xlimpnfvlem1 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimpnfvlem1		Structured version Visualization version GIF version

Theorem xlimpnfvlem1 44542

Description: Lemma for xlimpnfv 44544: the "only if" part of the biconditional. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
xlimpnfvlem1.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
xlimpnfvlem1.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
xlimpnfvlem1.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
xlimpnfvlem1.c	⊢ (𝜑 → 𝐹~~>*+∞)
xlimpnfvlem1.x	⊢ (𝜑 → 𝑋 ∈ ℝ)

**Assertion**
Ref	Expression
xlimpnfvlem1	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘))

Distinct variable groups: 𝑗,𝐹,𝑘 𝑗,𝑀 𝑗,𝑋,𝑘 𝑗,𝑍,𝑘 𝜑,𝑗,𝑘 Allowed substitution hint: 𝑀(𝑘)

Proof of Theorem xlimpnfvlem1 Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iocpnfordt 22718	. . . . . 6 ⊢ (𝑋(,]+∞) ∈ (ordTop‘ ≤ )
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑋(,]+∞) ∈ (ordTop‘ ≤ ))
3		xlimpnfvlem1.c	. . . . . . . 8 ⊢ (𝜑 → 𝐹~~>*+∞)
4		df-xlim 44525	. . . . . . . . 9 ⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
5	4	breqi 5154	. . . . . . . 8 ⊢ (𝐹~~>*+∞ ↔ 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))+∞)
6	3, 5	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝐹(⇝_𝑡‘(ordTop‘ ≤ ))+∞)
7		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑘𝐹
8		letopon 22708	. . . . . . . . 9 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
9	8	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*))
10	7, 9	lmbr3 44453	. . . . . . 7 ⊢ (𝜑 → (𝐹(⇝_𝑡‘(ordTop‘ ≤ ))+∞ ↔ (𝐹 ∈ (ℝ^* ↑_pm ℂ) ∧ +∞ ∈ ℝ^* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))))
11	6, 10	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (ℝ^* ↑_pm ℂ) ∧ +∞ ∈ ℝ^* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))
12	11	simp3d 1144	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ (ordTop‘ ≤ )(+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
13	2, 12	jca 512	. . . 4 ⊢ (𝜑 → ((𝑋(,]+∞) ∈ (ordTop‘ ≤ ) ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))
14		xlimpnfvlem1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ)
15	14	rexrd 11263	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
16	11	simp2d 1143	. . . . 5 ⊢ (𝜑 → +∞ ∈ ℝ^*)
17	14	ltpnfd 13100	. . . . 5 ⊢ (𝜑 → 𝑋 < +∞)
18		ubioc1 13376	. . . . 5 ⊢ ((𝑋 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑋 < +∞) → +∞ ∈ (𝑋(,]+∞))
19	15, 16, 17, 18	syl3anc 1371	. . . 4 ⊢ (𝜑 → +∞ ∈ (𝑋(,]+∞))
20		eleq2 2822	. . . . . 6 ⊢ (𝑢 = (𝑋(,]+∞) → (+∞ ∈ 𝑢 ↔ +∞ ∈ (𝑋(,]+∞)))
21		eleq2 2822	. . . . . . . . 9 ⊢ (𝑢 = (𝑋(,]+∞) → ((𝐹‘𝑘) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ (𝑋(,]+∞)))
22	21	anbi2d 629	. . . . . . . 8 ⊢ (𝑢 = (𝑋(,]+∞) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))))
23	22	ralbidv 3177	. . . . . . 7 ⊢ (𝑢 = (𝑋(,]+∞) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))))
24	23	rexbidv 3178	. . . . . 6 ⊢ (𝑢 = (𝑋(,]+∞) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))))
25	20, 24	imbi12d 344	. . . . 5 ⊢ (𝑢 = (𝑋(,]+∞) → ((+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)) ↔ (+∞ ∈ (𝑋(,]+∞) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞)))))
26	25	rspcva 3610	. . . 4 ⊢ (((𝑋(,]+∞) ∈ (ordTop‘ ≤ ) ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(+∞ ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) → (+∞ ∈ (𝑋(,]+∞) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))))
27	13, 19, 26	sylc 65	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞)))
28		nfv 1917	. . . 4 ⊢ Ⅎ𝑗𝜑
29		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑘𝜑
30	15	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))) → 𝑋 ∈ ℝ^*)
31		xlimpnfvlem1.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
32	31	ffdmd 6748	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ^*)
33	32	ffvelcdmda 7086	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ dom 𝐹) → (𝐹‘𝑘) ∈ ℝ^*)
34	33	adantrr 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))) → (𝐹‘𝑘) ∈ ℝ^*)
35	16	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))) → +∞ ∈ ℝ^*)
36		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))) → (𝐹‘𝑘) ∈ (𝑋(,]+∞))
37	30, 35, 36	iocgtlbd 44274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))) → 𝑋 < (𝐹‘𝑘))
38	30, 34, 37	xrltled 13128	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞))) → 𝑋 ≤ (𝐹‘𝑘))
39	38	ex 413	. . . . . . 7 ⊢ (𝜑 → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞)) → 𝑋 ≤ (𝐹‘𝑘)))
40	39	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞)) → 𝑋 ≤ (𝐹‘𝑘)))
41	29, 40	ralimdaa 3257	. . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘)))
42	41	a1d 25	. . . 4 ⊢ (𝜑 → (𝑗 ∈ ℤ → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘))))
43	28, 42	reximdai 3258	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ (𝑋(,]+∞)) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘)))
44	27, 43	mpd 15	. 2 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘))
45		xlimpnfvlem1.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46		xlimpnfvlem1.z	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
47	46	rexuz3 15294	. . 3 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘)))
48	45, 47	syl 17	. 2 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘)))
49	44, 48	mpbird 256	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝑋 ≤ (𝐹‘𝑘))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 class class class wbr 5148 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ↑_pm cpm 8820 ℂcc 11107 ℝcr 11108 +∞cpnf 11244 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 ℤcz 12557 ℤ_≥cuz 12821 (,]cioc 13324 ordTopcordt 17444 TopOnctopon 22411 ⇝_𝑡clm 22729 ~~>*clsxlim 44524

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 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

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-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-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 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-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-1o 8465 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-z 12558 df-uz 12822 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-topgen 17388 df-ordt 17446 df-ps 18518 df-tsr 18519 df-top 22395 df-topon 22412 df-bases 22448 df-lm 22732 df-xlim 44525

This theorem is referenced by: xlimpnfv 44544

W3C validator