Theorem limsupubuzlem 43943

Description: If the limsup is not +∞, then the function is bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupubuzlem.j	⊢ Ⅎ𝑗𝜑
limsupubuzlem.e	⊢ Ⅎ𝑗𝑋
limsupubuzlem.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
limsupubuzlem.z	⊢ 𝑍 = (ℤ≥‘𝑀)
limsupubuzlem.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
limsupubuzlem.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
limsupubuzlem.k	⊢ (𝜑 → 𝐾 ∈ ℝ)
limsupubuzlem.b	⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
limsupubuzlem.n	⊢ 𝑁 = if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾))
limsupubuzlem.w	⊢ 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < )
limsupubuzlem.x	⊢ 𝑋 = if(𝑊 ≤ 𝑌, 𝑌, 𝑊)

Assertion

Ref	Expression
limsupubuzlem	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)

Distinct variable groups:   𝑥,𝐹   𝑗,𝑀   𝑗,𝑁   𝑥,𝑋   𝑥,𝑍   𝑥,𝑗

Allowed substitution hints:   𝜑(𝑥,𝑗)   𝐹(𝑗)   𝐾(𝑥,𝑗)   𝑀(𝑥)   𝑁(𝑥)   𝑊(𝑥,𝑗)   𝑋(𝑗)   𝑌(𝑥,𝑗)   𝑍(𝑗)

Proof of Theorem limsupubuzlem

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupubuzlem.x	. . 3 ⊢ 𝑋 = if(𝑊 ≤ 𝑌, 𝑌, 𝑊)
2		limsupubuzlem.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ℝ)
3		limsupubuzlem.w	. . . . . 6 ⊢ 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < )
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ))
5		limsupubuzlem.j	. . . . . 6 ⊢ Ⅎ𝑗𝜑
6		ltso 11235	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		fzfid 13878	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
9		eqid 2736	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
10		limsupubuzlem.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
11		limsupubuzlem.n	. . . . . . . . . . 11 ⊢ 𝑁 = if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾))
12	11	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 = if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
13		limsupubuzlem.k	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℝ)
14		ceilcl 13747	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℝ → (⌈‘𝐾) ∈ ℤ)
15	13, 14	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℤ)
16	10, 15	ifcld 4532	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) ∈ ℤ)
17	12, 16	eqeltrd 2838	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	15	zred 12607	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℝ)
19	10	zred 12607	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
20		max2 13106	. . . . . . . . . . 11 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
21	18, 19, 20	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
22	12	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) = 𝑁)
23	21, 22	breqtrd 5131	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
24	9, 10, 17, 23	eluzd 43634	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
25		eluzfz2 13449	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
27	26	ne0d 4295	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ≠ ∅)
28		limsupubuzlem.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
29	28	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐹:𝑍⟶ℝ)
30	10	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
31		elfzelz 13441	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
32	31	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℤ)
33		elfzle1 13444	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑗)
34	33	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑗)
35	9, 30, 32, 34	eluzd 43634	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ (ℤ≥‘𝑀))
36		limsupubuzlem.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
37	35, 36	eleqtrrdi 2849	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ 𝑍)
38	29, 37	ffvelcdmd 7036	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)
39	5, 7, 8, 27, 38	fisupclrnmpt 43623	. . . . 5 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ) ∈ ℝ)
40	4, 39	eqeltrd 2838	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ)
41	2, 40	ifcld 4532	. . 3 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
42	1, 41	eqeltrid 2842	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
43	28	ffvelcdmda 7035	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
44	43	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ∈ ℝ)
45	40	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑊 ∈ ℝ)
46	42	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑋 ∈ ℝ)
47		simpll 765	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝜑)
48	10	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑀 ∈ ℤ)
49	17	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑁 ∈ ℤ)
50	36	eluzelz2 43628	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
51	50	ad2antlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℤ)
52	36	eleq2i 2829	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
53	52	biimpi 215	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀))
54		eluzle 12776	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑗)
55	53, 54	syl 17	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑀 ≤ 𝑗)
56	55	ad2antlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑀 ≤ 𝑗)
57		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ≤ 𝑁)
58	48, 49, 51, 56, 57	elfzd 13432	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ (𝑀...𝑁))
59	5, 8, 38	fimaxre4 43626	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑗 ∈ (𝑀...𝑁)(𝐹‘𝑗) ≤ 𝑏)
60	5, 38, 59	suprubrnmpt 43471	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ))
61	60, 3	breqtrrdi 5147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ 𝑊)
62	47, 58, 61	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑊)
63		max1 13104	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
64	40, 2, 63	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
65	64, 1	breqtrrdi 5147	. . . . . . 7 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
66	65	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑊 ≤ 𝑋)
67	44, 45, 46, 62, 66	letrd 11312	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑋)
68	13	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ∈ ℝ)
69		uzssre 12785	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
70	36, 69	eqsstri 3978	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℝ
71	70	sseli 3940	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ)
72	71	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℝ)
73	69, 24	sselid 3942	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
74	73	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 ∈ ℝ)
75		ceilge 13750	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℝ → 𝐾 ≤ (⌈‘𝐾))
76	13, 75	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ (⌈‘𝐾))
77		max1 13104	. . . . . . . . . . . 12 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
78	18, 19, 77	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
79	78, 22	breqtrd 5131	. . . . . . . . . 10 ⊢ (𝜑 → (⌈‘𝐾) ≤ 𝑁)
80	13, 18, 73, 76, 79	letrd 11312	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≤ 𝑁)
81	80	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑁)
82		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → ¬ 𝑗 ≤ 𝑁)
83	74, 72	ltnled 11302	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
84	82, 83	mpbird 256	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 < 𝑗)
85	68, 74, 72, 81, 84	lelttrd 11313	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 < 𝑗)
86	68, 72, 85	ltled 11303	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑗)
87	43	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ∈ ℝ)
88	2	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ∈ ℝ)
89	42	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑋 ∈ ℝ)
90		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐾 ≤ 𝑗)
91		limsupubuzlem.b	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
92	91	r19.21bi 3234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
93	92	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
94	90, 93	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑌)
95		max2 13106	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
96	40, 2, 95	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
97	96, 1	breqtrrdi 5147	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≤ 𝑋)
98	97	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ≤ 𝑋)
99	87, 88, 89, 94, 98	letrd 11312	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑋)
100	86, 99	syldan 591	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑋)
101	67, 100	pm2.61dan 811	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ≤ 𝑋)
102	101	ex 413	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → (𝐹‘𝑗) ≤ 𝑋))
103	5, 102	ralrimi 3240	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋)
104		nfv 1917	. . 3 ⊢ Ⅎ𝑥∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋
105		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑗𝑥
106		limsupubuzlem.e	. . . . 5 ⊢ Ⅎ𝑗𝑋
107	105, 106	nfeq 2920	. . . 4 ⊢ Ⅎ𝑗 𝑥 = 𝑋
108		breq2 5109	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑋))
109	107, 108	ralbid 3256	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋))
110	104, 109	rspce 3570	. 2 ⊢ ((𝑋 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
111	42, 103, 110	syl2anc 584	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106  Ⅎwnfc 2887  ∀wral 3064  ∃wrex 3073  ifcif 4486   class class class wbr 5105   ↦ cmpt 5188   Or wor 5544  ran crn 5634  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  supcsup 9376  ℝcr 11050   < clt 11189   ≤ cle 11190  ℤcz 12499  ℤ_≥cuz 12763  ...cfz 13424  ⌈cceil 13696

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fl 13697  df-ceil 13698

This theorem is referenced by:  limsupubuz  43944