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Theorem limsuplt 14828

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 14827	. . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)))
3	2	notbid 321	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (¬ 𝐴 ≤ (lim sup‘𝐹) ↔ ¬ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)))
4		rexnal 3201	. . 3 ⊢ (∃𝑗 ∈ ℝ ¬ 𝐴 ≤ (𝐺‘𝑗) ↔ ¬ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))
5	3, 4	syl6bbr 292	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (¬ 𝐴 ≤ (lim sup‘𝐹) ↔ ∃𝑗 ∈ ℝ ¬ 𝐴 ≤ (𝐺‘𝑗)))
6		simp2 1134	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐹:𝐵⟶ℝ^)
7		reex 10617	. . . . . . 7 ⊢ ℝ ∈ V
8	7	ssex 5189	. . . . . 6 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
9	8	3ad2ant1 1130	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐵 ∈ V)
10		xrex 12374	. . . . . 6 ⊢ ℝ^* ∈ V
11	10	a1i 11	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → ℝ^ ∈ V)
12		fex2 7620	. . . . 5 ⊢ ((𝐹:𝐵⟶ℝ^* ∧ 𝐵 ∈ V ∧ ℝ^* ∈ V) → 𝐹 ∈ V)
13	6, 9, 11, 12	syl3anc 1368	. . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐹 ∈ V)
14		limsupcl 14822	. . . 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 1135	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐴 ∈ ℝ^)
17		xrltnle 10697	. . 3 ⊢ (((lim sup‘𝐹) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((lim sup‘𝐹) < 𝐴 ↔ ¬ 𝐴 ≤ (lim sup‘𝐹)))
18	15, 16, 17	syl2anc 587	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((lim sup‘𝐹) < 𝐴 ↔ ¬ 𝐴 ≤ (lim sup‘𝐹)))
19	1	limsupgf 14824	. . . . 5 ⊢ 𝐺:ℝ⟶ℝ^*
20	19	ffvelrni 6827	. . . 4 ⊢ (𝑗 ∈ ℝ → (𝐺‘𝑗) ∈ ℝ^*)
21		xrltnle 10697	. . . 4 ⊢ (((𝐺‘𝑗) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((𝐺‘𝑗) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐺‘𝑗)))
22	20, 16, 21	syl2anr 599	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ 𝑗 ∈ ℝ) → ((𝐺‘𝑗) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐺‘𝑗)))
23	22	rexbidva 3255	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → (∃𝑗 ∈ ℝ (𝐺‘𝑗) < 𝐴 ↔ ∃𝑗 ∈ ℝ ¬ 𝐴 ≤ (𝐺‘𝑗)))
24	5, 18, 23	3bitr4d 314	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 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 “ cima 5522 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℝcr 10525 +∞cpnf 10661 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 [,)cico 12728 lim supclsp 14819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-limsup 14820

This theorem is referenced by: limsupgre 14830 limsuplt2 42395

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