. . . . 5 ⊢ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin
5	4	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℕ) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
6		eldifn 4086	. . . . . 6 ⊢ (𝑥 ∈ (ℕ ∖ (1...𝑁)) → ¬ 𝑥 ∈ (1...𝑁))
7	6	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → ¬ 𝑥 ∈ (1...𝑁))
8		iffalse 4490	. . . . 5 ⊢ (¬ 𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
9		eqimss 3994	. . . . 5 ⊢ (if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0} → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ⊆ {0})
10	7, 8, 9	3syl 18	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ⊆ {0})
11	1, 5, 10	ixpfi2 9262	. . 3 ⊢ (⊤ → X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
12	11	mptru 1549	. 2 ⊢ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin
13		eulerpart.p	. . . . 5 ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}
14	13	eulerpartleme 34540	. . . 4 ⊢ (𝑔 ∈ 𝑃 ↔ (𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁))
15		ffn 6670	. . . . . 6 ⊢ (𝑔:ℕ⟶ℕ0 → 𝑔 Fn ℕ)
16	15	3ad2ant1 1134	. . . . 5 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑔 Fn ℕ)
17		ffvelcdm 7035	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℕ0)
18	17	3ad2antl1 1187	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℕ0)
19	18	nn0red 12475	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℝ)
20		nnre 12164	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
21	20	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ)
22	19, 21	remulcld 11174	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ∈ ℝ)
23		cnvimass 6049	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑔 “ ℕ) ⊆ dom 𝑔
24		fdm 6679	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:ℕ⟶ℕ0 → dom 𝑔 = ℕ)
25	24	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → dom 𝑔 = ℕ)
26	23, 25	sseqtrid 3978	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ⊆ ℕ)
27	26	sselda 3935	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
28		ffvelcdm 7035	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ ℕ0)
29	28	adantlr 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ ℕ0)
30	27, 29	syldan 592	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → (𝑔‘𝑘) ∈ ℕ0)
31	27	nnnn0d 12474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ0)
32	30, 31	nn0mulcld 12479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
33	32	nn0cnd 12476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℂ)
34		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → 𝑔:ℕ⟶ℕ0)
35		nnex 12163	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ ∈ V
36		fcdmnn0supp 12470	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℕ ∈ V ∧ 𝑔:ℕ⟶ℕ0) → (𝑔 supp 0) = (◡𝑔 “ ℕ))
37	35, 36	mpan 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔:ℕ⟶ℕ0 → (𝑔 supp 0) = (◡𝑔 “ ℕ))
38	37	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) = (◡𝑔 “ ℕ))
39		eqimss 3994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 supp 0) = (◡𝑔 “ ℕ) → (𝑔 supp 0) ⊆ (◡𝑔 “ ℕ))
40	38, 39	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) ⊆ (◡𝑔 “ ℕ))
41	35	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → ℕ ∈ V)
42		0nn0 12428	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℕ0
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → 0 ∈ ℕ0)
44	34, 40, 41, 43	suppssr 8147	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → (𝑔‘𝑘) = 0)
45	44	oveq1d 7383	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = (0 · 𝑘))
46		eldifi 4085	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
47	46	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → 𝑘 ∈ ℕ)
48		nncn 12165	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
49		mul02 11323	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℂ → (0 · 𝑘) = 0)
50	47, 48, 49	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → (0 · 𝑘) = 0)
51	45, 50	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = 0)
52		nnuz 12802	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ≥‘1)
53	52	eqimssi 3996	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ ⊆ (ℤ≥‘1)
54	53	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → ℕ ⊆ (ℤ≥‘1))
55	26, 33, 51, 54	sumss 15659	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
56		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ∈ Fin)
57	56, 32	fsumnn0cl 15671	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
58	55, 57	eqeltrrd 2838	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
59		eleq1 2825	. . . . . . . . . . . . . . 15 ⊢ (Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁 → (Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
