Theorem eulerpartlemb 32968

Description: Lemma for eulerpart 32982. 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 13878	. . . 4 ⊢ (⊤ → (1...𝑁) ∈ Fin)
2		fzfi 13877	. . . . . 6 ⊢ (0...𝑁) ∈ Fin
3		snfi 8988	. . . . . 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 4087	. . . . . 6 ⊢ (𝑥 ∈ (ℕ ∖ (1...𝑁)) → ¬ 𝑥 ∈ (1...𝑁))
7	6	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → ¬ 𝑥 ∈ (1...𝑁))
8		iffalse 4495	. . . . 5 ⊢ (¬ 𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
9		eqimss 4000	. . . . 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 9294	. . 3 ⊢ (⊤ → X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
12	11	mptru 1548	. 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 32963	. . . 4 ⊢ (𝑔 ∈ 𝑃 ↔ (𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁))
15		ffn 6668	. . . . . 6 ⊢ (𝑔:ℕ⟶ℕ0 → 𝑔 Fn ℕ)
16	15	3ad2ant1 1133	. . . . 5 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑔 Fn ℕ)
17		ffvelcdm 7032	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℕ0)
18	17	3ad2antl1 1185	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℕ0)
19	18	nn0red 12474	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℝ)
20		nnre 12160	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
21	20	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ)
22	19, 21	remulcld 11185	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ∈ ℝ)
23		cnvimass 6033	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑔 “ ℕ) ⊆ dom 𝑔
24		fdm 6677	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:ℕ⟶ℕ0 → dom 𝑔 = ℕ)
25	24	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → dom 𝑔 = ℕ)
26	23, 25	sseqtrid 3996	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ⊆ ℕ)
27	26	sselda 3944	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
28		ffvelcdm 7032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ ℕ0)
29	28	adantlr 713	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ ℕ0)
30	27, 29	syldan 591	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → (𝑔‘𝑘) ∈ ℕ0)
31	27	nnnn0d 12473	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ0)
32	30, 31	nn0mulcld 12478	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
33	32	nn0cnd 12475	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℂ)
34		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → 𝑔:ℕ⟶ℕ0)
35		nnex 12159	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ ∈ V
36		fcdmnn0supp 12469	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℕ ∈ V ∧ 𝑔:ℕ⟶ℕ0) → (𝑔 supp 0) = (◡𝑔 “ ℕ))
37	35, 36	mpan 688	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔:ℕ⟶ℕ0 → (𝑔 supp 0) = (◡𝑔 “ ℕ))
38	37	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) = (◡𝑔 “ ℕ))
39		eqimss 4000	. . . . . . . . . . . . . . . . . . . . 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 8127	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → (𝑔‘𝑘) = 0)
45	44	oveq1d 7372	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = (0 · 𝑘))
46		eldifi 4086	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
47	46	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → 𝑘 ∈ ℕ)
48		nncn 12161	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
49		mul02 11333	. . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = 0)
52		nnuz 12806	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ≥‘1)
53	52	eqimssi 4002	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ ⊆ (ℤ≥‘1)
54	53	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → ℕ ⊆ (ℤ≥‘1))
55	26, 33, 51, 54	sumss 15609	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
56		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ∈ Fin)
57	56, 32	fsumnn0cl 15621	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
58	55, 57	eqeltrrd 2839	. . . . . . . . . . . . . . 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 244	. . . . . . . . . . . . . 14 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁 → 𝑁 ∈ ℕ0))
61	60	3impia 1117	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑁 ∈ ℕ0)
62	61	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℕ0)
63	62	nn0red 12474	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
64	18	nn0ge0d 12476	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝑔‘𝑥))
65		nnge1 12181	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 1 ≤ 𝑥)
66	65	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 1 ≤ 𝑥)
67	19, 21, 64, 66	lemulge11d 12092	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ ((𝑔‘𝑥) · 𝑥))
68	56	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → (◡𝑔 “ ℕ) ∈ Fin)
69	32	nn0red 12474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
70	69	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
71	32	nn0ge0d 12476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
72	71	adantlr 713	. . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → ((𝑔‘𝑘) · 𝑘) = ((𝑔‘𝑥) · 𝑥))
76		simprr 771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → 𝑥 ∈ (◡𝑔 “ ℕ))
77	68, 70, 72, 75, 76	fsumge1 15682	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
78	77	expr 457	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (◡𝑔 “ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)))
79		eldif 3920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) ↔ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ)))
80	51	ralrimiva 3143	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → ∀𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0)
