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Theorem eulerpartlemb 30558
Description: Lemma for eulerpart 30572. 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 12812 . . . 4 (⊤ → (1...𝑁) ∈ Fin)
2 fzfi 12811 . . . . . 6 (0...𝑁) ∈ Fin
3 snfi 8079 . . . . . 6 {0} ∈ Fin
42, 3keepel 4188 . . . . 5 if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin
54a1i 11 . . . 4 ((⊤ ∧ 𝑥 ∈ ℕ) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
6 eldifn 3766 . . . . . 6 (𝑥 ∈ (ℕ ∖ (1...𝑁)) → ¬ 𝑥 ∈ (1...𝑁))
76adantl 481 . . . . 5 ((⊤ ∧ 𝑥 ∈ (ℕ ∖ (1...𝑁))) → ¬ 𝑥 ∈ (1...𝑁))
8 iffalse 4128 . . . . 5 𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = {0})
9 eqimss 3690 . . . . 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 8305 . . 3 (⊤ → X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) ∈ Fin)
1211trud 1533 . 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 30553 . . . 4 (𝑔𝑃 ↔ (𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁))
15 ffn 6083 . . . . . 6 (𝑔:ℕ⟶ℕ0𝑔 Fn ℕ)
16153ad2ant1 1102 . . . . 5 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑔 Fn ℕ)
17 ffvelrn 6397 . . . . . . . . . . . . 13 ((𝑔:ℕ⟶ℕ0𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℕ0)
18173ad2antl1 1243 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℕ0)
1918nn0red 11390 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ ℝ)
20 nnre 11065 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
2120adantl 481 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℝ)
2219, 21remulcld 10108 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ∈ ℝ)
23 cnvimass 5520 . . . . . . . . . . . . . . . . . 18 (𝑔 “ ℕ) ⊆ dom 𝑔
24 fdm 6089 . . . . . . . . . . . . . . . . . . 19 (𝑔:ℕ⟶ℕ0 → dom 𝑔 = ℕ)
2524adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → dom 𝑔 = ℕ)
2623, 25syl5sseq 3686 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 “ ℕ) ⊆ ℕ)
2726sselda 3636 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
28 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:ℕ⟶ℕ0𝑘 ∈ ℕ) → (𝑔𝑘) ∈ ℕ0)
2928adantlr 751 . . . . . . . . . . . . . . . . . . . 20 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ ℕ) → (𝑔𝑘) ∈ ℕ0)
3027, 29syldan 486 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → (𝑔𝑘) ∈ ℕ0)
3127nnnn0d 11389 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ0)
3230, 31nn0mulcld 11394 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℕ0)
3332nn0cnd 11391 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℂ)
34 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → 𝑔:ℕ⟶ℕ0)
35 nnex 11064 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ ∈ V
36 frnnn0supp 11387 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℕ ∈ V ∧ 𝑔:ℕ⟶ℕ0) → (𝑔 supp 0) = (𝑔 “ ℕ))
3735, 36mpan 706 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔:ℕ⟶ℕ0 → (𝑔 supp 0) = (𝑔 “ ℕ))
3837adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 supp 0) = (𝑔 “ ℕ))
39 eqimss 3690 . . . . . . . . . . . . . . . . . . . . 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 11345 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℕ0
4342a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → 0 ∈ ℕ0)
4434, 40, 41, 43suppssr 7371 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → (𝑔𝑘) = 0)
4544oveq1d 6705 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑘) · 𝑘) = (0 · 𝑘))
46 eldifi 3765 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ)) → 𝑘 ∈ ℕ)
4746adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → 𝑘 ∈ ℕ)
48 nncn 11066 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
49 mul02 10252 . . . . . . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑘) · 𝑘) = 0)
52 nnuz 11761 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
5352eqimssi 3692 . . . . . . . . . . . . . . . . . 18 ℕ ⊆ (ℤ‘1)
5453a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → ℕ ⊆ (ℤ‘1))
5526, 33, 51, 54sumss 14499 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
56 simpr 476 . . . . . . . . . . . . . . . . 17 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (𝑔 “ ℕ) ∈ Fin)
5756, 32fsumnn0cl 14511 . . . . . . . . . . . . . . . 16 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) ∈ ℕ0)
5855, 57eqeltrrd 2731 . . . . . . . . . . . . . . 15 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) ∈ ℕ0)
59 eleq1 2718 . . . . . . . . . . . . . . 15 𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁 → (Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) ∈ ℕ0𝑁 ∈ ℕ0))
