Theorem limsupref 44973

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 5145	. . . . . 6 ⊢ (𝑏 = 𝑦 → ((abs‘(𝐹‘𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑗)) ≤ 𝑦))
6	5	imbi2d 340	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
7	6	ralbidv 3171	. . . 4 ⊢ (𝑏 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
8		breq1 5144	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗))
9	8	imbi1d 341	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
10	9	ralbidv 3171	. . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
11		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)
12		nfv 1909	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑥
13		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑗abs
14		limsupref.j	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝐹
15		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝑥
16	14, 15	nffv 6895	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝐹‘𝑥)
17	13, 16	nffv 6895	. . . . . . . . 9 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥))
18		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑗 ≤
19		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑦
20	17, 18, 19	nfbr 5188	. . . . . . . 8 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) ≤ 𝑦
21	12, 20	nfim 1891	. . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)
22		breq2 5145	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑥))
23		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑗 = 𝑥 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑥)))
24	23	breq1d 5151	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → ((abs‘(𝐹‘𝑗)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑦))
25	22, 24	imbi12d 344	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ((𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)))
26	11, 21, 25	cbvralw 3297	. . . . . 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 279	. . . 4 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)))
29	7, 28	cbvrex2vw 3233	. . 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 217	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))
31	1, 2, 3, 30	limsupre 44929	1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2877  ∀wral 3055  ∃wrex 3064   ⊆ wss 3943   class class class wbr 5141  ⟶wf 6533  ‘cfv 6537  supcsup 9437  ℝcr 11111  +∞cpnf 11249  ℝ^*cxr 11251   < clt 11252   ≤ cle 11253  abscabs 15187  lim supclsp 15420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12981  df-ico 13336  df-seq 13973  df-exp 14033  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-limsup 15421

This theorem is referenced by: (None)