Theorem limsupbnd1f 45670

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 5154	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗))
6	5	imbi1d 341	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
7	6	ralbidv 3178	. . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑗 ∈ 𝐵 (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
8		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑙(𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)
9		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑙
10		limsupbnd1f.1	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝐹
11		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑙
12	10, 11	nffv 6924	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝐹‘𝑙)
13		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑗 ≤
14		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐴
15	12, 13, 14	nfbr 5198	. . . . . . . 8 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝐴
16	9, 15	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)
17		breq2 5155	. . . . . . . 8 ⊢ (𝑗 = 𝑙 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑙))
18		fveq2 6914	. . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝐹‘𝑗) = (𝐹‘𝑙))
19	18	breq1d 5161	. . . . . . . 8 ⊢ (𝑗 = 𝑙 → ((𝐹‘𝑗) ≤ 𝐴 ↔ (𝐹‘𝑙) ≤ 𝐴))
20	17, 19	imbi12d 344	. . . . . . 7 ⊢ (𝑗 = 𝑙 → ((𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)))
21	8, 16, 20	cbvralw 3306	. . . . . 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 279	. . . 4 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)))
24	23	cbvrexvw 3238	. . 3 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))
25	4, 24	sylib 218	. 2 ⊢ (𝜑 → ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))
26	1, 2, 3, 25	limsupbnd1 15524	1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2108  Ⅎwnfc 2890  ∀wral 3061  ∃wrex 3070   ⊆ wss 3966   class class class wbr 5151  ⟶wf 6565  ‘cfv 6569  ℝcr 11161  ℝ^*cxr 11301   ≤ cle 11303  lim supclsp 15512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761  ax-cnex 11218  ax-resscn 11219  ax-1cn 11220  ax-icn 11221  ax-addcl 11222  ax-addrcl 11223  ax-mulcl 11224  ax-mulrcl 11225  ax-mulcom 11226  ax-addass 11227  ax-mulass 11228  ax-distr 11229  ax-i2m1 11230  ax-1ne0 11231  ax-1rid 11232  ax-rnegex 11233  ax-rrecex 11234  ax-cnre 11235  ax-pre-lttri 11236  ax-pre-lttrn 11237  ax-pre-ltadd 11238  ax-pre-mulgt0 11239  ax-pre-sup 11240

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-po 5601  df-so 5602  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-riota 7395  df-ov 7441  df-oprab 7442  df-mpo 7443  df-er 8753  df-en 8994  df-dom 8995  df-sdom 8996  df-sup 9489  df-inf 9490  df-pnf 11304  df-mnf 11305  df-xr 11306  df-ltxr 11307  df-le 11308  df-sub 11501  df-neg 11502  df-ico 13399  df-limsup 15513

This theorem is referenced by:  limsuppnflem  45694