Theorem eulerpartlemb 34887

Description: Lemma for eulerpart 34901. 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 14041	. . . 4 ⊢ (⊤ → (1...𝑁) ∈ Fin)
2		fzfi 14040	. . . . . 6 ⊢ (0...𝑁) ∈ Fin
3		snfi 9054	. . . . . 6 ⊢ {0} ∈ Fin
4	2, 3	ifcli 4533	. . . . 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 4082	. . . . . 6 ⊢ (𝑥 ∈ (ℕ ∖ (1...𝑁)) → ¬ 𝑥 ∈ (1...𝑁))
7	6	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → ¬ 𝑥 ∈ (1...𝑁))
8		iffalse 4494	. . . . 5 ⊢ (¬ 𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
9		eqimss 3992	. . . . 5 ⊢ (if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0} → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ⊆ {0})
10	7, 8, 9	3syl 19	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ⊆ {0})
11	1, 5, 10	ixpfi2 9321	. . 3 ⊢ (⊤ → X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
12	11	mptru 1577	. 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 34882	. . . 4 ⊢ (𝑔 ∈ 𝑃 ↔ (𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁))
15		ffn 6706	. . . . . 6 ⊢ (𝑔:ℕ⟶ℕ0 → 𝑔 Fn ℕ)
16	15	3ad2ant1 1151	. . . . 5 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑔 Fn ℕ)
17		ffvelcdm 7078	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℕ0)
18	17	3ad2antl1 1204	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℕ0)
19	18	nn0red 12594	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℝ)
20		nnre 12268	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
21	20	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ)
22	19, 21	remulcld 11267	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ∈ ℝ)
23		cnvimass 6082	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑔 “ ℕ) ⊆ dom 𝑔
24		fdm 6716	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:ℕ⟶ℕ0 → dom 𝑔 = ℕ)
25	24	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → dom 𝑔 = ℕ)
26	23, 25	sseqtrid 3976	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ⊆ ℕ)
27	26	sselda 3934	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
28		ffvelcdm 7078	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ ℕ0)
29	28	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ ℕ0)
30	27, 29	syldan 603	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → (𝑔‘𝑘) ∈ ℕ0)
31	27	nnnn0d 12593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ0)
32	30, 31	nn0mulcld 12598	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
33	32	nn0cnd 12595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℂ)
34		simpl 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → 𝑔:ℕ⟶ℕ0)
35		nnex 12267	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ ∈ V
36		fcdmnn0supp 12589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℕ ∈ V ∧ 𝑔:ℕ⟶ℕ0) → (𝑔 supp 0) = (◡𝑔 “ ℕ))
37	35, 36	mpan 703	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔:ℕ⟶ℕ0 → (𝑔 supp 0) = (◡𝑔 “ ℕ))
38	37	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) = (◡𝑔 “ ℕ))
39		eqimss 3992	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 supp 0) = (◡𝑔 “ ℕ) → (𝑔 supp 0) ⊆ (◡𝑔 “ ℕ))
40	38, 39	syl 18	. . . . . . . . . . . . . . . . . . . 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 12547	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℕ0
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → 0 ∈ ℕ0)
44	34, 40, 41, 43	suppssr 8197	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → (𝑔‘𝑘) = 0)
45	44	oveq1d 7432	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = (0 · 𝑘))
46		eldifi 4081	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
47	46	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → 𝑘 ∈ ℕ)
48		nncn 12269	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
49		mul02 11416	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℂ → (0 · 𝑘) = 0)
50	47, 48, 49	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → (0 · 𝑘) = 0)
51	45, 50	eqtrd 2797	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = 0)
52		nnuz 12930	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ≥‘1)
53	52	eqimssi 3994	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ ⊆ (ℤ≥‘1)
54	53	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → ℕ ⊆ (ℤ≥‘1))
55	26, 33, 51, 54	sumss 15814	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
56		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ∈ Fin)
57	56, 32	fsumnn0cl 15826	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
58	55, 57	eqeltrrd 2863	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
59		eleq1 2850	. . . . . . . . . . . . . . 15 ⊢ (Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁 → (Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
60	58, 59	syl5ibcom 248	. . . . . . . . . . . . . 14 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁 → 𝑁 ∈ ℕ0))
61	60	3impia 1135	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑁 ∈ ℕ0)
62	61	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℕ0)
63	62	nn0red 12594	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
64	18	nn0ge0d 12596	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝑔‘𝑥))
65		nnge1 12292	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 1 ≤ 𝑥)
66	65	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 1 ≤ 𝑥)
67	19, 21, 64, 66	lemulge11d 12180	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ ((𝑔‘𝑥) · 𝑥))
68	56	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → (◡𝑔 “ ℕ) ∈ Fin)
69	32	nn0red 12594	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
70	69	adantlr 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
71	32	nn0ge0d 12596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
72	71	adantlr 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
73		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑔‘𝑘) = (𝑔‘𝑥))
74		id 23	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → 𝑘 = 𝑥)
75	73, 74	oveq12d 7435	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → ((𝑔‘𝑘) · 𝑘) = ((𝑔‘𝑥) · 𝑥))
76		simprr 785	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → 𝑥 ∈ (◡𝑔 “ ℕ))
77	68, 70, 72, 75, 76	fsumge1 15888	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
78	77	expr 462	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (◡𝑔 “ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)))
79		eldif 3912	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) ↔ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ)))
