Theorem uzinico 39787

Description: An upper interval of integers is the intersection of the integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
uzinico.1	|- ( ph -> M e. ZZ )
uzinico.2	|- Z = ( ZZ>= ` M )

Assertion

Ref	Expression
uzinico	|- ( ph -> Z = ( ZZ i^i ( M [,) +oo ) ) )

Proof of Theorem uzinico

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzinico.2	. . . . . . . 8 |- Z = ( ZZ>= ` M )
2	1	eluzelz2 39627	. . . . . . 7 |- ( k e. Z -> k e. ZZ )
3	2	adantl 482	. . . . . 6 |- ( ( ph /\ k e. Z ) -> k e. ZZ )
4		uzinico.1	. . . . . . . . . 10 |- ( ph -> M e. ZZ )
5	4	zred 11482	. . . . . . . . 9 |- ( ph -> M e. RR )
6	5	rexrd 10089	. . . . . . . 8 |- ( ph -> M e. RR* )
7	6	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> M e. RR* )
8		pnfxr 10092	. . . . . . . 8 |- +oo e. RR*
9	8	a1i 11	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> +oo e. RR* )
10		zssre 11384	. . . . . . . . . 10 |- ZZ C_ RR
11		ressxr 10083	. . . . . . . . . 10 |- RR C_ RR*
12	10, 11	sstri 3612	. . . . . . . . 9 |- ZZ C_ RR*
13	12, 2	sseldi 3601	. . . . . . . 8 |- ( k e. Z -> k e. RR* )
14	13	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> k e. RR* )
15	1	eleq2i 2693	. . . . . . . . . 10 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
16	15	biimpi 206	. . . . . . . . 9 |- ( k e. Z -> k e. ( ZZ>= ` M ) )
17		eluzle 11700	. . . . . . . . 9 |- ( k e. ( ZZ>= ` M ) -> M <_ k )
18	16, 17	syl 17	. . . . . . . 8 |- ( k e. Z -> M <_ k )
19	18	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> M <_ k )
20	10, 2	sseldi 3601	. . . . . . . . 9 |- ( k e. Z -> k e. RR )
21	20	ltpnfd 11955	. . . . . . . 8 |- ( k e. Z -> k < +oo )
22	21	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> k < +oo )
23	7, 9, 14, 19, 22	elicod 12224	. . . . . 6 |- ( ( ph /\ k e. Z ) -> k e. ( M [,) +oo ) )
24	3, 23	elind 3798	. . . . 5 |- ( ( ph /\ k e. Z ) -> k e. ( ZZ i^i ( M [,) +oo ) ) )
25	24	ex 450	. . . 4 |- ( ph -> ( k e. Z -> k e. ( ZZ i^i ( M [,) +oo ) ) ) )
26	4	adantr 481	. . . . . 6 |- ( ( ph /\ k e. ( ZZ i^i ( M [,) +oo ) ) ) -> M e. ZZ )
27		elinel1 3799	. . . . . . 7 |- ( k e. ( ZZ i^i ( M [,) +oo ) ) -> k e. ZZ )
28	27	adantl 482	. . . . . 6 |- ( ( ph /\ k e. ( ZZ i^i ( M [,) +oo ) ) ) -> k e. ZZ )
29		elinel2 3800	. . . . . . . 8 |- ( k e. ( ZZ i^i ( M [,) +oo ) ) -> k e. ( M [,) +oo ) )
30	29	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ i^i ( M [,) +oo ) ) ) -> k e. ( M [,) +oo ) )
31	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. ( M [,) +oo ) ) -> M e. RR* )
32	8	a1i 11	. . . . . . . 8 |- ( ( ph /\ k e. ( M [,) +oo ) ) -> +oo e. RR* )
33		simpr 477	. . . . . . . 8 |- ( ( ph /\ k e. ( M [,) +oo ) ) -> k e. ( M [,) +oo ) )
34	31, 32, 33	icogelbd 39785	. . . . . . 7 |- ( ( ph /\ k e. ( M [,) +oo ) ) -> M <_ k )
35	30, 34	syldan 487	. . . . . 6 |- ( ( ph /\ k e. ( ZZ i^i ( M [,) +oo ) ) ) -> M <_ k )
36	1, 26, 28, 35	eluzd 39635	. . . . 5 |- ( ( ph /\ k e. ( ZZ i^i ( M [,) +oo ) ) ) -> k e. Z )
37	36	ex 450	. . . 4 |- ( ph -> ( k e. ( ZZ i^i ( M [,) +oo ) ) -> k e. Z ) )
38	25, 37	impbid 202	. . 3 |- ( ph -> ( k e. Z <-> k e. ( ZZ i^i ( M [,) +oo ) ) ) )
39	38	alrimiv 1855	. 2 |- ( ph -> A. k ( k e. Z <-> k e. ( ZZ i^i ( M [,) +oo ) ) ) )
40		dfcleq 2616	. 2 |- ( Z = ( ZZ i^i ( M [,) +oo ) ) <-> A. k ( k e. Z <-> k e. ( ZZ i^i ( M [,) +oo ) ) ) )
41	39, 40	sylibr 224	1 |- ( ph -> Z = ( ZZ i^i ( M [,) +oo ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   i^i cin 3573   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   [,)cico 12177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pnf 10076  df-xr 10078  df-ltxr 10079  df-neg 10269  df-z 11378  df-uz 11688  df-ico 12181

This theorem is referenced by:  uzinico2  39789  limsupresuz  39935  liminfresuz  40016