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Theorem limsuple 15387

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 1137	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐹:𝐵⟶ℝ^)
2		reex 11104	. . . . . . 7 ⊢ ℝ ∈ V
3	2	ssex 5261	. . . . . 6 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
4	3	3ad2ant1 1133	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐵 ∈ V)
5		xrex 12887	. . . . . 6 ⊢ ℝ^* ∈ V
6	5	a1i 11	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → ℝ^ ∈ V)
7		fex2 7872	. . . . 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 15383	. . . 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 5105	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ (lim sup‘𝐹) ↔ 𝐴 ≤ inf(ran 𝐺, ℝ^, < )))
13	9	limsupgf 15384	. . . . 5 ⊢ 𝐺:ℝ⟶ℝ^*
14		frn 6663	. . . . 5 ⊢ (𝐺:ℝ⟶ℝ^* → ran 𝐺 ⊆ ℝ^*)
15	13, 14	ax-mp 5	. . . 4 ⊢ ran 𝐺 ⊆ ℝ^*
16		simp3 1138	. . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐴 ∈ ℝ^)
17		infxrgelb 13237	. . . 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 6656	. . . . 5 ⊢ (𝐺:ℝ⟶ℝ^* → 𝐺 Fn ℝ)
20	13, 19	ax-mp 5	. . . 4 ⊢ 𝐺 Fn ℝ
21		breq2 5097	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑗) → (𝐴 ≤ 𝑥 ↔ 𝐴 ≤ (𝐺‘𝑗)))
22	21	ralrn 7027	. . . 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 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3048 Vcvv 3437 ∩ cin 3897 ⊆ wss 3898 class class class wbr 5093 ↦ cmpt 5174 ran crn 5620 “ cima 5622 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 supcsup 9331 infcinf 9332 ℝcr 11012 +∞cpnf 11150 ℝ^*cxr 11152 < clt 11153 ≤ cle 11154 [,)cico 13249 lim supclsp 15379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-sup 9333 df-inf 9334 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-limsup 15380

This theorem is referenced by: limsuplt 15388 limsupbnd1 15391 limsupbnd2 15392 mbflimsup 25595 limsupge 45883

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