80	51	ralrimiva 3156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → ∀𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0)
81	75	eqeq1d 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (((𝑔‘𝑘) · 𝑘) = 0 ↔ ((𝑔‘𝑥) · 𝑥) = 0))
82	81	rspccva 3578	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0 ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
83	80, 82	sylan 592	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
84	79, 83	sylan2br 607	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
85	56	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (◡𝑔 “ ℕ) ∈ Fin)
86	32	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
87	86	nn0red 12594	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
88	86	nn0ge0d 12596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
89	85, 87, 88	fsumge0 15886	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → 0 ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
90	89	adantrr 730	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → 0 ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
91	84, 90	eqbrtrd 5131	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
92	91	expr 462	. . . . . . . . . . . . . . 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 181	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
94	55	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
95	93, 94	breqtrd 5135	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
96	95	3adantl3 1187	. . . . . . . . . . . 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 1212	. . . . . . . . . . . 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 5135	. . . . . . . . . . 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 11395	. . . . . . . . . 10 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ 𝑁)
100		nn0uz 12929	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
101	18, 100	eleqtrdi 2872	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ (ℤ≥‘0))
102	62	nn0zd 12644	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
103		elfz5 13574	. . . . . . . . . . 11 ⊢ (((𝑔‘𝑥) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → ((𝑔‘𝑥) ∈ (0...𝑁) ↔ (𝑔‘𝑥) ≤ 𝑁))
104	101, 102, 103	syl2anc 596	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ (0...𝑁))
106	105	adantr 486	. . . . . . . 8 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ (0...𝑁))
107		iftrue 4491	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁))
108	107	adantl 487	. . . . . . . 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 2865	. . . . . . 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 12292	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑥) ∈ ℕ → 1 ≤ (𝑔‘𝑥))
111		nnnn0 12539	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
112	111	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
113	112	nn0ge0d 12596	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ 𝑥)
114		lemulge12 12106	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) ∧ (0 ≤ 𝑥 ∧ 1 ≤ (𝑔‘𝑥))) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥))
115	114	expr 462	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) ∧ 0 ≤ 𝑥) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥)))
116	21, 19, 113, 115	syl21anc 851	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥)))
117		letr 11332	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 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 707	. . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ 𝑁))
121	110, 120	syl5 35	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ → 𝑥 ≤ 𝑁))
122		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
123	122, 52	eleqtrdi 2872	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ≥‘1))
124		elfz5 13574	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥 ≤ 𝑁))
125	123, 102, 124	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥 ≤ 𝑁))
126	121, 125	sylibrd 262	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ → 𝑥 ∈ (1...𝑁)))
127	126	con3d 153	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → ¬ (𝑔‘𝑥) ∈ ℕ))
128		elnn0 12534	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑥) ∈ ℕ0 ↔ ((𝑔‘𝑥) ∈ ℕ ∨ (𝑔‘𝑥) = 0))
129	18, 128	sylib 221	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ ∨ (𝑔‘𝑥) = 0))
130	129	ord 878	. . . . . . . . . . 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 48	. . . . . . . . . 10 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → (𝑔‘𝑥) = 0))
132	131	imp 412	. . . . . . . . 9 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) = 0)
133		fvex 6895	. . . . . . . . . 10 ⊢ (𝑔‘𝑥) ∈ V
134	133	elsn 4602	. . . . . . . . 9 ⊢ ((𝑔‘𝑥) ∈ {0} ↔ (𝑔‘𝑥) = 0)
135	132, 134	sylibr 237	. . . . . . . 8 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ {0})
136	8	adantl 487	. . . . . . . 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 2865	. . . . . . 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 825	. . . . . 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 3156	. . . . 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 3457	. . . . . 6 ⊢ 𝑔 ∈ V
141	140	elixp 8915	. . . . 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 595	. . . 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 220	. . 3 ⊢ (𝑔 ∈ 𝑃 → 𝑔 ∈ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
144	143	ssriv 3938	. 2 ⊢ 𝑃 ⊆ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})
145		ssfi 9171	. 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 705	1 ⊢ 𝑃 ∈ Fin

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ifcif 4485  𝒫 cpw 4560  {csn 4587   class class class wbr 5107  {copab 5171   ↦ cmpt 5190  ^◡ccnv 5658  dom cdm 5659   “ cima 5662   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419   supp csupp 8162   ↑_m cmap 8830  Xcixp 8908  Fincfn 8956  ℂcc 11126  ℝcr 11127  0cc0 11128  1c1 11129   · cmul 11133   ≤ cle 11272  ℕcn 12261  2c2 12323  ℕ₀cn0 12532  ℤcz 12619  ℤ_≥cuz 12891  ...cfz 13565  ↑cexp 14129  Σcsu 15777   ∥ cdvds 16348

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  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-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-oi 9486  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-n0 12533  df-z 12620  df-uz 12892  df-rp 13047  df-ico 13408  df-fz 13566  df-fzo 13714  df-seq 14070  df-exp 14130  df-hash 14399  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-clim 15579  df-sum 15778

This theorem is used by: (None)