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Theorem eulerpartlemb 29560
Description: Lemma for eulerpart 29574. 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 12586 . . . 4 (⊤ → (1...𝑁) ∈ Fin)
2 fzfi 12585 . . . . . 6 (0...𝑁) ∈ Fin
3 snfi 7897 . . . . . 6 {0} ∈ Fin
42, 3keepel 4101 . . . . 5 if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin
54a1i 11 . . . 4 ((⊤ ∧ 𝑥 ∈ ℕ) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
6 eldifn 3691 . . . . . 6 (𝑥 ∈ (ℕ ∖ (1...𝑁)) → ¬ 𝑥 ∈ (1...𝑁))
76adantl 480 . . . . 5 ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → ¬ 𝑥 ∈ (1...𝑁))
8 iffalse 4041 . . . . 5 𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
9 eqimss 3616 . . . . 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 8121 . . 3 (⊤ → X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
1211trud 1483 . 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 29555 . . . 4 (𝑔𝑃 ↔ (𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁))
15 ffn 5941 . . . . . 6 (𝑔:ℕ⟶ℕ0𝑔 Fn ℕ)
16153ad2ant1 1074 . . . . 5 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑔 Fn ℕ)
17 ffvelrn 6247 . . . . . . . . . . . . 13 ((𝑔:ℕ⟶ℕ0𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℕ0)
18173ad2antl1 1215 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℕ0)
1918nn0red 11196 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℝ)
20 nnre 10871 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
2120adantl 480 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ)
2219, 21remulcld 9923 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ∈ ℝ)
23 cnvimass 5388 . . . . . . . . . . . . . . . . . 18 (𝑔 “ ℕ) ⊆ dom 𝑔
24 fdm 5947 . . . . . . . . . . . . . . . . . . 19 (𝑔:ℕ⟶ℕ0 → dom 𝑔 = ℕ)
2524adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → dom 𝑔 = ℕ)
2623, 25syl5sseq 3612 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 “ ℕ) ⊆ ℕ)
2726sselda 3564 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
28 ffvelrn 6247 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:ℕ⟶ℕ0𝑘 ∈ ℕ) → (𝑔𝑘) ∈ ℕ0)
2928adantlr 746 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ ℕ) → (𝑔𝑘) ∈ ℕ0)
3027, 29syldan 485 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → (𝑔𝑘) ∈ ℕ0)
3127nnnn0d 11195 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ0)
3230, 31nn0mulcld 11200 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℕ0)
3332nn0cnd 11197 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℂ)
34 simpl 471 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → 𝑔:ℕ⟶ℕ0)
35 nnex 10870 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ ∈ V
36 frnnn0supp 11193 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℕ ∈ V ∧ 𝑔:ℕ⟶ℕ0) → (𝑔 supp 0) = (𝑔 “ ℕ))
3735, 36mpan 701 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔:ℕ⟶ℕ0 → (𝑔 supp 0) = (𝑔 “ ℕ))
3837adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) = (𝑔 “ ℕ))
39 eqimss 3616 . . . . . . . . . . . . . . . . . . . . 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 11151 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℕ0
4342a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → 0 ∈ ℕ0)
4434, 40, 41, 43suppssr 7187 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → (𝑔𝑘) = 0)
4544oveq1d 6539 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑘) · 𝑘) = (0 · 𝑘))
46 eldifi 3690 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
4746adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → 𝑘 ∈ ℕ)
48 nncn 10872 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
49 mul02 10062 . . . . . . . . . . . . . . . . . . 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 2640 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑘) · 𝑘) = 0)
52 nnuz 11552 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
5352eqimssi 3618 . . . . . . . . . . . . . . . . . 18 ℕ ⊆ (ℤ‘1)
5453a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → ℕ ⊆ (ℤ‘1))
5526, 33, 51, 54sumss 14245 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
56 simpr 475 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 “ ℕ) ∈ Fin)
5756, 32fsumnn0cl 14257 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) ∈ ℕ0)
5855, 57eqeltrrd 2685 . . . . . . . . . . . . . . 15 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) ∈ ℕ0)
59 eleq1 2672 . . . . . . . . . . . . . . 15 𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁 → (Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) ∈ ℕ0𝑁 ∈ ℕ0))
6058, 59syl5ibcom 233 . . . . . . . . . . . . . 14 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁𝑁 ∈ ℕ0))