60	58, 59	syl5ibcom 245	. . . . . . . . . . . . . 14 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁 → 𝑁 ∈ ℕ0))
61	60	3impia 1118	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑁 ∈ ℕ0)
62	61	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℕ0)
63	62	nn0red 12475	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
64	18	nn0ge0d 12477	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝑔‘𝑥))
65		nnge1 12185	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 1 ≤ 𝑥)
66	65	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 1 ≤ 𝑥)
67	19, 21, 64, 66	lemulge11d 12091	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ ((𝑔‘𝑥) · 𝑥))
68	56	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → (◡𝑔 “ ℕ) ∈ Fin)
69	32	nn0red 12475	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
70	69	adantlr 716	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
71	32	nn0ge0d 12477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
72	71	adantlr 716	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
73		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑔‘𝑘) = (𝑔‘𝑥))
74		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → 𝑘 = 𝑥)
75	73, 74	oveq12d 7386	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → ((𝑔‘𝑘) · 𝑘) = ((𝑔‘𝑥) · 𝑥))
76		simprr 773	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → 𝑥 ∈ (◡𝑔 “ ℕ))
77	68, 70, 72, 75, 76	fsumge1 15732	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
78	77	expr 456	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (◡𝑔 “ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)))
79		eldif 3913	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) ↔ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ)))
80	51	ralrimiva 3130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → ∀𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0)
81	75	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (((𝑔‘𝑘) · 𝑘) = 0 ↔ ((𝑔‘𝑥) · 𝑥) = 0))
82	81	rspccva 3577	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0 ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
83	80, 82	sylan 581	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
84	79, 83	sylan2br 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
85	56	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (◡𝑔 “ ℕ) ∈ Fin)
86	32	adantlr 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
87	86	nn0red 12475	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
88	86	nn0ge0d 12477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
89	85, 87, 88	fsumge0 15730	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → 0 ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
90	89	adantrr 718	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → 0 ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
91	84, 90	eqbrtrd 5122	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
92	91	expr 456	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (◡𝑔 “ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)))
93	78, 92	pm2.61d 179	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
94	55	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
95	93, 94	breqtrd 5126	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
96	95	3adantl3 1170	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
97		simpl3 1195	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁)
98	96, 97	breqtrd 5126	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ 𝑁)
99	19, 22, 63, 67, 98	letrd 11302	. . . . . . . . . 10 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ 𝑁)
100		nn0uz 12801	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
101	18, 100	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ (ℤ≥‘0))
102	62	nn0zd 12525	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
103		elfz5 13444	. . . . . . . . . . 11 ⊢ (((𝑔‘𝑥) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → ((𝑔‘𝑥) ∈ (0...𝑁) ↔ (𝑔‘𝑥) ≤ 𝑁))
104	101, 102, 103	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ (0...𝑁) ↔ (𝑔‘𝑥) ≤ 𝑁))
105	99, 104	mpbird 257	. . . . . . . . 9 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ (0...𝑁))
106	105	adantr 480	. . . . . . . 8 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ (0...𝑁))
107		iftrue 4487	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁))
108	107	adantl 481	. . . . . . . 8 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁))
109	106, 108	eleqtrrd 2840	. . . . . . 7 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
