Theorem limsupref 46125

Description: If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupref.j	⊢ Ⅎ𝑗𝐹
limsupref.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupref.s	⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
limsupref.f	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
limsupref.b	⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))

Assertion

Ref	Expression
limsupref	⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

Distinct variable groups:   𝐴,𝑏,𝑗,𝑘   𝐹,𝑏,𝑘

Allowed substitution hints:   𝜑(𝑗,𝑘,𝑏)   𝐹(𝑗)

Proof of Theorem limsupref

Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupref.a	. 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		limsupref.s	. 2 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
3		limsupref.f	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4		limsupref.b	. . 3 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))
5		breq2 5079	. . . . . 6 ⊢ (𝑏 = 𝑦 → ((abs‘(𝐹‘𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑗)) ≤ 𝑦))
6	5	imbi2d 341	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
7	6	ralbidv 3159	. . . 4 ⊢ (𝑏 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
8		breq1 5078	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗))
9	8	imbi1d 342	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
10	9	ralbidv 3159	. . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
11		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)
12		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑥
13		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑗abs
14		limsupref.j	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝐹
15		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝑥
16	14, 15	nffv 6840	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝐹‘𝑥)
17	13, 16	nffv 6840	. . . . . . . . 9 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥))
18		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑗 ≤
19		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑦
20	17, 18, 19	nfbr 5122	. . . . . . . 8 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) ≤ 𝑦
21	12, 20	nfim 1899	. . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)
22		breq2 5079	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑥))
23		2fveq3 6835	. . . . . . . . 9 ⊢ (𝑗 = 𝑥 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑥)))
24	23	breq1d 5085	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → ((abs‘(𝐹‘𝑗)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑦))
25	22, 24	imbi12d 345	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ((𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)))
26	11, 21, 25	cbvralw 3278	. . . . . 6 ⊢ (∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))
27	26	a1i 11	. . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)))
28	10, 27	bitrd 280	. . . 4 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)))
29	7, 28	cbvrex2vw 3219	. . 3 ⊢ (∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))
30	4, 29	sylib 219	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))
31	1, 2, 3, 30	limsupre 46081	1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1543   ∈ wcel 2115  Ⅎwnfc 2883  ∀wral 3050  ∃wrex 3060   ⊆ wss 3886   class class class wbr 5075  ⟶wf 6484  ‘cfv 6488  supcsup 9346  ℝcr 11031  +∞cpnf 11170  ℝ^*cxr 11172   < clt 11173   ≤ cle 11174  abscabs 15190  lim supclsp 15426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7681  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3or 1089  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3061  df-rmo 3341  df-reu 3342  df-rab 3389  df-v 3430  df-sbc 3727  df-csb 3835  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3906  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-2nd 7935  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-inf 9349  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-n0 12432  df-z 12519  df-uz 12783  df-rp 12937  df-ico 13298  df-seq 13958  df-exp 14018  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-limsup 15427

This theorem is referenced by: (None)