61603impia 1252 . . . . . . . . . . . . 13 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑁 ∈ ℕ0)
6261adantr 479 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℕ0)
6362nn0red 11196 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
6418nn0ge0d 11198 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝑔𝑥))
65 nnge1 10890 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 1 ≤ 𝑥)
6665adantl 480 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 1 ≤ 𝑥)
6719, 21, 64, 66lemulge11d 10807 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ≤ ((𝑔𝑥) · 𝑥))
6856adantr 479 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → (𝑔 “ ℕ) ∈ Fin)
6932nn0red 11196 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
7069adantlr 746 . . . . . . . . . . . . . . . . 17 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
7132nn0ge0d 11198 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
7271adantlr 746 . . . . . . . . . . . . . . . . 17 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
73 fveq2 6085 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝑔𝑘) = (𝑔𝑥))
74 id 22 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥𝑘 = 𝑥)
7573, 74oveq12d 6542 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → ((𝑔𝑘) · 𝑘) = ((𝑔𝑥) · 𝑥))
76 simprr 791 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → 𝑥 ∈ (𝑔 “ ℕ))
7768, 70, 72, 75, 76fsumge1 14313 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
7877expr 640 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (𝑔 “ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘)))
79 eldif 3546 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ)) ↔ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ)))
8051ralrimiva 2945 . . . . . . . . . . . . . . . . . . 19 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → ∀𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))((𝑔𝑘) · 𝑘) = 0)
8175eqeq1d 2608 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑥 → (((𝑔𝑘) · 𝑘) = 0 ↔ ((𝑔𝑥) · 𝑥) = 0))
8281rspccva 3277 . . . . . . . . . . . . . . . . . . 19 ((∀𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))((𝑔𝑘) · 𝑘) = 0 ∧ 𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8380, 82sylan 486 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8479, 83sylan2br 491 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8556adantr 479 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑔 “ ℕ) ∈ Fin)
8632adantlr 746 . . . . . . . . . . . . . . . . . . . 20 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℕ0)
8786nn0red 11196 . . . . . . . . . . . . . . . . . . 19 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
8886nn0ge0d 11198 . . . . . . . . . . . . . . . . . . 19 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
8985, 87, 88fsumge0 14311 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → 0 ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9089adantrr 748 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → 0 ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9184, 90eqbrtrd 4596 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9291expr 640 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (𝑔 “ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘)))
9378, 92pm2.61d 168 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9455adantr 479 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
9593, 94breqtrd 4600 . . . . . . . . . . . . 13 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
96953adantl3 1211 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
97 simpl3 1058 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁)
9896, 97breqtrd 4600 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ 𝑁)
9919, 22, 63, 67, 98letrd 10042 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ≤ 𝑁)
100 nn0uz 11551 . . . . . . . . . . . 12 0 = (ℤ‘0)
10118, 100syl6eleq 2694 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ (ℤ‘0))
10262nn0zd 11309 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
103 elfz5 12157 . . . . . . . . . . 11 (((𝑔𝑥) ∈ (ℤ‘0) ∧ 𝑁 ∈ ℤ) → ((𝑔𝑥) ∈ (0...𝑁) ↔ (𝑔𝑥) ≤ 𝑁))
104101, 102, 103syl2anc 690 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ (0...𝑁) ↔ (𝑔𝑥) ≤ 𝑁))
10599, 104mpbird 245 . . . . . . . . 9 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ (0...𝑁))
106105adantr 479 . . . . . . . 8 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) ∈ (0...𝑁))
107 iftrue 4038 . . . . . . . . 9 (𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁))
