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Theorem limsupgle 15423

Description: The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)

**Hypothesis**
Ref	Expression
limsupval.1	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))

**Assertion**
Ref	Expression
limsupgle	⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝐺‘𝐶) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝐶 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))

Distinct variable groups: 𝑘,𝐹 𝐴,𝑗 𝐵,𝑗 𝐶,𝑗,𝑘 𝑗,𝐹 Allowed substitution hints: 𝐴(𝑘) 𝐵(𝑘) 𝐺(𝑗,𝑘)

Proof of Theorem limsupgle Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupval.1	. . . . 5 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
2	1	limsupgval 15422	. . . 4 ⊢ (𝐶 ∈ ℝ → (𝐺‘𝐶) = sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ))
3	2	3ad2ant2 1134	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝐺‘𝐶) = sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ))
4	3	breq1d 5158	. 2 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝐺‘𝐶) ≤ 𝐴 ↔ sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝐴))
5		inss2 4229	. . 3 ⊢ ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^) ⊆ ℝ^
6		simp3 1138	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → 𝐴 ∈ ℝ^*)
7		supxrleub 13307	. . 3 ⊢ ((((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^) ⊆ ℝ^ ∧ 𝐴 ∈ ℝ^) → (sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^)𝑥 ≤ 𝐴))
8	5, 6, 7	sylancr 587	. 2 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^*)𝑥 ≤ 𝐴))
9		imassrn 6070	. . . . . . 7 ⊢ (𝐹 “ (𝐶[,)+∞)) ⊆ ran 𝐹
10		simp1r 1198	. . . . . . . 8 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → 𝐹:𝐵⟶ℝ^*)
11	10	frnd 6725	. . . . . . 7 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ran 𝐹 ⊆ ℝ^*)
12	9, 11	sstrid 3993	. . . . . 6 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝐹 “ (𝐶[,)+∞)) ⊆ ℝ^*)
13		df-ss 3965	. . . . . 6 ⊢ ((𝐹 “ (𝐶[,)+∞)) ⊆ ℝ^* ↔ ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^*) = (𝐹 “ (𝐶[,)+∞)))
14	12, 13	sylib 217	. . . . 5 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^*) = (𝐹 “ (𝐶[,)+∞)))
15		imadmres 6233	. . . . 5 ⊢ (𝐹 “ dom (𝐹 ↾ (𝐶[,)+∞))) = (𝐹 “ (𝐶[,)+∞))
16	14, 15	eqtr4di 2790	. . . 4 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^*) = (𝐹 “ dom (𝐹 ↾ (𝐶[,)+∞))))
17	16	raleqdv 3325	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (∀𝑥 ∈ ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^*)𝑥 ≤ 𝐴 ↔ ∀𝑥 ∈ (𝐹 “ dom (𝐹 ↾ (𝐶[,)+∞)))𝑥 ≤ 𝐴))
18	10	ffnd 6718	. . . 4 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → 𝐹 Fn 𝐵)
19	10	fdmd 6728	. . . . . . 7 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → dom 𝐹 = 𝐵)
20	19	ineq2d 4212	. . . . . 6 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝐶[,)+∞) ∩ dom 𝐹) = ((𝐶[,)+∞) ∩ 𝐵))
21		dmres 6003	. . . . . 6 ⊢ dom (𝐹 ↾ (𝐶[,)+∞)) = ((𝐶[,)+∞) ∩ dom 𝐹)
22		incom 4201	. . . . . 6 ⊢ (𝐵 ∩ (𝐶[,)+∞)) = ((𝐶[,)+∞) ∩ 𝐵)
23	20, 21, 22	3eqtr4g 2797	. . . . 5 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → dom (𝐹 ↾ (𝐶[,)+∞)) = (𝐵 ∩ (𝐶[,)+∞)))
24		inss1 4228	. . . . 5 ⊢ (𝐵 ∩ (𝐶[,)+∞)) ⊆ 𝐵
25	23, 24	eqsstrdi 4036	. . . 4 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → dom (𝐹 ↾ (𝐶[,)+∞)) ⊆ 𝐵)
26		breq1 5151	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑗) → (𝑥 ≤ 𝐴 ↔ (𝐹‘𝑗) ≤ 𝐴))
