Theorem zssinfcl 10322

Description: The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.)

Hypotheses

Ref	Expression
zssinfcl.ex	|- ( ph -> E. x e. RR ( A. y e. B -. y < x /\ A. y e. RR ( x < y -> E. z e. B z < y ) ) )
zssinfcl.ss	|- ( ph -> B C_ ZZ )
zssinfcl.zz	|- ( ph -> inf ( B , RR , < ) e. ZZ )

Assertion

Ref	Expression
zssinfcl	|- ( ph -> inf ( B , RR , < ) e. B )

Distinct variable groups:   x, B, y, z   ph, x, y, z

Proof of Theorem zssinfcl

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zssinfcl.zz	. . . . 5 |- ( ph -> inf ( B , RR , < ) e. ZZ )
2	1	zred 9448	. . . 4 |- ( ph -> inf ( B , RR , < ) e. RR )
3		1red 8041	. . . 4 |- ( ph -> 1 e. RR )
4	2, 3	readdcld 8056	. . 3 |- ( ph -> (inf ( B , RR , < ) + 1 ) e. RR )
5	2	ltp1d 8957	. . 3 |- ( ph -> inf ( B , RR , < ) < (inf ( B , RR , < ) + 1 ) )
6		lttri3 8106	. . . . 5 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
7	6	adantl 277	. . . 4 |- ( ( ph /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
8		zssinfcl.ex	. . . 4 |- ( ph -> E. x e. RR ( A. y e. B -. y < x /\ A. y e. RR ( x < y -> E. z e. B z < y ) ) )
9	7, 8	infglbti 7091	. . 3 |- ( ph -> ( ( (inf ( B , RR , < ) + 1 ) e. RR /\ inf ( B , RR , < ) < (inf ( B , RR , < ) + 1 ) ) -> E. z e. B z < (inf ( B , RR , < ) + 1 ) ) )
10	4, 5, 9	mp2and 433	. 2 |- ( ph -> E. z e. B z < (inf ( B , RR , < ) + 1 ) )
11	2	adantr 276	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) e. RR )
12		zssinfcl.ss	. . . . . . . 8 |- ( ph -> B C_ ZZ )
13	12	adantr 276	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> B C_ ZZ )
14		simprl 529	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. B )
15	13, 14	sseldd 3184	. . . . . 6 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. ZZ )
16	15	zred 9448	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. RR )
17	7, 8	inflbti 7090	. . . . . . 7 |- ( ph -> ( z e. B -> -. z < inf ( B , RR , < ) ) )
18	17	imp 124	. . . . . 6 |- ( ( ph /\ z e. B ) -> -. z < inf ( B , RR , < ) )
19	18	adantrr 479	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> -. z < inf ( B , RR , < ) )
20	11, 16, 19	nltled 8147	. . . 4 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) <_ z )
21		simprr 531	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z < (inf ( B , RR , < ) + 1 ) )
22	1	adantr 276	. . . . . 6 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) e. ZZ )
23		zleltp1 9381	. . . . . 6 |- ( ( z e. ZZ /\ inf ( B , RR , < ) e. ZZ ) -> ( z <_ inf ( B , RR , < ) <-> z < (inf ( B , RR , < ) + 1 ) ) )
24	15, 22, 23	syl2anc 411	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> ( z <_ inf ( B , RR , < ) <-> z < (inf ( B , RR , < ) + 1 ) ) )
25	21, 24	mpbird 167	. . . 4 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z <_ inf ( B , RR , < ) )
26	11, 16	letri3d 8142	. . . 4 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> (inf ( B , RR , < ) = z <-> (inf ( B , RR , < ) <_ z /\ z <_ inf ( B , RR , < ) ) ) )
27	20, 25, 26	mpbir2and 946	. . 3 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) = z )
28	27, 14	eqeltrd 2273	. 2 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) e. B )
29	10, 28	rexlimddv 2619	1 |- ( ph -> inf ( B , RR , < ) e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   C_ wss 3157   class class class wbr 4033  (class class class)co 5922  infcinf 7049   RRcr 7878   1c1 7880   + caddc 7882   < clt 8061   <_ cle 8062   ZZcz 9326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327

This theorem is referenced by: (None)