Theorem limsupubuzlem 45633

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 11370	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		fzfid 14024	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
9		eqid 2740	. . . . . . . . 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 13893	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℝ → (⌈‘𝐾) ∈ ℤ)
15	13, 14	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℤ)
16	10, 15	ifcld 4594	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) ∈ ℤ)
17	12, 16	eqeltrd 2844	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	15	zred 12747	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℝ)
19	10	zred 12747	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
20		max2 13249	. . . . . . . . . . 11 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
21	18, 19, 20	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
22	12	eqcomd 2746	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) = 𝑁)
23	21, 22	breqtrd 5192	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
24	9, 10, 17, 23	eluzd 45324	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
25		eluzfz2 13592	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
27	26	ne0d 4365	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ≠ ∅)
28		limsupubuzlem.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐹:𝑍⟶ℝ)
30	10	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
31		elfzelz 13584	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℤ)
33		elfzle1 13587	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑗)
34	33	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑗)
35	9, 30, 32, 34	eluzd 45324	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ (ℤ≥‘𝑀))
36		limsupubuzlem.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
37	35, 36	eleqtrrdi 2855	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ 𝑍)
38	29, 37	ffvelcdmd 7119	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)
39	5, 7, 8, 27, 38	fisupclrnmpt 45313	. . . . 5 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ) ∈ ℝ)
40	4, 39	eqeltrd 2844	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ)
41	2, 40	ifcld 4594	. . 3 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
42	1, 41	eqeltrid 2848	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
43	28	ffvelcdmda 7118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
44	43	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ∈ ℝ)
45	40	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑊 ∈ ℝ)
46	42	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑋 ∈ ℝ)
47		simpll 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝜑)
48	10	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑀 ∈ ℤ)
49	17	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑁 ∈ ℤ)
50	36	eluzelz2 45318	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
51	50	ad2antlr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℤ)
52	36	eleq2i 2836	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
53	52	biimpi 216	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀))
54		eluzle 12916	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑗)
55	53, 54	syl 17	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑀 ≤ 𝑗)
56	55	ad2antlr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑀 ≤ 𝑗)
57		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ≤ 𝑁)
58	48, 49, 51, 56, 57	elfzd 13575	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ (𝑀...𝑁))
59	5, 8, 38	fimaxre4 45316	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑗 ∈ (𝑀...𝑁)(𝐹‘𝑗) ≤ 𝑏)
60	5, 38, 59	suprubrnmpt 45162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ))
61	60, 3	breqtrrdi 5208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ 𝑊)
62	47, 58, 61	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑊)
63		max1 13247	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
64	40, 2, 63	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
65	64, 1	breqtrrdi 5208	. . . . . . 7 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
66	65	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑊 ≤ 𝑋)
67	44, 45, 46, 62, 66	letrd 11447	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑋)
68	13	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ∈ ℝ)
69		uzssre 12925	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
70	36, 69	eqsstri 4043	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℝ
71	70	sseli 4004	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ)
72	71	ad2antlr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℝ)
73	69, 24	sselid 4006	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
74	73	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 ∈ ℝ)
75		ceilge 13896	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℝ → 𝐾 ≤ (⌈‘𝐾))
76	13, 75	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ (⌈‘𝐾))
77		max1 13247	. . . . . . . . . . . 12 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
78	18, 19, 77	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
79	78, 22	breqtrd 5192	. . . . . . . . . 10 ⊢ (𝜑 → (⌈‘𝐾) ≤ 𝑁)
80	13, 18, 73, 76, 79	letrd 11447	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≤ 𝑁)
81	80	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑁)
82		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → ¬ 𝑗 ≤ 𝑁)
83	74, 72	ltnled 11437	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
84	82, 83	mpbird 257	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 < 𝑗)
85	68, 74, 72, 81, 84	lelttrd 11448	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 < 𝑗)
86	68, 72, 85	ltled 11438	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑗)
87	43	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ∈ ℝ)
88	2	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ∈ ℝ)
89	42	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑋 ∈ ℝ)
90		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐾 ≤ 𝑗)
91		limsupubuzlem.b	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
92	91	r19.21bi 3257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
93	92	adantr 480	. . . . . . . 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 13249	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
96	40, 2, 95	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
97	96, 1	breqtrrdi 5208	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≤ 𝑋)
98	97	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ≤ 𝑋)
99	87, 88, 89, 94, 98	letrd 11447	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑋)
100	86, 99	syldan 590	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑋)
101	67, 100	pm2.61dan 812	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ≤ 𝑋)
102	101	ex 412	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → (𝐹‘𝑗) ≤ 𝑋))
103	5, 102	ralrimi 3263	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋)
104		nfv 1913	. . 3 ⊢ Ⅎ𝑥∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋
105		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑗𝑥
106		limsupubuzlem.e	. . . . 5 ⊢ Ⅎ𝑗𝑋
107	105, 106	nfeq 2922	. . . 4 ⊢ Ⅎ𝑗 𝑥 = 𝑋
108		breq2 5170	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑋))
109	107, 108	ralbid 3279	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋))
110	104, 109	rspce 3624	. 2 ⊢ ((𝑋 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
111	42, 103, 110	syl2anc 583	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893  ∀wral 3067  ∃wrex 3076  ifcif 4548   class class class wbr 5166   ↦ cmpt 5249   Or wor 5606  ran crn 5701  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  supcsup 9509  ℝcr 11183   < clt 11324   ≤ cle 11325  ℤcz 12639  ℤ_≥cuz 12903  ...cfz 13567  ⌈cceil 13842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fl 13843  df-ceil 13844

This theorem is referenced by:  limsupubuz  45634