81	75	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (((𝑔‘𝑘) · 𝑘) = 0 ↔ ((𝑔‘𝑥) · 𝑥) = 0))
82	81	rspccva 3580	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0 ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
83	80, 82	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
84	79, 83	sylan2br 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
85	56	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (◡𝑔 “ ℕ) ∈ Fin)
86	32	adantlr 713	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
87	86	nn0red 12474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
88	86	nn0ge0d 12476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
89	85, 87, 88	fsumge0 15680	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → 0 ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
90	89	adantrr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → 0 ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
91	84, 90	eqbrtrd 5127	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
92	91	expr 457	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
95	93, 94	breqtrd 5131	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
96	95	3adantl3 1168	. . . . . . . . . . . 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 1193	. . . . . . . . . . . 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 5131	. . . . . . . . . . 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 11312	. . . . . . . . . 10 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ 𝑁)
100		nn0uz 12805	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
101	18, 100	eleqtrdi 2848	. . . . . . . . . . 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 13433	. . . . . . . . . . 11 ⊢ (((𝑔‘𝑥) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → ((𝑔‘𝑥) ∈ (0...𝑁) ↔ (𝑔‘𝑥) ≤ 𝑁))
104	101, 102, 103	syl2anc 584	. . . . . . . . . 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 256	. . . . . . . . 9 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ (0...𝑁))
106	105	adantr 481	. . . . . . . 8 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ (0...𝑁))
107		iftrue 4492	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁))
108	107	adantl 482	. . . . . . . 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 2841	. . . . . . 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 12181	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑥) ∈ ℕ → 1 ≤ (𝑔‘𝑥))
111		nnnn0 12420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
112	111	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
113	112	nn0ge0d 12476	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ 𝑥)
114		lemulge12 12018	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) ∧ (0 ≤ 𝑥 ∧ 1 ≤ (𝑔‘𝑥))) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥))
115	114	expr 457	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) ∧ 0 ≤ 𝑥) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥)))
116	21, 19, 113, 115	syl21anc 836	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔‘𝑥) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥)))
117		letr 11249	. . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
123	122, 52	eleqtrdi 2848	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ≥‘1))
124		elfz5 13433	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥 ≤ 𝑁))
125	123, 102, 124	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥 ≤ 𝑁))
126	121, 125	sylibrd 258	. . . . . . . . . . . 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 217	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) ∈ ℕ ∨ (𝑔‘𝑥) = 0))
130	129	ord 862	. . . . . . . . . . 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 407	. . . . . . . . 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 4601	. . . . . . . . 9 ⊢ ((𝑔‘𝑥) ∈ {0} ↔ (𝑔‘𝑥) = 0)
135	132, 134	sylibr 233	. . . . . . . 8 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ {0})
136	8	adantl 482	. . . . . . . 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 2841	. . . . . . 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 811	. . . . . 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 3143	. . . . 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 3449	. . . . . 6 ⊢ 𝑔 ∈ V
141	140	elixp 8842	. . . . 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 583	. . . 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 216	. . 3 ⊢ (𝑔 ∈ 𝑃 → 𝑔 ∈ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
144	143	ssriv 3948	. 2 ⊢ 𝑃 ⊆ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})
145		ssfi 9117	. 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 690	1 ⊢ 𝑃 ∈ Fin

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541  ⊤wtru 1542   ∈ wcel 2106  ∀wral 3064  {crab 3407  Vcvv 3445   ∖ cdif 3907   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  ifcif 4486  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  {copab 5167   ↦ cmpt 5188  ^◡ccnv 5632  dom cdm 5633   “ cima 5636   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359   supp csupp 8092   ↑_m cmap 8765  Xcixp 8835  Fincfn 8883  ℂcc 11049  ℝcr 11050  0cc0 11051  1c1 11052   · cmul 11056   ≤ cle 11190  ℕcn 12153  2c2 12208  ℕ₀cn0 12413  ℤcz 12499  ℤ_≥cuz 12763  ...cfz 13424  ↑cexp 13967  Σcsu 15570   ∥ cdvds 16136

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ico 13270  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571

This theorem is referenced by: (None)