110		nnge1 12185	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑥) ∈ ℕ → 1 ≤ (𝑔‘𝑥))
111		nnnn0 12420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
112	111	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
113	112	nn0ge0d 12477	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ 𝑥)
114		lemulge12 12017	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) ∧ (0 ≤ 𝑥 ∧ 1 ≤ (𝑔‘𝑥))) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥))
115	114	expr 456	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) ∧ 0 ≤ 𝑥) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥)))
116	21, 19, 113, 115	syl21anc 838	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥)))
117		letr 11239	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ ((𝑔‘𝑥) · 𝑥) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑥 ≤ ((𝑔‘𝑥) · 𝑥) ∧ ((𝑔‘𝑥) · 𝑥) ≤ 𝑁) → 𝑥 ≤ 𝑁))
118	21, 22, 63, 117	syl3anc 1374	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑥 ≤ ((𝑔‘𝑥) · 𝑥) ∧ ((𝑔‘𝑥) · 𝑥) ≤ 𝑁) → 𝑥 ≤ 𝑁))
119	98, 118	mpan2d 695	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ ((𝑔‘𝑥) · 𝑥) → 𝑥 ≤ 𝑁))
120	116, 119	syld 47	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ 𝑁))
121	110, 120	syl5 34	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ → 𝑥 ≤ 𝑁))
122		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
123	122, 52	eleqtrdi 2847	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ≥‘1))
124		elfz5 13444	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥 ≤ 𝑁))
125	123, 102, 124	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥 ≤ 𝑁))
126	121, 125	sylibrd 259	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ → 𝑥 ∈ (1...𝑁)))
127	126	con3d 152	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → ¬ (𝑔‘𝑥) ∈ ℕ))
128		elnn0 12415	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑥) ∈ ℕ0 ↔ ((𝑔‘𝑥) ∈ ℕ ∨ (𝑔‘𝑥) = 0))
129	18, 128	sylib 218	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ ∨ (𝑔‘𝑥) = 0))
130	129	ord 865	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ (𝑔‘𝑥) ∈ ℕ → (𝑔‘𝑥) = 0))
131	127, 130	syld 47	. . . . . . . . . 10 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → (𝑔‘𝑥) = 0))
132	131	imp 406	. . . . . . . . 9 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) = 0)
133		fvex 6855	. . . . . . . . . 10 ⊢ (𝑔‘𝑥) ∈ V
134	133	elsn 4597	. . . . . . . . 9 ⊢ ((𝑔‘𝑥) ∈ {0} ↔ (𝑔‘𝑥) = 0)
135	132, 134	sylibr 234	. . . . . . . 8 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ {0})
136	8	adantl 481	. . . . . . . 8 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
137	135, 136	eleqtrrd 2840	. . . . . . 7 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
138	109, 137	pm2.61dan 813	. . . . . 6 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
139	138	ralrimiva 3130	. . . . 5 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → ∀𝑥 ∈ ℕ (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
140		vex 3446	. . . . . 6 ⊢ 𝑔 ∈ V
141	140	elixp 8854	. . . . 5 ⊢ (𝑔 ∈ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ↔ (𝑔 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑔‘𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})))
142	16, 139, 141	sylanbrc 584	. . . 4 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑔 ∈ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
143	14, 142	sylbi 217	. . 3 ⊢ (𝑔 ∈ 𝑃 → 𝑔 ∈ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
144	143	ssriv 3939	. 2 ⊢ 𝑃 ⊆ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})
145		ssfi 9109	. 2 ⊢ ((X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin ∧ 𝑃 ⊆ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})) → 𝑃 ∈ Fin)
146	12, 144, 145	mp2an 693	1 ⊢ 𝑃 ∈ Fin

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  ∀wral 3052  {crab 3401  Vcvv 3442   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  ifcif 4481  𝒫 cpw 4556  {csn 4582   class class class wbr 5100  {copab 5162   ↦ cmpt 5181  ^◡ccnv 5631  dom cdm 5632   “ cima 5635   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370   supp csupp 8112   ↑_m cmap 8775  Xcixp 8847  Fincfn 8895  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   · cmul 11043   ≤ cle 11179  ℕcn 12157  2c2 12212  ℕ₀cn0 12413  ℤcz 12500  ℤ_≥cuz 12763  ...cfz 13435  ↑cexp 13996  Σcsu 15621   ∥ cdvds 16191

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-rp 12918  df-ico 13279  df-fz 13436  df-fzo 13583  df-seq 13937  df-exp 13997  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-sum 15622

This theorem is referenced by: (None)