Theorem limsupubuzlem 41991

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 10720	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		fzfid 13340	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
9		eqid 2821	. . . . . . . . 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 13211	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℝ → (⌈‘𝐾) ∈ ℤ)
15	13, 14	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℤ)
16	10, 15	ifcld 4511	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) ∈ ℤ)
17	12, 16	eqeltrd 2913	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	15	zred 12086	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ∈ ℝ)
19	10	zred 12086	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
20		max2 12579	. . . . . . . . . . 11 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
21	18, 19, 20	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
22	12	eqcomd 2827	. . . . . . . . . 10 ⊢ (𝜑 → if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) = 𝑁)
23	21, 22	breqtrd 5091	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
24	9, 10, 17, 23	eluzd 41680	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
25		eluzfz2 12914	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
27	26	ne0d 4300	. . . . . 6 ⊢ (𝜑 → (𝑀...𝑁) ≠ ∅)
28		limsupubuzlem.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
29	28	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐹:𝑍⟶ℝ)
30	10	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
31		elfzelz 12907	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑗 ∈ ℤ)
32	31	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ ℤ)
33		elfzle1 12909	. . . . . . . . . 10 ⊢ (𝑗 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑗)
34	33	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑀 ≤ 𝑗)
35	9, 30, 32, 34	eluzd 41680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ (ℤ≥‘𝑀))
36		limsupubuzlem.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
37	35, 36	eleqtrrdi 2924	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝑗 ∈ 𝑍)
38	29, 37	ffvelrnd 6851	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)
39	5, 7, 8, 27, 38	fisupclrnmpt 41669	. . . . 5 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ) ∈ ℝ)
40	4, 39	eqeltrd 2913	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ)
41	2, 40	ifcld 4511	. . 3 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
42	1, 41	eqeltrid 2917	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
43	28	ffvelrnda 6850	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
44	43	adantr 483	. . . . . 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 41674	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
51	50	ad2antlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℤ)
52	36	eleq2i 2904	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
53	52	biimpi 218	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀))
54		eluzle 12255	. . . . . . . . . 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 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ≤ 𝑁)
58	48, 49, 51, 56, 57	elfzd 41681	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑗 ∈ (𝑀...𝑁))
59	5, 8, 38	fimaxre4 41672	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑗 ∈ (𝑀...𝑁)(𝐹‘𝑗) ≤ 𝑏)
60	5, 38, 59	suprubrnmpt 41523	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ))
61	60, 3	breqtrrdi 5107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ≤ 𝑊)
62	47, 58, 61	syl2anc 586	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑊)
63		max1 12577	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
64	40, 2, 63	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
65	64, 1	breqtrrdi 5107	. . . . . . 7 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
66	65	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → 𝑊 ≤ 𝑋)
67	44, 45, 46, 62, 66	letrd 10796	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 ≤ 𝑁) → (𝐹‘𝑗) ≤ 𝑋)
68	13	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ∈ ℝ)
69		uzssre 41667	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
70	36, 69	eqsstri 4000	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℝ
71	70	sseli 3962	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ)
72	71	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑗 ∈ ℝ)
73	69, 24	sseldi 3964	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
74	73	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 ∈ ℝ)
75		ceilge 13213	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℝ → 𝐾 ≤ (⌈‘𝐾))
76	13, 75	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ (⌈‘𝐾))
77		max1 12577	. . . . . . . . . . . 12 ⊢ (((⌈‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
78	18, 19, 77	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘𝐾) ≤ if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)))
79	78, 22	breqtrd 5091	. . . . . . . . . 10 ⊢ (𝜑 → (⌈‘𝐾) ≤ 𝑁)
80	13, 18, 73, 76, 79	letrd 10796	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≤ 𝑁)
81	80	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑁)
82		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → ¬ 𝑗 ≤ 𝑁)
83	74, 72	ltnled 10786	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
84	82, 83	mpbird 259	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝑁 < 𝑗)
85	68, 74, 72, 81, 84	lelttrd 10797	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 < 𝑗)
86	68, 72, 85	ltled 10787	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝑗 ≤ 𝑁) → 𝐾 ≤ 𝑗)
87	43	adantr 483	. . . . . . 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 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐾 ≤ 𝑗)
91		limsupubuzlem.b	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
92	91	r19.21bi 3208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌))
93	92	adantr 483	. . . . . . . 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 12579	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
96	40, 2, 95	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
97	96, 1	breqtrrdi 5107	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≤ 𝑋)
98	97	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ≤ 𝑋)
99	87, 88, 89, 94, 98	letrd 10796	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑋)
100	86, 99	syldan 593	. . . . 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 415	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → (𝐹‘𝑗) ≤ 𝑋))
103	5, 102	ralrimi 3216	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋)
104		nfv 1911	. . 3 ⊢ Ⅎ𝑥∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋
105		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑗𝑥
106		limsupubuzlem.e	. . . . 5 ⊢ Ⅎ𝑗𝑋
107	105, 106	nfeq 2991	. . . 4 ⊢ Ⅎ𝑗 𝑥 = 𝑋
108		breq2 5069	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑋))
109	107, 108	ralbid 3231	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋))
110	104, 109	rspce 3611	. 2 ⊢ ((𝑋 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑋) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
111	42, 103, 110	syl2anc 586	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  Ⅎwnfc 2961  ∀wral 3138  ∃wrex 3139  ifcif 4466   class class class wbr 5065   ↦ cmpt 5145   Or wor 5472  ran crn 5555  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  supcsup 8903  ℝcr 10535   < clt 10674   ≤ cle 10675  ℤcz 11980  ℤ_≥cuz 12242  ...cfz 12891  ⌈cceil 13160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-inf 8906  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-fl 13161  df-ceil 13162

This theorem is referenced by:  limsupubuz  41992