Theorem infssuzcldc 11680

Description: The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.)

Hypotheses

Ref	Expression
infssuzledc.m	|- ( ph -> M e. ZZ )
infssuzledc.s	|- S = { n e. ( ZZ>= ` M ) | ps }
infssuzledc.a	|- ( ph -> A e. S )
infssuzledc.dc	|- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )

Assertion

Ref	Expression
infssuzcldc	|- ( ph -> inf ( S , RR , < ) e. S )

Distinct variable groups:   A, n   n, M   ph, n

Allowed substitution hints:   ps( n)   S( n)

Proof of Theorem infssuzcldc

Dummy variables y w x z u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infssuzledc.m	. . . 4 |- ( ph -> M e. ZZ )
2		infssuzledc.s	. . . 4 |- S = { n e. ( ZZ>= ` M ) | ps }
3		infssuzledc.a	. . . 4 |- ( ph -> A e. S )
4		infssuzledc.dc	. . . 4 |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )
5	1, 2, 3, 4	infssuzex 11678	. . 3 |- ( ph -> E. x e. RR ( A. y e. S -. y < x /\ A. y e. RR ( x < y -> E. w e. S w < y ) ) )
6		ssrab2 3187	. . . . . . 7 |- { n e. ( ZZ>= ` M ) | ps } C_ ( ZZ>= ` M )
7	2, 6	eqsstri 3134	. . . . . 6 |- S C_ ( ZZ>= ` M )
8		uzssz 9369	. . . . . 6 |- ( ZZ>= ` M ) C_ ZZ
9	7, 8	sstri 3111	. . . . 5 |- S C_ ZZ
10		zssre 9085	. . . . 5 |- ZZ C_ RR
11	9, 10	sstri 3111	. . . 4 |- S C_ RR
12	11	a1i 9	. . 3 |- ( ph -> S C_ RR )
13	5, 12	infrenegsupex 9416	. 2 |- ( ph -> inf ( S , RR , < ) = -u sup ( { w e. RR | -u w e. S } , RR , < ) )
14	1, 2, 3, 4	infssuzex 11678	. . . . . 6 |- ( ph -> E. x e. RR ( A. y e. S -. y < x /\ A. y e. RR ( x < y -> E. z e. S z < y ) ) )
15	14, 12	infsupneg 9418	. . . . 5 |- ( ph -> E. x e. RR ( A. y e. { w e. RR | -u w e. S } -. x < y /\ A. y e. RR ( y < x -> E. z e. { w e. RR | -u w e. S } y < z ) ) )
16		negeq 7979	. . . . . . . . . 10 |- ( w = u -> -u w = -u u )
17	16	eleq1d 2209	. . . . . . . . 9 |- ( w = u -> ( -u w e. S <-> -u u e. S ) )
18	17	elrab 2844	. . . . . . . 8 |- ( u e. { w e. RR | -u w e. S } <-> ( u e. RR /\ -u u e. S ) )
19	9	sseli 3098	. . . . . . . . . 10 |- ( -u u e. S -> -u u e. ZZ )
20	19	adantl 275	. . . . . . . . 9 |- ( ( u e. RR /\ -u u e. S ) -> -u u e. ZZ )
21		simpl 108	. . . . . . . . . . 11 |- ( ( u e. RR /\ -u u e. S ) -> u e. RR )
22	21	recnd 7818	. . . . . . . . . 10 |- ( ( u e. RR /\ -u u e. S ) -> u e. CC )
23		znegclb 9111	. . . . . . . . . 10 |- ( u e. CC -> ( u e. ZZ <-> -u u e. ZZ ) )
24	22, 23	syl 14	. . . . . . . . 9 |- ( ( u e. RR /\ -u u e. S ) -> ( u e. ZZ <-> -u u e. ZZ ) )
25	20, 24	mpbird 166	. . . . . . . 8 |- ( ( u e. RR /\ -u u e. S ) -> u e. ZZ )
26	18, 25	sylbi 120	. . . . . . 7 |- ( u e. { w e. RR | -u w e. S } -> u e. ZZ )
27	26	ssriv 3106	. . . . . 6 |- { w e. RR | -u w e. S } C_ ZZ
28	27	a1i 9	. . . . 5 |- ( ph -> { w e. RR | -u w e. S } C_ ZZ )
29	15, 28	suprzclex 9173	. . . 4 |- ( ph -> sup ( { w e. RR | -u w e. S } , RR , < ) e. { w e. RR | -u w e. S } )
30		nfrab1 2613	. . . . . 6 |- F/_ w { w e. RR | -u w e. S }
31		nfcv 2282	. . . . . 6 |- F/_ w RR
32		nfcv 2282	. . . . . 6 |- F/_ w <
33	30, 31, 32	nfsup 6887	. . . . 5 |- F/_ w sup ( { w e. RR | -u w e. S } , RR , < )
34	33	nfneg 7983	. . . . . 6 |- F/_ w -u sup ( { w e. RR | -u w e. S } , RR , < )
35	34	nfel1 2293	. . . . 5 |- F/ w -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S
36		negeq 7979	. . . . . 6 |- ( w = sup ( { w e. RR | -u w e. S } , RR , < ) -> -u w = -u sup ( { w e. RR | -u w e. S } , RR , < ) )
37	36	eleq1d 2209	. . . . 5 |- ( w = sup ( { w e. RR | -u w e. S } , RR , < ) -> ( -u w e. S <-> -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S ) )
38	33, 31, 35, 37	elrabf 2842	. . . 4 |- ( sup ( { w e. RR | -u w e. S } , RR , < ) e. { w e. RR | -u w e. S } <-> ( sup ( { w e. RR | -u w e. S } , RR , < ) e. RR /\ -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S ) )
39	29, 38	sylib 121	. . 3 |- ( ph -> ( sup ( { w e. RR | -u w e. S } , RR , < ) e. RR /\ -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S ) )
40	39	simprd 113	. 2 |- ( ph -> -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S )
41	13, 40	eqeltrd 2217	1 |- ( ph -> inf ( S , RR , < ) e. S )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 820   = wceq 1332   e. wcel 1481   {crab 2421   C_ wss 3076   ` cfv 5131  (class class class)co 5782   supcsup 6877  infcinf 6878   CCcc 7642   RRcr 7643   < clt 7824   -ucneg 7958   ZZcz 9078   ZZ>=cuz 9350   ...cfz 9821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-sup 6879  df-inf 6880  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-fz 9822  df-fzo 9951

This theorem is referenced by:  lcmval  11780  lcmcllem  11784