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Theorem limsuple 15514

Description: The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

**Hypothesis**
Ref	Expression
limsupval.1	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))

**Assertion**
Ref	Expression
limsuple	⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)))

Distinct variable groups: 𝑘,𝐹 𝐴,𝑗 𝐵,𝑗 𝑗,𝑘,𝐹 𝑗,𝐺 Allowed substitution hints: 𝐴(𝑘) 𝐵(𝑘) 𝐺(𝑘)

Proof of Theorem limsuple Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1138	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐹:𝐵⟶ℝ^)
2		reex 11246	. . . . . . 7 ⊢ ℝ ∈ V
3	2	ssex 5321	. . . . . 6 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
4	3	3ad2ant1 1134	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐵 ∈ V)
5		xrex 13029	. . . . . 6 ⊢ ℝ^* ∈ V
6	5	a1i 11	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → ℝ^ ∈ V)
7		fex2 7958	. . . . 5 ⊢ ((𝐹:𝐵⟶ℝ^* ∧ 𝐵 ∈ V ∧ ℝ^* ∈ V) → 𝐹 ∈ V)
8	1, 4, 6, 7	syl3anc 1373	. . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐹 ∈ V)
9		limsupval.1	. . . . 5 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
10	9	limsupval 15510	. . . 4 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) = inf(ran 𝐺, ℝ^*, < ))
11	8, 10	syl 17	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → (lim sup‘𝐹) = inf(ran 𝐺, ℝ^, < ))
12	11	breq2d 5155	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ (lim sup‘𝐹) ↔ 𝐴 ≤ inf(ran 𝐺, ℝ^, < )))
13	9	limsupgf 15511	. . . . 5 ⊢ 𝐺:ℝ⟶ℝ^*
14		frn 6743	. . . . 5 ⊢ (𝐺:ℝ⟶ℝ^* → ran 𝐺 ⊆ ℝ^*)
15	13, 14	ax-mp 5	. . . 4 ⊢ ran 𝐺 ⊆ ℝ^*
16		simp3 1139	. . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐴 ∈ ℝ^)
17		infxrgelb 13377	. . . 4 ⊢ ((ran 𝐺 ⊆ ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ inf(ran 𝐺, ℝ^, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥))
18	15, 16, 17	sylancr 587	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ inf(ran 𝐺, ℝ^, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥))
19		ffn 6736	. . . . 5 ⊢ (𝐺:ℝ⟶ℝ^* → 𝐺 Fn ℝ)
20	13, 19	ax-mp 5	. . . 4 ⊢ 𝐺 Fn ℝ
21		breq2 5147	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑗) → (𝐴 ≤ 𝑥 ↔ 𝐴 ≤ (𝐺‘𝑗)))
22	21	ralrn 7108	. . . 4 ⊢ (𝐺 Fn ℝ → (∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥 ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)))
23	20, 22	ax-mp 5	. . 3 ⊢ (∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥 ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))
24	18, 23	bitrdi 287	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ inf(ran 𝐺, ℝ^, < ) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)))
25	12, 24	bitrd 279	1 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 infcinf 9481 ℝcr 11154 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 [,)cico 13389 lim supclsp 15506

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 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 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-limsup 15507

This theorem is referenced by: limsuplt 15515 limsupbnd1 15518 limsupbnd2 15519 mbflimsup 25701 limsupge 45776

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