Theorem limsupref 46515

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 5111	. . . . . 6 ⊢ (𝑏 = 𝑦 → ((abs‘(𝐹‘𝑗)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑗)) ≤ 𝑦))
6	5	imbi2d 343	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
7	6	ralbidv 3187	. . . 4 ⊢ (𝑏 = 𝑦 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
8		breq1 5110	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗))
9	8	imbi1d 344	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
10	9	ralbidv 3187	. . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)))
11		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑥(𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦)
12		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑥
13		nfcv 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑗abs
14		limsupref.j	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝐹
15		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝑥
16	14, 15	nffv 6892	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝐹‘𝑥)
17	13, 16	nffv 6892	. . . . . . . . 9 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥))
18		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑗 ≤
19		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑦
20	17, 18, 19	nfbr 5156	. . . . . . . 8 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑥)) ≤ 𝑦
21	12, 20	nfim 1929	. . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)
22		breq2 5111	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑥))
23		2fveq3 6887	. . . . . . . . 9 ⊢ (𝑗 = 𝑥 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑥)))
24	23	breq1d 5117	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → ((abs‘(𝐹‘𝑗)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑦))
25	22, 24	imbi12d 347	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ((𝑖 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)))
26	11, 21, 25	cbvralw 3306	. . . . . 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 282	. . . 4 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦)))
29	7, 28	cbvrex2vw 3247	. . 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 221	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑖 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))
31	1, 2, 3, 30	limsupre 46471	1 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2909  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902   class class class wbr 5107  ⟶wf 6533  ‘cfv 6537  supcsup 9413  ℝcr 11126  +∞cpnf 11267  ℝ^*cxr 11269   < clt 11270   ≤ cle 11271  abscabs 15323  lim supclsp 15559

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9415  df-inf 9416  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-z 12619  df-uz 12891  df-rp 13045  df-ico 13406  df-seq 14068  df-exp 14128  df-cj 15188  df-re 15189  df-im 15190  df-sqrt 15324  df-abs 15325  df-limsup 15560

This theorem is used by: (None)