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Theorem limsuple 15431

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 11120	. . . . . . 7 ⊢ ℝ ∈ V
3	2	ssex 5258	. . . . . 6 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V)
4	3	3ad2ant1 1134	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^*) → 𝐵 ∈ V)
5		xrex 12928	. . . . . 6 ⊢ ℝ^* ∈ V
6	5	a1i 11	. . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → ℝ^ ∈ V)
7		fex2 7880	. . . . 5 ⊢ ((𝐹:𝐵⟶ℝ^* ∧ 𝐵 ∈ V ∧ ℝ^* ∈ V) → 𝐹 ∈ V)
8	1, 4, 6, 7	syl3anc 1374	. . . 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 15427	. . . 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 5098	. 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ (lim sup‘𝐹) ↔ 𝐴 ≤ inf(ran 𝐺, ℝ^, < )))
13	9	limsupgf 15428	. . . . 5 ⊢ 𝐺:ℝ⟶ℝ^*
14		frn 6669	. . . . 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 13279	. . . 4 ⊢ ((ran 𝐺 ⊆ ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ inf(ran 𝐺, ℝ^, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥))
18	15, 16, 17	sylancr 588	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ inf(ran 𝐺, ℝ^, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥))
19		ffn 6662	. . . . 5 ⊢ (𝐺:ℝ⟶ℝ^* → 𝐺 Fn ℝ)
20	13, 19	ax-mp 5	. . . 4 ⊢ 𝐺 Fn ℝ
21		breq2 5090	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑗) → (𝐴 ≤ 𝑥 ↔ 𝐴 ≤ (𝐺‘𝑗)))
22	21	ralrn 7034	. . . 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 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ran crn 5625 “ cima 5627 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 supcsup 9346 infcinf 9347 ℝcr 11028 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 [,)cico 13291 lim supclsp 15423

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 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9348 df-inf 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-limsup 15424

This theorem is referenced by: limsuplt 15432 limsupbnd1 15435 limsupbnd2 15436 mbflimsup 25643 limsupge 46207

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