Theorem limsupubuzlem 46155

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 11217	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		fzfid 13926	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
9		eqid 2739	. . . . . . . . 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 13792	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℝ → (⌈‘𝐾) ∈ ℤ)
15	13, 14	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℤ)
16	10, 15	ifcld 4501	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) ∈ ℤ)
17	12, 16	eqeltrd 2839	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	15	zred 12624	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℝ)
19	10	zred 12624	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
20		max2 13130	. . . . . . . . . . 11 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
21	18, 19, 20	syl2anc 590	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
22	12	eqcomd 2745	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) = 𝑁)
23	21, 22	breqtrd 5098	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
24	9, 10, 17, 23	eluzd 45852	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
25		eluzfz2 13477	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
27	26	ne0d 4270	. . . . . 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 13469	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
32	31	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℤ)
33		elfzle1 13472	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑗)
34	33	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑗)
35	9, 30, 32, 34	eluzd 45852	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ (ℤ≥‘𝑀))
36		limsupubuzlem.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
37	35, 36	eleqtrrdi 2850	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ 𝑍)
38	29, 37	ffvelcdmd 7026	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)
39	5, 7, 8, 27, 38	fisupclrnmpt 45842	. . . . 5 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ) ∈ ℝ)
40	4, 39	eqeltrd 2839	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ)
41	2, 40	ifcld 4501	. . 3 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
42	1, 41	eqeltrid 2843	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
43	28	ffvelcdmda 7025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
44	43	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ∈ ℝ)
45	40	ad2antrr 732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑊 ∈ ℝ)
46	42	ad2antrr 732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑋 ∈ ℝ)
47		simpll 772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝜑)
48	10	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑀 ∈ ℤ)
49	17	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑁 ∈ ℤ)
50	36	eluzelz2 45846	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
51	50	ad2antlr 733	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℤ)
52	36	eleq2i 2831	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
53	52	biimpi 217	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀))
54		eluzle 12792	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑗)
55	53, 54	syl 17	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑀 ≤ 𝑗)
56	55	ad2antlr 733	. . . . . . . 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 13460	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ (𝑀...𝑁))
59	5, 8, 38	fimaxre4 45844	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑗 ∈ (𝑀...𝑁)(𝐹‘𝑗) ≤ 𝑏)
60	5, 38, 59	suprubrnmpt 45697	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ))
61	60, 3	breqtrrdi 5114	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ 𝑊)
62	47, 58, 61	syl2anc 590	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑊)
63		max1 13128	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
64	40, 2, 63	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
65	64, 1	breqtrrdi 5114	. . . . . . 7 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
66	65	ad2antrr 732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑊 ≤ 𝑋)
67	44, 45, 46, 62, 66	letrd 11294	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑋)
68	13	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ∈ ℝ)
69		uzssre 12801	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
70	36, 69	eqsstri 3961	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℝ
71	70	sseli 3911	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ)
72	71	ad2antlr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℝ)
73	69, 24	sselid 3913	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
74	73	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 ∈ ℝ)
75		ceilge 13795	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℝ → 𝐾 ≤ (⌈‘𝐾))
76	13, 75	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ (⌈‘𝐾))
77		max1 13128	. . . . . . . . . . . 12 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
78	18, 19, 77	syl2anc 590	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
79	78, 22	breqtrd 5098	. . . . . . . . . 10 ⊢ (𝜑 → (⌈‘𝐾) ≤ 𝑁)
80	13, 18, 73, 76, 79	letrd 11294	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≤ 𝑁)
81	80	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑁)
82		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → ¬ 𝑗 ≤ 𝑁)
83	74, 72	ltnled 11284	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
84	82, 83	mpbird 258	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 < 𝑗)
85	68, 74, 72, 81, 84	lelttrd 11295	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 < 𝑗)
86	68, 72, 85	ltled 11285	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑗)
87	43	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ∈ ℝ)
88	2	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ∈ ℝ)
89	42	ad2antrr 732	. . . . . . 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 3231	. . . . . . . . 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 13130	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
96	40, 2, 95	syl2anc 590	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
97	96, 1	breqtrrdi 5114	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≤ 𝑋)
98	97	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ≤ 𝑋)
99	87, 88, 89, 94, 98	letrd 11294	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑋)
100	86, 99	syldan 597	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑋)
101	67, 100	pm2.61dan 818	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ≤ 𝑋)
102	101	ex 413	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → (𝐹‘𝑗) ≤ 𝑋))
103	5, 102	ralrimi 3237	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋)
104		nfv 1921	. . 3 ⊢ Ⅎ𝑥∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋
105		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑗𝑥
106		limsupubuzlem.e	. . . . 5 ⊢ Ⅎ𝑗𝑋
107	105, 106	nfeq 2914	. . . 4 ⊢ Ⅎ𝑗 𝑥 = 𝑋
108		breq2 5076	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑋))
109	107, 108	ralbid 3252	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋))
110	104, 109	rspce 3549	. 2 ⊢ ((𝑋 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
111	42, 103, 110	syl2anc 590	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  Ⅎwnfc 2886  ∀wral 3053  ∃wrex 3063  ifcif 4454   class class class wbr 5072   ↦ cmpt 5153   Or wor 5525  ran crn 5619  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  supcsup 9343  ℝcr 11028   < clt 11170   ≤ cle 11171  ℤcz 12515  ℤ_≥cuz 12779  ...cfz 13452  ⌈cceil 13741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-fl 13742  df-ceil 13743

This theorem is referenced by:  limsupubuz  46156