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Theorem limsuplt 15525

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
limsuplt	⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑗 ∈ ℝ (𝐺‘𝑗) < 𝐴))

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 limsuplt**
Step	Hyp	Ref	Expression
1		limsupval.1	. . . . 5 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
2	1	limsuple 15524	. . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)))
3	2	notbid 318	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (¬ 𝐴 ≤ (lim sup‘𝐹) ↔ ¬ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)))
4		rexnal 3106	. . 3 ⊢ (∃𝑗 ∈ ℝ ¬ 𝐴 ≤ (𝐺‘𝑗) ↔ ¬ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))
5	3, 4	bitr4di 289	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (¬ 𝐴 ≤ (lim sup‘𝐹) ↔ ∃𝑗 ∈ ℝ ¬ 𝐴 ≤ (𝐺‘𝑗)))
6		simp2 1137	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐹:𝐵⟶ℝ^)
7		reex 11275	. . . . . . 7 ⊢ ℝ ∈ V
8	7	ssex 5339	. . . . . 6 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
9	8	3ad2ant1 1133	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐵 ∈ V)
10		xrex 13052	. . . . . 6 ⊢ ℝ^* ∈ V
11	10	a1i 11	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → ℝ^ ∈ V)
12		fex2 7974	. . . . 5 ⊢ ((𝐹:𝐵⟶ℝ^* ∧ 𝐵 ∈ V ∧ ℝ^* ∈ V) → 𝐹 ∈ V)
13	6, 9, 11, 12	syl3anc 1371	. . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐹 ∈ V)
14		limsupcl 15519	. . . 4 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ^*)
15	13, 14	syl 17	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → (lim sup‘𝐹) ∈ ℝ^)
16		simp3 1138	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐴 ∈ ℝ^)
17		xrltnle 11357	. . 3 ⊢ (((lim sup‘𝐹) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((lim sup‘𝐹) < 𝐴 ↔ ¬ 𝐴 ≤ (lim sup‘𝐹)))
18	15, 16, 17	syl2anc 583	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((lim sup‘𝐹) < 𝐴 ↔ ¬ 𝐴 ≤ (lim sup‘𝐹)))
19	1	limsupgf 15521	. . . . 5 ⊢ 𝐺:ℝ⟶ℝ^*
20	19	ffvelcdmi 7117	. . . 4 ⊢ (𝑗 ∈ ℝ → (𝐺‘𝑗) ∈ ℝ^*)
21		xrltnle 11357	. . . 4 ⊢ (((𝐺‘𝑗) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((𝐺‘𝑗) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐺‘𝑗)))
22	20, 16, 21	syl2anr 596	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ 𝑗 ∈ ℝ) → ((𝐺‘𝑗) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐺‘𝑗)))
23	22	rexbidva 3183	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (∃𝑗 ∈ ℝ (𝐺‘𝑗) < 𝐴 ↔ ∃𝑗 ∈ ℝ ¬ 𝐴 ≤ (𝐺‘𝑗)))
24	5, 18, 23	3bitr4d 311	1 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑗 ∈ ℝ (𝐺‘𝑗) < 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 class class class wbr 5166 ↦ cmpt 5249 “ cima 5703 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 supcsup 9509 ℝcr 11183 +∞cpnf 11321 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 [,)cico 13409 lim supclsp 15516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-limsup 15517

This theorem is referenced by: limsupgre 15527 limsuplt2 45674

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