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Theorem eulerpartlemb 31683
Description: Lemma for eulerpart 31697. The set of all partitions of 𝑁 is finite. (Contributed by Mario Carneiro, 26-Jan-2015.)
Hypotheses
Ref Expression
eulerpart.p 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
eulerpart.o 𝑂 = {𝑔𝑃 ∣ ∀𝑛 ∈ (𝑔 “ ℕ) ¬ 2 ∥ 𝑛}
eulerpart.d 𝐷 = {𝑔𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔𝑛) ≤ 1}
eulerpart.j 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
eulerpart.f 𝐹 = (𝑥𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))
eulerpart.h 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}
eulerpart.m 𝑀 = (𝑟𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐽𝑦 ∈ (𝑟𝑥))})
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
StepHypRef Expression
1 fzfid 13345 . . . 4 (⊤ → (1...𝑁) ∈ Fin)
2 fzfi 13344 . . . . . 6 (0...𝑁) ∈ Fin
3 snfi 8590 . . . . . 6 {0} ∈ Fin
42, 3ifcli 4496 . . . . 5 if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin
54a1i 11 . . . 4 ((⊤ ∧ 𝑥 ∈ ℕ) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
6 eldifn 4090 . . . . . 6 (𝑥 ∈ (ℕ ∖ (1...𝑁)) → ¬ 𝑥 ∈ (1...𝑁))
76adantl 485 . . . . 5 ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → ¬ 𝑥 ∈ (1...𝑁))
8 iffalse 4459 . . . . 5 𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
9 eqimss 4009 . . . . 5 (if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0} → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ⊆ {0})
107, 8, 93syl 18 . . . 4 ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ⊆ {0})
111, 5, 10ixpfi2 8819 . . 3 (⊤ → X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
1211mptru 1545 . 2 X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin
13 eulerpart.p . . . . 5 𝑃 = {𝑓 ∈ (ℕ0m ℕ) ∣ ((𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓𝑘) · 𝑘) = 𝑁)}
1413eulerpartleme 31678 . . . 4 (𝑔𝑃 ↔ (𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁))
15 ffn 6503 . . . . . 6 (𝑔:ℕ⟶ℕ0𝑔 Fn ℕ)
16153ad2ant1 1130 . . . . 5 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑔 Fn ℕ)
17 ffvelrn 6840 . . . . . . . . . . . . 13 ((𝑔:ℕ⟶ℕ0𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℕ0)
18173ad2antl1 1182 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℕ0)
1918nn0red 11953 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℝ)
20 nnre 11641 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
2120adantl 485 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ)
2219, 21remulcld 10669 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ∈ ℝ)
23 cnvimass 5936 . . . . . . . . . . . . . . . . . 18 (𝑔 “ ℕ) ⊆ dom 𝑔
24 fdm 6511 . . . . . . . . . . . . . . . . . . 19 (𝑔:ℕ⟶ℕ0 → dom 𝑔 = ℕ)
2524adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → dom 𝑔 = ℕ)
2623, 25sseqtrid 4005 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 “ ℕ) ⊆ ℕ)
2726sselda 3953 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
28 ffvelrn 6840 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:ℕ⟶ℕ0𝑘 ∈ ℕ) → (𝑔𝑘) ∈ ℕ0)
2928adantlr 714 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ ℕ) → (𝑔𝑘) ∈ ℕ0)
3027, 29syldan 594 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → (𝑔𝑘) ∈ ℕ0)
3127nnnn0d 11952 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ0)
3230, 31nn0mulcld 11957 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℕ0)
3332nn0cnd 11954 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℂ)
34 simpl 486 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → 𝑔:ℕ⟶ℕ0)
35 nnex 11640 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ ∈ V
36 frnnn0supp 11950 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℕ ∈ V ∧ 𝑔:ℕ⟶ℕ0) → (𝑔 supp 0) = (𝑔 “ ℕ))
3735, 36mpan 689 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔:ℕ⟶ℕ0 → (𝑔 supp 0) = (𝑔 “ ℕ))
3837adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) = (𝑔 “ ℕ))
39 eqimss 4009 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔 supp 0) = (𝑔 “ ℕ) → (𝑔 supp 0) ⊆ (𝑔 “ ℕ))
4038, 39syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) ⊆ (𝑔 “ ℕ))
4135a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → ℕ ∈ V)
42 0nn0 11909 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℕ0
4342a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → 0 ∈ ℕ0)
4434, 40, 41, 43suppssr 7857 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → (𝑔𝑘) = 0)
4544oveq1d 7164 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑘) · 𝑘) = (0 · 𝑘))
46 eldifi 4089 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
4746adantl 485 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → 𝑘 ∈ ℕ)
48 nncn 11642 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
49 mul02 10816 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℂ → (0 · 𝑘) = 0)
5047, 48, 493syl 18 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → (0 · 𝑘) = 0)
5145, 50eqtrd 2859 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑘) · 𝑘) = 0)