27	26	ralima 7242	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ dom (𝐹 ↾ (𝐶[,)+∞)) ⊆ 𝐵) → (∀𝑥 ∈ (𝐹 “ dom (𝐹 ↾ (𝐶[,)+∞)))𝑥 ≤ 𝐴 ↔ ∀𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞))(𝐹‘𝑗) ≤ 𝐴))
28	18, 25, 27	syl2anc 584	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (∀𝑥 ∈ (𝐹 “ dom (𝐹 ↾ (𝐶[,)+∞)))𝑥 ≤ 𝐴 ↔ ∀𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞))(𝐹‘𝑗) ≤ 𝐴))
29	23	eleq2d 2819	. . . . . . . 8 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞)) ↔ 𝑗 ∈ (𝐵 ∩ (𝐶[,)+∞))))
30		elin 3964	. . . . . . . 8 ⊢ (𝑗 ∈ (𝐵 ∩ (𝐶[,)+∞)) ↔ (𝑗 ∈ 𝐵 ∧ 𝑗 ∈ (𝐶[,)+∞)))
31	29, 30	bitrdi 286	. . . . . . 7 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞)) ↔ (𝑗 ∈ 𝐵 ∧ 𝑗 ∈ (𝐶[,)+∞))))
32		simpl2 1192	. . . . . . . . 9 ⊢ ((((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ ℝ)
33		simp1l 1197	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → 𝐵 ⊆ ℝ)
34	33	sselda 3982	. . . . . . . . 9 ⊢ ((((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ ℝ)
35		elicopnf 13424	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℝ → (𝑗 ∈ (𝐶[,)+∞) ↔ (𝑗 ∈ ℝ ∧ 𝐶 ≤ 𝑗)))
36	35	baibd 540	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑗 ∈ (𝐶[,)+∞) ↔ 𝐶 ≤ 𝑗))
37	32, 34, 36	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) ∧ 𝑗 ∈ 𝐵) → (𝑗 ∈ (𝐶[,)+∞) ↔ 𝐶 ≤ 𝑗))
38	37	pm5.32da 579	. . . . . . 7 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝑗 ∈ 𝐵 ∧ 𝑗 ∈ (𝐶[,)+∞)) ↔ (𝑗 ∈ 𝐵 ∧ 𝐶 ≤ 𝑗)))
39	31, 38	bitrd 278	. . . . . 6 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞)) ↔ (𝑗 ∈ 𝐵 ∧ 𝐶 ≤ 𝑗)))
40	39	imbi1d 341	. . . . 5 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞)) → (𝐹‘𝑗) ≤ 𝐴) ↔ ((𝑗 ∈ 𝐵 ∧ 𝐶 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝐴)))
41		impexp 451	. . . . 5 ⊢ (((𝑗 ∈ 𝐵 ∧ 𝐶 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑗 ∈ 𝐵 → (𝐶 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
42	40, 41	bitrdi 286	. . . 4 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞)) → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑗 ∈ 𝐵 → (𝐶 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))))
43	42	ralbidv2 3173	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (∀𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞))(𝐹‘𝑗) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝐶 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
44	17, 28, 43	3bitrd 304	. 2 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (∀𝑥 ∈ ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^*)𝑥 ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝐶 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
45	4, 8, 44	3bitrd 304	1 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝐺‘𝐶) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝐶 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∩ cin 3947 ⊆ wss 3948 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 ↾ cres 5678 “ cima 5679 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 supcsup 9437 ℝcr 11111 +∞cpnf 11247 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 [,)cico 13328

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-ico 13332

This theorem is referenced by: limsupgre 15427 limsupbnd1 15428 limsupbnd2 15429 mbflimsup 25190

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