Theorem zssinfcl 10643

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 9747	. . . 4 |- ( ph -> inf ( B , RR , < ) e. RR )
3		1red 8331	. . . 4 |- ( ph -> 1 e. RR )
4	2, 3	readdcld 8345	. . 3 |- ( ph -> (inf ( B , RR , < ) + 1 ) e. RR )
5	2	ltp1d 9250	. . 3 |- ( ph -> inf ( B , RR , < ) < (inf ( B , RR , < ) + 1 ) )
6		lttri3 8395	. . . . 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 7355	. . 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 437	. 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 535	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. B )
15	13, 14	sseldd 3249	. . . . . 6 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. ZZ )
16	15	zred 9747	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. RR )
17	7, 8	inflbti 7354	. . . . . . 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 483	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> -. z < inf ( B , RR , < ) )
20	11, 16, 19	nltled 8437	. . . 4 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) <_ z )
21		simprr 537	. . . . 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 9679	. . . . . 6 |- ( ( z e. ZZ /\ inf ( B , RR , < ) e. ZZ ) -> ( z <_ inf ( B , RR , < ) <-> z < (inf ( B , RR , < ) + 1 ) ) )
24	15, 22, 23	syl2anc 415	. . . . 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 8431	. . . 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 957	. . 3 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) = z )
28	27, 14	eqeltrd 2315	. 2 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) e. B )
29	10, 28	rexlimddv 2673	1 |- ( ph -> inf ( B , RR , < ) e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   C_ wss 3220   class class class wbr 4125  (class class class)co 6075  infcinf 7313   RRcr 8168   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   ZZcz 9623

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624

This theorem is referenced by: (None)