Theorem zssinfcl 10592

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 9700	. . . 4 |- ( ph -> inf ( B , RR , < ) e. RR )
3		1red 8289	. . . 4 |- ( ph -> 1 e. RR )
4	2, 3	readdcld 8303	. . 3 |- ( ph -> (inf ( B , RR , < ) + 1 ) e. RR )
5	2	ltp1d 9204	. . 3 |- ( ph -> inf ( B , RR , < ) < (inf ( B , RR , < ) + 1 ) )
6		lttri3 8353	. . . . 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 7316	. . 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 3239	. . . . . 6 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. ZZ )
16	15	zred 9700	. . . . 5 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> z e. RR )
17	7, 8	inflbti 7315	. . . . . . 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 8394	. . . 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 9633	. . . . . 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 8389	. . . 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 2309	. 2 |- ( ( ph /\ ( z e. B /\ z < (inf ( B , RR , < ) + 1 ) ) ) -> inf ( B , RR , < ) e. B )
29	10, 28	rexlimddv 2665	1 |- ( ph -> inf ( B , RR , < ) e. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   C_ wss 3211   class class class wbr 4109  (class class class)co 6050  infcinf 7274   RRcr 8126   1c1 8128   + caddc 8130   < clt 8308   <_ cle 8309   ZZcz 9577

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578

This theorem is referenced by: (None)