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Theorem limsuplt 15414

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 15413	. . . 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 3090	. . 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 1138	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐹:𝐵⟶ℝ^)
7		reex 11129	. . . . . . 7 ⊢ ℝ ∈ V
8	7	ssex 5268	. . . . . 6 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
9	8	3ad2ant1 1134	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐵 ∈ V)
10		xrex 12912	. . . . . 6 ⊢ ℝ^* ∈ V
11	10	a1i 11	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → ℝ^ ∈ V)
12		fex2 7888	. . . . 5 ⊢ ((𝐹:𝐵⟶ℝ^* ∧ 𝐵 ∈ V ∧ ℝ^* ∈ V) → 𝐹 ∈ V)
13	6, 9, 11, 12	syl3anc 1374	. . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐹 ∈ V)
14		limsupcl 15408	. . . 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 1139	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → 𝐴 ∈ ℝ^)
17		xrltnle 11211	. . 3 ⊢ (((lim sup‘𝐹) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((lim sup‘𝐹) < 𝐴 ↔ ¬ 𝐴 ≤ (lim sup‘𝐹)))
18	15, 16, 17	syl2anc 585	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((lim sup‘𝐹) < 𝐴 ↔ ¬ 𝐴 ≤ (lim sup‘𝐹)))
19	1	limsupgf 15410	. . . . 5 ⊢ 𝐺:ℝ⟶ℝ^*
20	19	ffvelcdmi 7037	. . . 4 ⊢ (𝑗 ∈ ℝ → (𝐺‘𝑗) ∈ ℝ^*)
21		xrltnle 11211	. . . 4 ⊢ (((𝐺‘𝑗) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((𝐺‘𝑗) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐺‘𝑗)))
22	20, 16, 21	syl2anr 598	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ 𝑗 ∈ ℝ) → ((𝐺‘𝑗) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐺‘𝑗)))
23	22	rexbidva 3160	. 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 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3442 ∩ cin 3902 ⊆ wss 3903 class class class wbr 5100 ↦ cmpt 5181 “ cima 5635 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 supcsup 9355 ℝcr 11037 +∞cpnf 11175 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 [,)cico 13275 lim supclsp 15405

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-limsup 15406

This theorem is referenced by: limsupgre 15416 limsuplt2 46105

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