52 nnuz 12278 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
5352eqimssi 4011 . . . . . . . . . . . . . . . . . 18 ℕ ⊆ (ℤ‘1)
5453a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → ℕ ⊆ (ℤ‘1))
5526, 33, 51, 54sumss 15081 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
56 simpr 488 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 “ ℕ) ∈ Fin)
5756, 32fsumnn0cl 15093 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) ∈ ℕ0)
5855, 57eqeltrrd 2917 . . . . . . . . . . . . . . 15 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) ∈ ℕ0)
59 eleq1 2903 . . . . . . . . . . . . . . 15 𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁 → (Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) ∈ ℕ0𝑁 ∈ ℕ0))
6058, 59syl5ibcom 248 . . . . . . . . . . . . . 14 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁𝑁 ∈ ℕ0))
61603impia 1114 . . . . . . . . . . . . 13 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑁 ∈ ℕ0)
6261adantr 484 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℕ0)
6362nn0red 11953 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
6418nn0ge0d 11955 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝑔𝑥))
65 nnge1 11662 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 1 ≤ 𝑥)
6665adantl 485 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 1 ≤ 𝑥)
6719, 21, 64, 66lemulge11d 11575 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ≤ ((𝑔𝑥) · 𝑥))
6856adantr 484 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → (𝑔 “ ℕ) ∈ Fin)
6932nn0red 11953 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
7069adantlr 714 . . . . . . . . . . . . . . . . 17 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
7132nn0ge0d 11955 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
7271adantlr 714 . . . . . . . . . . . . . . . . 17 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
73 fveq2 6661 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝑔𝑘) = (𝑔𝑥))
74 id 22 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥𝑘 = 𝑥)
7573, 74oveq12d 7167 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → ((𝑔𝑘) · 𝑘) = ((𝑔𝑥) · 𝑥))
76 simprr 772 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → 𝑥 ∈ (𝑔 “ ℕ))
7768, 70, 72, 75, 76fsumge1 15152 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
7877expr 460 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (𝑔 “ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘)))
79 eldif 3929 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ)) ↔ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ)))
8051ralrimiva 3177 . . . . . . . . . . . . . . . . . . 19 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → ∀𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))((𝑔𝑘) · 𝑘) = 0)
8175eqeq1d 2826 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑥 → (((𝑔𝑘) · 𝑘) = 0 ↔ ((𝑔𝑥) · 𝑥) = 0))
8281rspccva 3608 . . . . . . . . . . . . . . . . . . 19 ((∀𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))((𝑔𝑘) · 𝑘) = 0 ∧ 𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8380, 82sylan 583 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8479, 83sylan2br 597 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8556adantr 484 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑔 “ ℕ) ∈ Fin)
8632adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℕ0)
8786nn0red 11953 . . . . . . . . . . . . . . . . . . 19 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
8886nn0ge0d 11955 . . . . . . . . . . . . . . . . . . 19 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
8985, 87, 88fsumge0 15150 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → 0 ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9089adantrr 716 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → 0 ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9184, 90eqbrtrd 5074 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9291expr 460 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (𝑔 “ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘)))
9378, 92pm2.61d 182 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9455adantr 484 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
9593, 94breqtrd 5078 . . . . . . . . . . . . 13 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
96953adantl3 1165 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
97 simpl3 1190 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁)
9896, 97breqtrd 5078 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ 𝑁)
9919, 22, 63, 67, 98letrd 10795 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ≤ 𝑁)
100 nn0uz 12277 . . . . . . . . . . . 12 0 = (ℤ‘0)
10118, 100eleqtrdi 2926 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ (ℤ‘0))
10262nn0zd 12082 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
103 elfz5 12903 . . . . . . . . . . 11 (((𝑔𝑥) ∈ (ℤ‘0) ∧ 𝑁 ∈ ℤ) → ((𝑔𝑥) ∈ (0...𝑁) ↔ (𝑔𝑥) ≤ 𝑁))