6058, 59syl5ibcom 235 . . . . . . . . . . . . . 14 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → (Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁𝑁 ∈ ℕ0))
61603impia 1280 . . . . . . . . . . . . 13 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑁 ∈ ℕ0)
6261adantr 480 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℕ0)
6362nn0red 11390 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℝ)
6418nn0ge0d 11392 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ (𝑔𝑥))
65 nnge1 11084 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → 1 ≤ 𝑥)
6665adantl 481 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 1 ≤ 𝑥)
6719, 21, 64, 66lemulge11d 10999 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ≤ ((𝑔𝑥) · 𝑥))
6856adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → (𝑔 “ ℕ) ∈ Fin)
6932nn0red 11390 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
7069adantlr 751 . . . . . . . . . . . . . . . . 17 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
7132nn0ge0d 11392 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
7271adantlr 751 . . . . . . . . . . . . . . . . 17 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
73 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝑔𝑘) = (𝑔𝑥))
74 id 22 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥𝑘 = 𝑥)
7573, 74oveq12d 6708 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → ((𝑔𝑘) · 𝑘) = ((𝑔𝑥) · 𝑥))
76 simprr 811 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → 𝑥 ∈ (𝑔 “ ℕ))
7768, 70, 72, 75, 76fsumge1 14573 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
7877expr 642 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (𝑔 “ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘)))
79 eldif 3617 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ)) ↔ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ)))
8051ralrimiva 2995 . . . . . . . . . . . . . . . . . . 19 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) → ∀𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))((𝑔𝑘) · 𝑘) = 0)
8175eqeq1d 2653 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑥 → (((𝑔𝑘) · 𝑘) = 0 ↔ ((𝑔𝑥) · 𝑥) = 0))
8281rspccva 3339 . . . . . . . . . . . . . . . . . . 19 ((∀𝑘 ∈ (ℕ ∖ (𝑔 “ ℕ))((𝑔𝑘) · 𝑘) = 0 ∧ 𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8380, 82sylan 487 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ (ℕ ∖ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8479, 83sylan2br 492 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) = 0)
8556adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (𝑔 “ ℕ) ∈ Fin)
8632adantlr 751 . . . . . . . . . . . . . . . . . . . 20 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℕ0)
8786nn0red 11390 . . . . . . . . . . . . . . . . . . 19 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → ((𝑔𝑘) · 𝑘) ∈ ℝ)
8886nn0ge0d 11392 . . . . . . . . . . . . . . . . . . 19 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) ∧ 𝑘 ∈ (𝑔 “ ℕ)) → 0 ≤ ((𝑔𝑘) · 𝑘))
8985, 87, 88fsumge0 14571 . . . . . . . . . . . . . . . . . 18 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → 0 ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9089adantrr 753 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → 0 ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9184, 90eqbrtrd 4707 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ (𝑥 ∈ ℕ ∧ ¬ 𝑥 ∈ (𝑔 “ ℕ))) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9291expr 642 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (𝑔 “ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘)))
9378, 92pm2.61d 170 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘))
9455adantr 480 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ (𝑔 “ ℕ)((𝑔𝑘) · 𝑘) = Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
9593, 94breqtrd 4711 . . . . . . . . . . . . 13 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
96953adantl3 1239 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘))
97 simpl3 1086 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁)
9896, 97breqtrd 4711 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) · 𝑥) ≤ 𝑁)
9919, 22, 63, 67, 98letrd 10232 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ≤ 𝑁)
100 nn0uz 11760 . . . . . . . . . . . 12 0 = (ℤ‘0)
10118, 100syl6eleq 2740 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ (ℤ‘0))
10262nn0zd 11518 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
103 elfz5 12372 . . . . . . . . . . 11 (((𝑔𝑥) ∈ (ℤ‘0) ∧ 𝑁 ∈ ℤ) → ((𝑔𝑥) ∈ (0...𝑁) ↔ (𝑔𝑥) ≤ 𝑁))
