Theorem limsupmnf 45726

Description: The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to an upper interval of real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupmnf.j	⊢ Ⅎ𝑗𝐹
limsupmnf.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupmnf.f	⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Assertion

Ref	Expression
limsupmnf	⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))

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

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

Proof of Theorem limsupmnf

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

Step	Hyp	Ref	Expression
1		limsupmnf.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		limsupmnf.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
3		eqid 2730	. . 3 ⊢ (𝑖 ∈ ℝ ↦ sup((𝐹 “ (𝑖[,)+∞)), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup((𝐹 “ (𝑖[,)+∞)), ℝ*, < ))
4	1, 2, 3	limsupmnflem 45725	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦)))
5		breq2 5114	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥))
6	5	imbi2d 340	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
7	6	ralbidv 3157	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
8	7	rexbidv 3158	. . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
9		breq1 5113	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑖 ≤ 𝑙 ↔ 𝑘 ≤ 𝑙))
10	9	imbi1d 341	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
11	10	ralbidv 3157	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)))
12		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑙
13		limsupmnf.j	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹
14		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙
15	13, 14	nffv 6871	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙)
16		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤
17		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑥
18	15, 16, 17	nfbr 5157	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
19	12, 18	nfim 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥)
20		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)
21		breq2 5114	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑗))
22		fveq2 6861	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
23	22	breq1d 5120	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
24	21, 23	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → ((𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
25	19, 20, 24	cbvralw 3282	. . . . . . . . 9 ⊢ (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
26	25	a1i 11	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
27	11, 26	bitrd 279	. . . . . . 7 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
28	27	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
29	28	a1i 11	. . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
30	8, 29	bitrd 279	. . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
31	30	cbvralvw 3216	. . 3 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
32	31	a1i 11	. 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
33	4, 32	bitrd 279	1 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  Ⅎwnfc 2877  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917   class class class wbr 5110   ↦ cmpt 5191   “ cima 5644  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  supcsup 9398  ℝcr 11074  +∞cpnf 11212  -∞cmnf 11213  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216  [,)cico 13315  lim supclsp 15443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-ico 13319  df-limsup 15444

This theorem is referenced by:  limsupre2lem  45729  limsupmnfuzlem  45731