Theorem zssinfcl 10614

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 9718	. . . 4 |- ( ph -> inf ( B , RR , < ) e. RR )
3		1red 8305	. . . 4 |- ( ph -> 1 e. RR )
4	2, 3	readdcld 8319	. . 3 |- ( ph -> (inf ( B , RR , < ) + 1 ) e. RR )
5	2	ltp1d 9221	. . 3 |- ( ph -> inf ( B , RR , < ) < (inf ( B , RR , < ) + 1 ) )
6		lttri3 8369	. . . . 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 7329	. . 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 3243	. . . . . 6 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. ZZ )
16	15	zred 9718	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. RR )
17	7, 8	inflbti 7328	. . . . . . 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 8410	. . . 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 9650	. . . . . 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 8405	. . . 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 953	. . 3 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) = z )
28	27, 14	eqeltrd 2311	. 2 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) e. B )
29	10, 28	rexlimddv 2667	1 |- ( ph -> inf ( B , RR , < ) e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   C_ wss 3214   class class class wbr 4114  (class class class)co 6058  infcinf 7287   RRcr 8142   1c1 8144   + caddc 8146   < clt 8324   <_ cle 8325   ZZcz 9594

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595

This theorem is referenced by: (None)