Theorem zssinfcl 10491

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 9601	. . . 4 |- ( ph -> inf ( B , RR , < ) e. RR )
3		1red 8193	. . . 4 |- ( ph -> 1 e. RR )
4	2, 3	readdcld 8208	. . 3 |- ( ph -> (inf ( B , RR , < ) + 1 ) e. RR )
5	2	ltp1d 9109	. . 3 |- ( ph -> inf ( B , RR , < ) < (inf ( B , RR , < ) + 1 ) )
6		lttri3 8258	. . . . 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 7223	. . 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 531	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. B )
15	13, 14	sseldd 3228	. . . . . 6 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. ZZ )
16	15	zred 9601	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. RR )
17	7, 8	inflbti 7222	. . . . . . 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 8299	. . . 4 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) <_ z )
21		simprr 533	. . . . 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 9534	. . . . . 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 8294	. . . 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 952	. . 3 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) = z )
28	27, 14	eqeltrd 2308	. 2 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) e. B )
29	10, 28	rexlimddv 2655	1 |- ( ph -> inf ( B , RR , < ) e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   C_ wss 3200   class class class wbr 4088  (class class class)co 6017  infcinf 7181   RRcr 8030   1c1 8032   + caddc 8034   < clt 8213   <_ cle 8214   ZZcz 9478

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479

This theorem is referenced by: (None)