Theorem limsupbnd1f 46143

Description: If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupbnd1f.1	⊢ Ⅎ𝑗𝐹
limsupbnd1f.2	⊢ (𝜑 → 𝐵 ⊆ ℝ)
limsupbnd1f.3	⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)
limsupbnd1f.4	⊢ (𝜑 → 𝐴 ∈ ℝ*)
limsupbnd1f.5	⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))

Assertion

Ref	Expression
limsupbnd1f	⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

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

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

Proof of Theorem limsupbnd1f

Dummy variables 𝑖 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupbnd1f.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
2		limsupbnd1f.3	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)
3		limsupbnd1f.4	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
4		limsupbnd1f.5	. . 3 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))
5		breq1 5078	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗))
6	5	imbi1d 343	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
7	6	ralbidv 3164	. . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑗 ∈ 𝐵 (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
8		nfv 1922	. . . . . . 7 ⊢ Ⅎ𝑙(𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)
9		nfv 1922	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑙
10		limsupbnd1f.1	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝐹
11		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑙
12	10, 11	nffv 6841	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝐹‘𝑙)
13		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑗 ≤
14		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐴
15	12, 13, 14	nfbr 5122	. . . . . . . 8 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝐴
16	9, 15	nfim 1904	. . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)
17		breq2 5079	. . . . . . . 8 ⊢ (𝑗 = 𝑙 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑙))
18		fveq2 6831	. . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝐹‘𝑗) = (𝐹‘𝑙))
19	18	breq1d 5085	. . . . . . . 8 ⊢ (𝑗 = 𝑙 → ((𝐹‘𝑗) ≤ 𝐴 ↔ (𝐹‘𝑙) ≤ 𝐴))
20	17, 19	imbi12d 346	. . . . . . 7 ⊢ (𝑗 = 𝑙 → ((𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)))
21	8, 16, 20	cbvralw 3283	. . . . . 6 ⊢ (∀𝑗 ∈ 𝐵 (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))
22	21	a1i 11	. . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)))
23	7, 22	bitrd 281	. . . 4 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)))
24	23	cbvrexvw 3220	. . 3 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))
25	4, 24	sylib 220	. 2 ⊢ (𝜑 → ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))
26	1, 2, 3, 25	limsupbnd1 15439	1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548   ∈ wcel 2121  Ⅎwnfc 2888  ∀wral 3055  ∃wrex 3065   ⊆ wss 3885   class class class wbr 5075  ⟶wf 6485  ‘cfv 6489  ℝcr 11032  ℝ^*cxr 11173   ≤ cle 11175  lim supclsp 15427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-po 5529  df-so 5530  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-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-inf 9350  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-ico 13299  df-limsup 15428

This theorem is referenced by:  limsuppnflem  46167