104101, 102, 103syl2anc 694 . . . . . . . . . 10 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ (0...𝑁) ↔ (𝑔𝑥) ≤ 𝑁))
10599, 104mpbird 247 . . . . . . . . 9 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ (0...𝑁))
106105adantr 480 . . . . . . . 8 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) ∈ (0...𝑁))
107 iftrue 4125 . . . . . . . . 9 (𝑥 ∈ (1...𝑁) → if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}) = (0...𝑁))
108107adantl 481 . . . . . . . 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 2733 . . . . . . 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 11084 . . . . . . . . . . . . . 14 ((𝑔𝑥) ∈ ℕ → 1 ≤ (𝑔𝑥))
111 nnnn0 11337 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
112111adantl 481 . . . . . . . . . . . . . . . . 17 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
113112nn0ge0d 11392 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 0 ≤ 𝑥)
114 lemulge12 10924 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ ∧ (𝑔𝑥) ∈ ℝ) ∧ (0 ≤ 𝑥 ∧ 1 ≤ (𝑔𝑥))) → 𝑥 ≤ ((𝑔𝑥) · 𝑥))
115114expr 642 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℝ ∧ (𝑔𝑥) ∈ ℝ) ∧ 0 ≤ 𝑥) → (1 ≤ (𝑔𝑥) → 𝑥 ≤ ((𝑔𝑥) · 𝑥)))
11621, 19, 113, 115syl21anc 1365 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (1 ≤ (𝑔𝑥) → 𝑥 ≤ ((𝑔𝑥) · 𝑥)))
117 letr 10169 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ ((𝑔𝑥) · 𝑥) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑥 ≤ ((𝑔𝑥) · 𝑥) ∧ ((𝑔𝑥) · 𝑥) ≤ 𝑁) → 𝑥𝑁))
11821, 22, 63, 117syl3anc 1366 . . . . . . . . . . . . . . . 16 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑥 ≤ ((𝑔𝑥) · 𝑥) ∧ ((𝑔𝑥) · 𝑥) ≤ 𝑁) → 𝑥𝑁))
11998, 118mpan2d 710 . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . 15 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
123122, 52syl6eleq 2740 . . . . . . . . . . . . . 14 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ (ℤ‘1))
124 elfz5 12372 . . . . . . . . . . . . . 14 ((𝑥 ∈ (ℤ‘1) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥𝑁))
125123, 102, 124syl2anc 694 . . . . . . . . . . . . 13 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑥 ∈ (1...𝑁) ↔ 𝑥𝑁))
126121, 125sylibrd 249 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ → 𝑥 ∈ (1...𝑁)))
127126con3d 148 . . . . . . . . . . 11 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (¬ 𝑥 ∈ (1...𝑁) → ¬ (𝑔𝑥) ∈ ℕ))
128 elnn0 11332 . . . . . . . . . . . . 13 ((𝑔𝑥) ∈ ℕ0 ↔ ((𝑔𝑥) ∈ ℕ ∨ (𝑔𝑥) = 0))
12918, 128sylib 208 . . . . . . . . . . . 12 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → ((𝑔𝑥) ∈ ℕ ∨ (𝑔𝑥) = 0))
130129ord 391 . . . . . . . . . . 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 444 . . . . . . . . 9 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) = 0)
133 fvex 6239 . . . . . . . . . 10 (𝑔𝑥) ∈ V
134133elsn 4225 . . . . . . . . 9 ((𝑔𝑥) ∈ {0} ↔ (𝑔𝑥) = 0)
135132, 134sylibr 224 . . . . . . . 8 ((((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ∈ (1...𝑁)) → (𝑔𝑥) ∈ {0})
1368adantl 481 . . . . . . . 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 2733 . . . . . . 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 849 . . . . . 6 (((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) ∧ 𝑥 ∈ ℕ) → (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
139138ralrimiva 2995 . . . . 5 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → ∀𝑥 ∈ ℕ (𝑔𝑥) ∈ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
140 vex 3234 . . . . . 6 𝑔 ∈ V
141140elixp 7957 . . . . 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 699 . . . 4 ((𝑔:ℕ⟶ℕ0 ∧ (𝑔 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑔𝑘) · 𝑘) = 𝑁) → 𝑔X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
14314, 142sylbi 207 . . 3 (𝑔𝑃𝑔X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0}))
144143ssriv 3640 . 2 𝑃X𝑥 ∈ ℕ if(𝑥 ∈ (1...𝑁), (0...𝑁), {0})
145 ssfi 8221 . 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 708 1 𝑃 ∈ Fin
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wtru 1524  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210   class class class wbr 4685  {copab 4745  cmpt 4762  ccnv 5142  dom cdm 5143  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692   supp csupp 7340  𝑚 cmap 7899  Xcixp 7950  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   · cmul 9979  cle 10113  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  cexp 12900  Σcsu 14460  cdvds 15027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461
This theorem is referenced by: (None)
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