Theorem eulerpartlemb 31736

Description: Lemma for eulerpart 31750. 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 13336	. . . 4 ⊢ (⊤ → (1...𝑁) ∈ Fin)
2		fzfi 13335	. . . . . 6 ⊢ (0...𝑁) ∈ Fin
3		snfi 8577	. . . . . 6 ⊢ {0} ∈ Fin
4	2, 3	ifcli 4471	. . . . 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 4055	. . . . . 6 ⊢ (𝑥 ∈ (ℕ ∖ (1...𝑁)) → ¬ 𝑥 ∈ (1...𝑁))
7	6	adantl 485	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → ¬ 𝑥 ∈ (1...𝑁))
8		iffalse 4434	. . . . 5 ⊢ (¬ 𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
9		eqimss 3971	. . . . 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 8806	. . 3 ⊢ (⊤ → X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
12	11	mptru 1545	. 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 31731	. . . 4 ⊢ (𝑔 ∈ 𝑃 ↔ (𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁))
15		ffn 6487	. . . . . 6 ⊢ (𝑔:ℕ⟶ℕ0 → 𝑔 Fn ℕ)
16	15	3ad2ant1 1130	. . . . 5 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑔 Fn ℕ)
17		ffvelrn 6826	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℕ0)
18	17	3ad2antl1 1182	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℕ0)
19	18	nn0red 11944	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ ℝ)
20		nnre 11632	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
21	20	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ)
22	19, 21	remulcld 10660	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ∈ ℝ)
23		cnvimass 5916	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑔 “ ℕ) ⊆ dom 𝑔
24		fdm 6495	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔:ℕ⟶ℕ0 → dom 𝑔 = ℕ)
25	24	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → dom 𝑔 = ℕ)
26	23, 25	sseqtrid 3967	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ⊆ ℕ)
27	26	sselda 3915	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
28		ffvelrn 6826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ ℕ0)
29	28	adantlr 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ ℕ0)
30	27, 29	syldan 594	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → (𝑔‘𝑘) ∈ ℕ0)
31	27	nnnn0d 11943	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ0)
32	30, 31	nn0mulcld 11948	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
33	32	nn0cnd 11945	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℂ)
34		simpl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → 𝑔:ℕ⟶ℕ0)
35		nnex 11631	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ ∈ V
36		frnnn0supp 11941	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℕ ∈ V ∧ 𝑔:ℕ⟶ℕ0) → (𝑔 supp 0) = (◡𝑔 “ ℕ))
37	35, 36	mpan 689	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔:ℕ⟶ℕ0 → (𝑔 supp 0) = (◡𝑔 “ ℕ))
38	37	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) = (◡𝑔 “ ℕ))
39		eqimss 3971	. . . . . . . . . . . . . . . . . . . . 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 11900	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℕ0
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → 0 ∈ ℕ0)
44	34, 40, 41, 43	suppssr 7844	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → (𝑔‘𝑘) = 0)
45	44	oveq1d 7150	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = (0 · 𝑘))
46		eldifi 4054	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
47	46	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → 𝑘 ∈ ℕ)
48		nncn 11633	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
49		mul02 10807	. . . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑘) · 𝑘) = 0)
52		nnuz 12269	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ≥‘1)
53	52	eqimssi 3973	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ ⊆ (ℤ≥‘1)
54	53	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → ℕ ⊆ (ℤ≥‘1))
55	26, 33, 51, 54	sumss 15073	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
56		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → (◡𝑔 “ ℕ) ∈ Fin)
57	56, 32	fsumnn0cl 15085	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
58	55, 57	eqeltrrd 2891	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
59		eleq1 2877	. . . . . . . . . . . . . . 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 1114	. . . . . . . . . . . . 13 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) → 𝑁 ∈ ℕ0)
62	61	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℕ0)
63	62	nn0red 11944	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
64	18	nn0ge0d 11946	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝑔‘𝑥))
65		nnge1 11653	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → 1 ≤ 𝑥)
66	65	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 1 ≤ 𝑥)
67	19, 21, 64, 66	lemulge11d 11566	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ ((𝑔‘𝑥) · 𝑥))
68	56	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → (◡𝑔 “ ℕ) ∈ Fin)
69	32	nn0red 11944	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
70	69	adantlr 714	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
71	32	nn0ge0d 11946	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
72	71	adantlr 714	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
73		fveq2 6645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → (𝑔‘𝑘) = (𝑔‘𝑥))
74		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑥 → 𝑘 = 𝑥)
75	73, 74	oveq12d 7153	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑥 → ((𝑔‘𝑘) · 𝑘) = ((𝑔‘𝑥) · 𝑥))
76		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → 𝑥 ∈ (◡𝑔 “ ℕ))
77	68, 70, 72, 75, 76	fsumge1 15144	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
78	77	expr 460	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (◡𝑔 “ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘)))