104101, 102, 103syl2anc 587 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ (0...𝑁) ↔ (𝑔𝑥) ≤ 𝑁))
10599, 104mpbird 260 . . . . . . . . 9 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ (0...𝑁))
106105adantr 484 . . . . . . . 8 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) ∈ (0...𝑁))
107 iftrue 4456 . . . . . . . . 9 (𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁))
108107adantl 485 . . . . . . . 8 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁))
109106, 108eleqtrrd 2919 . . . . . . 7 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
110 nnge1 11662 . . . . . . . . . . . . . 14 ((𝑔𝑥) ∈ ℕ → 1 ≤ (𝑔𝑥))
111 nnnn0 11901 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
112111adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
113112nn0ge0d 11955 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ 𝑥)
114 lemulge12 11501 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ ∧ (𝑔𝑥) ∈ ℝ) ∧ (0 ≤ 𝑥 ∧ 1 ≤ (𝑔𝑥))) → 𝑥 ≤ ((𝑔𝑥) · 𝑥))
115114expr 460 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℝ ∧ (𝑔𝑥) ∈ ℝ) ∧ 0 ≤ 𝑥) → (1 ≤ (𝑔𝑥) → 𝑥 ≤ ((𝑔𝑥) · 𝑥)))
11621, 19, 113, 115syl21anc 836 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔𝑥) → 𝑥 ≤ ((𝑔𝑥) · 𝑥)))
117 letr 10732 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ ((𝑔𝑥) · 𝑥) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑥 ≤ ((𝑔𝑥) · 𝑥) ∧ ((𝑔𝑥) · 𝑥) ≤ 𝑁) → 𝑥𝑁))
11821, 22, 63, 117syl3anc 1368 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑥 ≤ ((𝑔𝑥) · 𝑥) ∧ ((𝑔𝑥) · 𝑥) ≤ 𝑁) → 𝑥𝑁))
11998, 118mpan2d 693 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ ((𝑔𝑥) · 𝑥) → 𝑥𝑁))
120116, 119syld 47 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔𝑥) → 𝑥𝑁))
121110, 120syl5 34 . . . . . . . . . . . . 13 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ → 𝑥𝑁))
122 simpr 488 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
123122, 52eleqtrdi 2926 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ‘1))
124 elfz5 12903 . . . . . . . . . . . . . 14 ((𝑥 ∈ (ℤ‘1) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥𝑁))
125123, 102, 124syl2anc 587 . . . . . . . . . . . . 13 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥𝑁))
126121, 125sylibrd 262 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ → 𝑥 ∈ (1...𝑁)))
127126con3d 155 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → ¬ (𝑔𝑥) ∈ ℕ))
128 elnn0 11896 . . . . . . . . . . . . 13 ((𝑔𝑥) ∈ ℕ0 ↔ ((𝑔𝑥) ∈ ℕ ∨ (𝑔𝑥) = 0))
12918, 128sylib 221 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ ∨ (𝑔𝑥) = 0))
130129ord 861 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ (𝑔𝑥) ∈ ℕ → (𝑔𝑥) = 0))
131127, 130syld 47 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → (𝑔𝑥) = 0))
132131imp 410 . . . . . . . . 9 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) = 0)
133 fvex 6674 . . . . . . . . . 10 (𝑔𝑥) ∈ V
134133elsn 4565 . . . . . . . . 9 ((𝑔𝑥) ∈ {0} ↔ (𝑔𝑥) = 0)
135132, 134sylibr 237 . . . . . . . 8 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) ∈ {0})
1368adantl 485 . . . . . . . 8 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
137135, 136eleqtrrd 2919 . . . . . . 7 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
138109, 137pm2.61dan 812 . . . . . 6 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
139138ralrimiva 3177 . . . . 5 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → ∀𝑥 ∈ ℕ (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
140 vex 3483 . . . . . 6 𝑔 ∈ V
141140elixp 8464 . . . . 5 (𝑔X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ↔ (𝑔 Fn ℕ ∧ ∀𝑥 ∈ ℕ (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})))
14216, 139, 141sylanbrc 586 . . . 4 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑔X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
14314, 142sylbi 220 . . 3 (𝑔𝑃𝑔X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
144143ssriv 3957 . 2 𝑃X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})
145 ssfi 8735 . 2 ((X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin ∧ 𝑃X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})) → 𝑃 ∈ Fin)
14612, 144, 145mp2an 691 1 𝑃 ∈ Fin
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wtru 1539  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  cdif 3916  cin 3918  wss 3919  c0 4276  ifcif 4450  𝒫 cpw 4522  {csn 4550   class class class wbr 5052  {copab 5114  cmpt 5132  ccnv 5541  dom cdm 5542  cima 5545   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  cmpo 7151   supp csupp 7826  m cmap 8402  Xcixp 8457  Fincfn 8505  cc 10533  cr 10534  0cc0 10535  1c1 10536   · cmul 10540  cle 10674  cn 11634  2c2 11689  0cn0 11894  cz 11978  cuz 12240  ...cfz 12894  cexp 13434  Σcsu 15042  cdvds 15607
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-ico 12741  df-fz 12895  df-fzo 13038  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043
This theorem is referenced by: (None)
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