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Theorem eulerpartlemb 34202
Description: Lemma for eulerpart 34216. 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 13993 . . . 4 (⊤ → (1...𝑁) ∈ Fin)
2 fzfi 13992 . . . . . 6 (0...𝑁) ∈ Fin
3 snfi 9081 . . . . . 6 {0} ∈ Fin
42, 3ifcli 4580 . . . . 5 if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin
54a1i 11 . . . 4 ((⊤ ∧ 𝑥 ∈ ℕ) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
6 eldifn 4127 . . . . . 6 (𝑥 ∈ (ℕ ∖ (1...𝑁)) → ¬ 𝑥 ∈ (1...𝑁))
76adantl 480 . . . . 5 ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → ¬ 𝑥 ∈ (1...𝑁))
8 iffalse 4542 . . . . 5 𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
9 eqimss 4038 . . . . 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 9394 . . 3 (⊤ → X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
1211mptru 1541 . 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 34197 . . . 4 (𝑔𝑃 ↔ (𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁))
15 ffn 6728 . . . . . 6 (𝑔:ℕ⟶ℕ0𝑔 Fn ℕ)
16153ad2ant1 1130 . . . . 5 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑔 Fn ℕ)
17 ffvelcdm 7095 . . . . . . . . . . . . 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 12585 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℝ)
20 nnre 12271 . . . . . . . . . . . . 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 11294 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ∈ ℝ)
23 cnvimass 6091 . . . . . . . . . . . . . . . . . 18 (𝑔 “ ℕ) ⊆ dom 𝑔
24 fdm 6737 . . . . . . . . . . . . . . . . . . 19 (𝑔:ℕ⟶ℕ0 → dom 𝑔 = ℕ)
2524adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → dom 𝑔 = ℕ)
2623, 25sseqtrid 4032 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 “ ℕ) ⊆ ℕ)
2726sselda 3979 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
28 ffvelcdm 7095 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:ℕ⟶ℕ0𝑘 ∈ ℕ) → (𝑔𝑘) ∈ ℕ0)
2928adantlr 713 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ ℕ) → (𝑔𝑘) ∈ ℕ0)
3027, 29syldan 589 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → (𝑔𝑘) ∈ ℕ0)
3127nnnn0d 12584 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ0)
3230, 31nn0mulcld 12589 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℕ0)
3332nn0cnd 12586 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℂ)
34 simpl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → 𝑔:ℕ⟶ℕ0)
35 nnex 12270 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ ∈ V
36 fcdmnn0supp 12580 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℕ ∈ V ∧ 𝑔:ℕ⟶ℕ0) → (𝑔 supp 0) = (𝑔 “ ℕ))
3735, 36mpan 688 . . . . . . . . . . . . . . . . . . . . . 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 4038 . . . . . . . . . . . . . . . . . . . . 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 12539 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℕ0
4342a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → 0 ∈ ℕ0)
4434, 40, 41, 43suppssr 8210 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → (𝑔𝑘) = 0)
4544oveq1d 7439 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑘) · 𝑘) = (0 · 𝑘))
46 eldifi 4126 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
4746adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → 𝑘 ∈ ℕ)
48 nncn 12272 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
49 mul02 11442 . . . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑘) · 𝑘) = 0)
52 nnuz 12917 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
5352eqimssi 4040 . . . . . . . . . . . . . . . . . 18 ℕ ⊆ (ℤ‘1)
5453a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → ℕ ⊆ (ℤ‘1))
5526, 33, 51, 54sumss 15728 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
56 simpr 483 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 “ ℕ) ∈ Fin)
5756, 32fsumnn0cl 15740 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) ∈ ℕ0)
5855, 57eqeltrrd 2827 . . . . . . . . . . . . . . 15 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) ∈ ℕ0)
59 eleq1 2814 . . . . . . . . . . . . . . 15 𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁 → (Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) ∈ ℕ0𝑁 ∈ ℕ0))
6058, 59syl5ibcom 244 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℕ0)
6362nn0red 12585 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
6418nn0ge0d 12587 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝑔𝑥))
65 nnge1 12292 . . . . . . . . . . . . 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 12203 . . . . . . . . . . 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 12585 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
7069adantlr 713 . . . . . . . . . . . . . . . . 17 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
7132nn0ge0d 12587 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
7271adantlr 713 . . . . . . . . . . . . . . . . 17 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
73 fveq2 6901 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝑔𝑘) = (𝑔𝑥))
74 id 22 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥𝑘 = 𝑥)
7573, 74oveq12d 7442 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → ((𝑔𝑘) · 𝑘) = ((𝑔𝑥) · 𝑥))
76 simprr 771 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → 𝑥 ∈ (𝑔 “ ℕ))
7768, 70, 72, 75, 76fsumge1 15801 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
7877expr 455 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (𝑔 “ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘)))
79 eldif 3957 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ)) ↔ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ)))