79		eldif 3891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ)) ↔ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ)))
80	51	ralrimiva 3149	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) → ∀𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0)
81	75	eqeq1d 2800	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (((𝑔‘𝑘) · 𝑘) = 0 ↔ ((𝑔‘𝑥) · 𝑥) = 0))
82	81	rspccva 3570	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑘 ∈ (ℕ ∖ (◡𝑔 “ ℕ))((𝑔‘𝑘) · 𝑘) = 0 ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
83	80, 82	sylan 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ (ℕ ∖ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
84	79, 83	sylan2br 597	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) = 0)
85	56	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (◡𝑔 “ ℕ) ∈ Fin)
86	32	adantlr 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℕ0)
87	86	nn0red 11944	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → ((𝑔‘𝑘) · 𝑘) ∈ ℝ)
88	86	nn0ge0d 11946	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (◡𝑔 “ ℕ)) → 0 ≤ ((𝑔‘𝑘) · 𝑘))
89	85, 87, 88	fsumge0 15142	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → 0 ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
90	89	adantrr 716	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → 0 ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
91	84, 90	eqbrtrd 5052	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (◡𝑔 “ ℕ))) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
92	91	expr 460	. . . . . . . . . . . . . . 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 182	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘))
94	55	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ (◡𝑔 “ ℕ)((𝑔‘𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
95	93, 94	breqtrd 5056	. . . . . . . . . . . . 13 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔‘𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘))
96	95	3adantl3 1165	. . . . . . . . . . . 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 1190	. . . . . . . . . . . 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 5056	. . . . . . . . . . 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 10786	. . . . . . . . . 10 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ≤ 𝑁)
100		nn0uz 12268	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
101	18, 100	eleqtrdi 2900	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔‘𝑥) ∈ (ℤ≥‘0))
102	62	nn0zd 12073	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
103		elfz5 12894	. . . . . . . . . . 11 ⊢ (((𝑔‘𝑥) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → ((𝑔‘𝑥) ∈ (0...𝑁) ↔ (𝑔‘𝑥) ≤ 𝑁))
104	101, 102, 103	syl2anc 587	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) ∈ (0...𝑁))
107		iftrue 4431	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁))
108	107	adantl 485	. . . . . . . 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 2893	. . . . . . 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 11653	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑥) ∈ ℕ → 1 ≤ (𝑔‘𝑥))
111		nnnn0 11892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
112	111	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
113	112	nn0ge0d 11946	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ 𝑥)
114		lemulge12 11492	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ (𝑔‘𝑥) ∈ ℝ) ∧ (0 ≤ 𝑥 ∧ 1 ≤ (𝑔‘𝑥))) → 𝑥 ≤ ((𝑔‘𝑥) · 𝑥))
115	114	expr 460	. . . . . . . . . . . . . . . 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 10723	. . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
123	122, 52	eleqtrdi 2900	. . . . . . . . . . . . . 14 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ≥‘1))
124		elfz5 12894	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥 ≤ 𝑁))
125	123, 102, 124	syl2anc 587	. . . . . . . . . . . . 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 155	. . . . . . . . . . 11 ⊢ (((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → ¬ (𝑔‘𝑥) ∈ ℕ))
128		elnn0 11887	. . . . . . . . . . . . 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 861	. . . . . . . . . . 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 410	. . . . . . . . 9 ⊢ ((((𝑔:ℕ⟶ℕ0 ∧ (◡𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔‘𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔‘𝑥) = 0)
133		fvex 6658	. . . . . . . . . 10 ⊢ (𝑔‘𝑥) ∈ V
134	133	elsn 4540	. . . . . . . . 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 485	. . . . . . . 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 2893	. . . . . . 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 812	. . . . . 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 3149	. . . . 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 3444	. . . . . 6 ⊢ 𝑔 ∈ V
141	140	elixp 8451	. . . . 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 586	. . . 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 3919	. 2 ⊢ 𝑃 ⊆ X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})
145		ssfi 8722	. 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 691	1 ⊢ 𝑃 ∈ Fin

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ifcif 4425  𝒫 cpw 4497  {csn 4525   class class class wbr 5030  {copab 5092   ↦ cmpt 5110  ^◡ccnv 5518  dom cdm 5519   “ cima 5522   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   supp csupp 7813   ↑_m cmap 8389  Xcixp 8444  Fincfn 8492  ℂcc 10524  ℝcr 10525  0cc0 10526  1c1 10527   · cmul 10531   ≤ cle 10665  ℕcn 11625  2c2 11680  ℕ₀cn0 11885  ℤcz 11969  ℤ_≥cuz 12231  ...cfz 12885  ↑cexp 13425  Σcsu 15034   ∥ cdvds 15599

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ico 12732  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035

This theorem is referenced by: (None)