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Theorem limsupgle 15421

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 15420	. . . 4 ⊢ (𝐶 ∈ ℝ → (𝐺‘𝐶) = sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ))
3	2	3ad2ant2 1135	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝐺‘𝐶) = sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ))
4	3	breq1d 5159	. 2 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝐺‘𝐶) ≤ 𝐴 ↔ sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝐴))
5		inss2 4230	. . 3 ⊢ ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^) ⊆ ℝ^
6		simp3 1139	. . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → 𝐴 ∈ ℝ^*)
7		supxrleub 13305	. . 3 ⊢ ((((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^) ⊆ ℝ^ ∧ 𝐴 ∈ ℝ^) → (sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^)𝑥 ≤ 𝐴))
8	5, 6, 7	sylancr 588	. 2 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (sup(((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^*)𝑥 ≤ 𝐴))
9		imassrn 6071	. . . . . . 7 ⊢ (𝐹 “ (𝐶[,)+∞)) ⊆ ran 𝐹
10		simp1r 1199	. . . . . . . 8 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → 𝐹:𝐵⟶ℝ^*)
11	10	frnd 6726	. . . . . . 7 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ran 𝐹 ⊆ ℝ^*)
12	9, 11	sstrid 3994	. . . . . 6 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝐹 “ (𝐶[,)+∞)) ⊆ ℝ^*)
13		df-ss 3966	. . . . . 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 6234	. . . . 5 ⊢ (𝐹 “ dom (𝐹 ↾ (𝐶[,)+∞))) = (𝐹 “ (𝐶[,)+∞))
16	14, 15	eqtr4di 2791	. . . 4 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝐹 “ (𝐶[,)+∞)) ∩ ℝ^*) = (𝐹 “ dom (𝐹 ↾ (𝐶[,)+∞))))
17	16	raleqdv 3326	. . 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 6719	. . . 4 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → 𝐹 Fn 𝐵)
19	10	fdmd 6729	. . . . . . 7 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → dom 𝐹 = 𝐵)
20	19	ineq2d 4213	. . . . . 6 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝐶[,)+∞) ∩ dom 𝐹) = ((𝐶[,)+∞) ∩ 𝐵))
21		dmres 6004	. . . . . 6 ⊢ dom (𝐹 ↾ (𝐶[,)+∞)) = ((𝐶[,)+∞) ∩ dom 𝐹)
22		incom 4202	. . . . . 6 ⊢ (𝐵 ∩ (𝐶[,)+∞)) = ((𝐶[,)+∞) ∩ 𝐵)
23	20, 21, 22	3eqtr4g 2798	. . . . 5 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → dom (𝐹 ↾ (𝐶[,)+∞)) = (𝐵 ∩ (𝐶[,)+∞)))
24		inss1 4229	. . . . 5 ⊢ (𝐵 ∩ (𝐶[,)+∞)) ⊆ 𝐵
25	23, 24	eqsstrdi 4037	. . . 4 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → dom (𝐹 ↾ (𝐶[,)+∞)) ⊆ 𝐵)
26		breq1 5152	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑗) → (𝑥 ≤ 𝐴 ↔ (𝐹‘𝑗) ≤ 𝐴))
27	26	ralima 7240	. . . 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 585	. . 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 2820	. . . . . . . 8 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞)) ↔ 𝑗 ∈ (𝐵 ∩ (𝐶[,)+∞))))
30		elin 3965	. . . . . . . 8 ⊢ (𝑗 ∈ (𝐵 ∩ (𝐶[,)+∞)) ↔ (𝑗 ∈ 𝐵 ∧ 𝑗 ∈ (𝐶[,)+∞)))
31	29, 30	bitrdi 287	. . . . . . 7 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞)) ↔ (𝑗 ∈ 𝐵 ∧ 𝑗 ∈ (𝐶[,)+∞))))
32		simpl2 1193	. . . . . . . . 9 ⊢ ((((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ ℝ)
33		simp1l 1198	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → 𝐵 ⊆ ℝ)
34	33	sselda 3983	. . . . . . . . 9 ⊢ ((((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ ℝ)
35		elicopnf 13422	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℝ → (𝑗 ∈ (𝐶[,)+∞) ↔ (𝑗 ∈ ℝ ∧ 𝐶 ≤ 𝑗)))
36	35	baibd 541	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑗 ∈ (𝐶[,)+∞) ↔ 𝐶 ≤ 𝑗))
37	32, 34, 36	syl2anc 585	. . . . . . . 8 ⊢ ((((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) ∧ 𝑗 ∈ 𝐵) → (𝑗 ∈ (𝐶[,)+∞) ↔ 𝐶 ≤ 𝑗))
38	37	pm5.32da 580	. . . . . . 7 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → ((𝑗 ∈ 𝐵 ∧ 𝑗 ∈ (𝐶[,)+∞)) ↔ (𝑗 ∈ 𝐵 ∧ 𝐶 ≤ 𝑗)))
39	31, 38	bitrd 279	. . . . . 6 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ^) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ^) → (𝑗 ∈ dom (𝐹 ↾ (𝐶[,)+∞)) ↔ (𝑗 ∈ 𝐵 ∧ 𝐶 ≤ 𝑗)))
40	39	imbi1d 342	. . . . 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 452	. . . . 5 ⊢ (((𝑗 ∈ 𝐵 ∧ 𝐶 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑗 ∈ 𝐵 → (𝐶 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
42	40, 41	bitrdi 287	. . . 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 3174	. . 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 305	. 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 305	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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∩ cin 3948 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ran crn 5678 ↾ cres 5679 “ cima 5680 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 supcsup 9435 ℝcr 11109 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 [,)cico 13326

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-ico 13330

This theorem is referenced by: limsupgre 15425 limsupbnd1 15426 limsupbnd2 15427 mbflimsup 25183

W3C validator