108107adantl 480 . . . . . . . 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 2687 . . . . . . 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 10890 . . . . . . . . . . . . . 14 ((𝑔𝑥) ∈ ℕ → 1 ≤ (𝑔𝑥))
111 nnnn0 11143 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
112111adantl 480 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
113112nn0ge0d 11198 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ 𝑥)
114 lemulge12 10732 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ ∧ (𝑔𝑥) ∈ ℝ) ∧ (0 ≤ 𝑥 ∧ 1 ≤ (𝑔𝑥))) → 𝑥 ≤ ((𝑔𝑥) · 𝑥))
115114expr 640 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℝ ∧ (𝑔𝑥) ∈ ℝ) ∧ 0 ≤ 𝑥) → (1 ≤ (𝑔𝑥) → 𝑥 ≤ ((𝑔𝑥) · 𝑥)))
11621, 19, 113, 115syl21anc 1316 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔𝑥) → 𝑥 ≤ ((𝑔𝑥) · 𝑥)))
117 letr 9979 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ ((𝑔𝑥) · 𝑥) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑥 ≤ ((𝑔𝑥) · 𝑥) ∧ ((𝑔𝑥) · 𝑥) ≤ 𝑁) → 𝑥𝑁))
11821, 22, 63, 117syl3anc 1317 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑥 ≤ ((𝑔𝑥) · 𝑥) ∧ ((𝑔𝑥) · 𝑥) ≤ 𝑁) → 𝑥𝑁))
11998, 118mpan2d 705 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ ((𝑔𝑥) · 𝑥) → 𝑥𝑁))
120116, 119syld 45 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔𝑥) → 𝑥𝑁))
121110, 120syl5 33 . . . . . . . . . . . . 13 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ → 𝑥𝑁))
122 simpr 475 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
123122, 52syl6eleq 2694 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ‘1))
124 elfz5 12157 . . . . . . . . . . . . . 14 ((𝑥 ∈ (ℤ‘1) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥𝑁))
125123, 102, 124syl2anc 690 . . . . . . . . . . . . 13 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥𝑁))
126121, 125sylibrd 247 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ → 𝑥 ∈ (1...𝑁)))
127126con3d 146 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → ¬ (𝑔𝑥) ∈ ℕ))
128 elnn0 11138 . . . . . . . . . . . . 13 ((𝑔𝑥) ∈ ℕ0 ↔ ((𝑔𝑥) ∈ ℕ ∨ (𝑔𝑥) = 0))
12918, 128sylib 206 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ ∨ (𝑔𝑥) = 0))
130129ord 390 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ (𝑔𝑥) ∈ ℕ → (𝑔𝑥) = 0))
131127, 130syld 45 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → (𝑔𝑥) = 0))
132131imp 443 . . . . . . . . 9 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) = 0)
133 fvex 6095 . . . . . . . . . 10 (𝑔𝑥) ∈ V
134133elsn 4136 . . . . . . . . 9 ((𝑔𝑥) ∈ {0} ↔ (𝑔𝑥) = 0)
135132, 134sylibr 222 . . . . . . . 8 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) ∈ {0})
1368adantl 480 . . . . . . . 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 2687 . . . . . . 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 827 . . . . . 6 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
139138ralrimiva 2945 . . . . 5 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → ∀𝑥 ∈ ℕ (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
140 vex 3172 . . . . . 6 𝑔 ∈ V
141140elixp 7775 . . . . 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 694 . . . 4 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑔X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
14314, 142sylbi 205 . . 3 (𝑔𝑃𝑔X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
144143ssriv 3568 . 2 𝑃X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})
145 ssfi 8039 . 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 703 1 𝑃 ∈ Fin
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wtru 1475  wcel 1976  wral 2892  {crab 2896  Vcvv 3169  cdif 3533  cin 3535  wss 3536  c0 3870  ifcif 4032  𝒫 cpw 4104  {csn 4121   class class class wbr 4574  {copab 4633  cmpt 4634  ccnv 5024  dom cdm 5025  cima 5028   Fn wfn 5782  wf 5783  cfv 5787  (class class class)co 6524  cmpt2 6526   supp csupp 7156  𝑚 cmap 7718  Xcixp 7768  Fincfn 7815  cc 9787  cr 9788  0cc0 9789  1c1 9790   · cmul 9794  cle 9928  cn 10864  2c2 10914  0cn0 11136  cz 11207  cuz 11516  ...cfz 12149  cexp 12674  Σcsu 14207  cdvds 14764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-supp 7157  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-2o 7422  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-ixp 7769  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-sup 8205  df-oi 8272  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-rp 11662  df-ico 12005  df-fz 12150  df-fzo 12287  df-seq 12616  df-exp 12675  df-hash 12932  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-clim 14010  df-sum 14208
This theorem is referenced by: (None)
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