Theorem limsupubuzlem 42795

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 10799	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		fzfid 13432	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
9		eqid 2738	. . . . . . . . 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 13303	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℝ → (⌈‘𝐾) ∈ ℤ)
15	13, 14	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℤ)
16	10, 15	ifcld 4460	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) ∈ ℤ)
17	12, 16	eqeltrd 2833	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	15	zred 12168	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℝ)
19	10	zred 12168	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
20		max2 12663	. . . . . . . . . . 11 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
21	18, 19, 20	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
22	12	eqcomd 2744	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) = 𝑁)
23	21, 22	breqtrd 5056	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
24	9, 10, 17, 23	eluzd 42487	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
25		eluzfz2 13006	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
27	26	ne0d 4224	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ≠ ∅)
28		limsupubuzlem.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
29	28	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐹:𝑍⟶ℝ)
30	10	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
31		elfzelz 12998	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
32	31	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℤ)
33		elfzle1 13001	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑗)
34	33	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑗)
35	9, 30, 32, 34	eluzd 42487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ (ℤ≥‘𝑀))
36		limsupubuzlem.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
37	35, 36	eleqtrrdi 2844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ 𝑍)
38	29, 37	ffvelrnd 6862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)
39	5, 7, 8, 27, 38	fisupclrnmpt 42476	. . . . 5 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ) ∈ ℝ)
40	4, 39	eqeltrd 2833	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ)
41	2, 40	ifcld 4460	. . 3 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
42	1, 41	eqeltrid 2837	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
43	28	ffvelrnda 6861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
44	43	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ∈ ℝ)
45	40	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑊 ∈ ℝ)
46	42	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑋 ∈ ℝ)
47		simpll 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝜑)
48	10	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑀 ∈ ℤ)
49	17	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑁 ∈ ℤ)
50	36	eluzelz2 42481	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
51	50	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℤ)
52	36	eleq2i 2824	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
53	52	biimpi 219	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀))
54		eluzle 12337	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑗)
55	53, 54	syl 17	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑀 ≤ 𝑗)
56	55	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑀 ≤ 𝑗)
57		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ≤ 𝑁)
58	48, 49, 51, 56, 57	elfzd 12989	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ (𝑀...𝑁))
59	5, 8, 38	fimaxre4 42479	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑗 ∈ (𝑀...𝑁)(𝐹‘𝑗) ≤ 𝑏)
60	5, 38, 59	suprubrnmpt 42336	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ))
61	60, 3	breqtrrdi 5072	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ 𝑊)
62	47, 58, 61	syl2anc 587	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑊)
63		max1 12661	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
64	40, 2, 63	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
65	64, 1	breqtrrdi 5072	. . . . . . 7 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
66	65	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑊 ≤ 𝑋)
67	44, 45, 46, 62, 66	letrd 10875	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑋)
68	13	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ∈ ℝ)
69		uzssre 12346	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
70	36, 69	eqsstri 3911	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℝ
71	70	sseli 3873	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ)
72	71	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℝ)
73	69, 24	sseldi 3875	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
74	73	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 ∈ ℝ)
75		ceilge 13305	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℝ → 𝐾 ≤ (⌈‘𝐾))
76	13, 75	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ (⌈‘𝐾))
77		max1 12661	. . . . . . . . . . . 12 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
78	18, 19, 77	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
79	78, 22	breqtrd 5056	. . . . . . . . . 10 ⊢ (𝜑 → (⌈‘𝐾) ≤ 𝑁)
80	13, 18, 73, 76, 79	letrd 10875	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≤ 𝑁)
81	80	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑁)
82		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → ¬ 𝑗 ≤ 𝑁)
83	74, 72	ltnled 10865	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
84	82, 83	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 < 𝑗)
85	68, 74, 72, 81, 84	lelttrd 10876	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 < 𝑗)
86	68, 72, 85	ltled 10866	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑗)
87	43	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ∈ ℝ)
88	2	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ∈ ℝ)
89	42	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑋 ∈ ℝ)
90		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐾 ≤ 𝑗)
91		limsupubuzlem.b	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
92	91	r19.21bi 3121	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
93	92	adantr 484	. . . . . . . 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 12663	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
96	40, 2, 95	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
97	96, 1	breqtrrdi 5072	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≤ 𝑋)
98	97	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ≤ 𝑋)
99	87, 88, 89, 94, 98	letrd 10875	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑋)
100	86, 99	syldan 594	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑋)
101	67, 100	pm2.61dan 813	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ≤ 𝑋)
102	101	ex 416	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → (𝐹‘𝑗) ≤ 𝑋))
103	5, 102	ralrimi 3128	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋)
104		nfv 1921	. . 3 ⊢ Ⅎ𝑥∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋
105		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑗𝑥
106		limsupubuzlem.e	. . . . 5 ⊢ Ⅎ𝑗𝑋
107	105, 106	nfeq 2912	. . . 4 ⊢ Ⅎ𝑗 𝑥 = 𝑋
108		breq2 5034	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑋))
109	107, 108	ralbid 3145	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋))
110	104, 109	rspce 3515	. 2 ⊢ ((𝑋 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
111	42, 103, 110	syl2anc 587	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1542  Ⅎwnf 1790   ∈ wcel 2114  Ⅎwnfc 2879  ∀wral 3053  ∃wrex 3054  ifcif 4414   class class class wbr 5030   ↦ cmpt 5110   Or wor 5441  ran crn 5526  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170  supcsup 8977  ℝcr 10614   < clt 10753   ≤ cle 10754  ℤcz 12062  ℤ_≥cuz 12324  ...cfz 12981  ⌈cceil 13252

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-inf 8980  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-n0 11977  df-z 12063  df-uz 12325  df-fz 12982  df-fl 13253  df-ceil 13254

This theorem is referenced by:  limsupubuz  42796