8051ralrimiva 3136 . . . . . . . . . . . . . . . . . . 19 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → ∀𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))((𝑔𝑘) · 𝑘) = 0)
8175eqeq1d 2728 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑥 → (((𝑔𝑘) · 𝑘) = 0 ↔ ((𝑔𝑥) · 𝑥) = 0))
8281rspccva 3607 . . . . . . . . . . . . . . . . . . 19 ((∀𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))((𝑔𝑘) · 𝑘) = 0 ∧ 𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8380, 82sylan 578 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8479, 83sylan2br 593 . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . 20 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℕ0)
8786nn0red 12585 . . . . . . . . . . . . . . . . . . 19 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
8886nn0ge0d 12587 . . . . . . . . . . . . . . . . . . 19 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
8985, 87, 88fsumge0 15799 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → 0 ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9089adantrr 715 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → 0 ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9184, 90eqbrtrd 5175 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9291expr 455 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (𝑔 “ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘)))
9378, 92pm2.61d 179 . . . . . . . . . . . . . 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 5179 . . . . . . . . . . . . 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 5179 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ 𝑁)
9919, 22, 63, 67, 98letrd 11421 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ≤ 𝑁)
100 nn0uz 12916 . . . . . . . . . . . 12 0 = (ℤ‘0)
10118, 100eleqtrdi 2836 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ (ℤ‘0))
10262nn0zd 12636 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
103 elfz5 13547 . . . . . . . . . . 11 (((𝑔𝑥) ∈ (ℤ‘0) ∧ 𝑁 ∈ ℤ) → ((𝑔𝑥) ∈ (0...𝑁) ↔ (𝑔𝑥) ≤ 𝑁))
104101, 102, 103syl2anc 582 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ (0...𝑁) ↔ (𝑔𝑥) ≤ 𝑁))
10599, 104mpbird 256 . . . . . . . . 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 4539 . . . . . . . . 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 2829 . . . . . . 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 12292 . . . . . . . . . . . . . 14 ((𝑔𝑥) ∈ ℕ → 1 ≤ (𝑔𝑥))
111 nnnn0 12531 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
112111adantl 480 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
113112nn0ge0d 12587 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ 𝑥)
114 lemulge12 12129 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ ∧ (𝑔𝑥) ∈ ℝ) ∧ (0 ≤ 𝑥 ∧ 1 ≤ (𝑔𝑥))) → 𝑥 ≤ ((𝑔𝑥) · 𝑥))
115114expr 455 . . . . . . . . . . . . . . . 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 11358 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
123122, 52eleqtrdi 2836 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ‘1))
124 elfz5 13547 . . . . . . . . . . . . . 14 ((𝑥 ∈ (ℤ‘1) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥𝑁))
125123, 102, 124syl2anc 582 . . . . . . . . . . . . 13 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥𝑁))
126121, 125sylibrd 258 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ → 𝑥 ∈ (1...𝑁)))
127126con3d 152 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → ¬ (𝑔𝑥) ∈ ℕ))
128 elnn0 12526 . . . . . . . . . . . . 13 ((𝑔𝑥) ∈ ℕ0 ↔ ((𝑔𝑥) ∈ ℕ ∨ (𝑔𝑥) = 0))
12918, 128sylib 217 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ ∨ (𝑔𝑥) = 0))
130129ord 862 . . . . . . . . . . 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 405 . . . . . . . . 9 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) = 0)
133 fvex 6914 . . . . . . . . . 10 (𝑔𝑥) ∈ V
134133elsn 4648 . . . . . . . . 9 ((𝑔𝑥) ∈ {0} ↔ (𝑔𝑥) = 0)
135132, 134sylibr 233 . . . . . . . 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 2829 . . . . . . 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 811 . . . . . 6 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
139138ralrimiva 3136 . . . . 5 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → ∀𝑥 ∈ ℕ (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
140 vex 3466 . . . . . 6 𝑔 ∈ V
141140elixp 8933 . . . . 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 581 . . . 4 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑔X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
14314, 142sylbi 216 . . 3 (𝑔𝑃𝑔X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
144143ssriv 3983 . 2 𝑃X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})
145 ssfi 9211 . 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 690 1 𝑃 ∈ Fin
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1534  wtru 1535  wcel 2099  wral 3051  {crab 3419  Vcvv 3462  cdif 3944  cin 3946  wss 3947  c0 4325  ifcif 4533  𝒫 cpw 4607  {csn 4633   class class class wbr 5153  {copab 5215  cmpt 5236  ccnv 5681  dom cdm 5682  cima 5685   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  cmpo 7426   supp csupp 8174  m cmap 8855  Xcixp 8926  Fincfn 8974  cc 11156  cr 11157  0cc0 11158  1c1 11159   · cmul 11163  cle 11299  cn 12264  2c2 12319  0cn0 12524  cz 12610  cuz 12874  ...cfz 13538  cexp 14081  Σcsu 15690  cdvds 16256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-pm 8858  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-rp 13029  df-ico 13384  df-fz 13539  df-fzo 13682  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-sum 15691
This theorem is referenced by: (None)
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