Theorem infssuzcldc 10325

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 10323	. . 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 3268	. . . . . . 7 |- { n e. ( ZZ>= ` M ) | ps } C_ ( ZZ>= ` M )
7	2, 6	eqsstri 3215	. . . . . 6 |- S C_ ( ZZ>= ` M )
8		uzssz 9621	. . . . . 6 |- ( ZZ>= ` M ) C_ ZZ
9	7, 8	sstri 3192	. . . . 5 |- S C_ ZZ
10		zssre 9333	. . . . 5 |- ZZ C_ RR
11	9, 10	sstri 3192	. . . 4 |- S C_ RR
12	11	a1i 9	. . 3 |- ( ph -> S C_ RR )
13	5, 12	infrenegsupex 9668	. 2 |- ( ph -> inf ( S , RR , < ) = -u sup ( { w e. RR | -u w e. S } , RR , < ) )
14	1, 2, 3, 4	infssuzex 10323	. . . . . 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 9670	. . . . 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 8219	. . . . . . . . . 10 |- ( w = u -> -u w = -u u )
17	16	eleq1d 2265	. . . . . . . . 9 |- ( w = u -> ( -u w e. S <-> -u u e. S ) )
18	17	elrab 2920	. . . . . . . 8 |- ( u e. { w e. RR | -u w e. S } <-> ( u e. RR /\ -u u e. S ) )
19	9	sseli 3179	. . . . . . . . . 10 |- ( -u u e. S -> -u u e. ZZ )
20	19	adantl 277	. . . . . . . . 9 |- ( ( u e. RR /\ -u u e. S ) -> -u u e. ZZ )
21		simpl 109	. . . . . . . . . . 11 |- ( ( u e. RR /\ -u u e. S ) -> u e. RR )
22	21	recnd 8055	. . . . . . . . . 10 |- ( ( u e. RR /\ -u u e. S ) -> u e. CC )
23		znegclb 9359	. . . . . . . . . 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 167	. . . . . . . 8 |- ( ( u e. RR /\ -u u e. S ) -> u e. ZZ )
26	18, 25	sylbi 121	. . . . . . 7 |- ( u e. { w e. RR | -u w e. S } -> u e. ZZ )
27	26	ssriv 3187	. . . . . 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 9424	. . . 4 |- ( ph -> sup ( { w e. RR | -u w e. S } , RR , < ) e. { w e. RR | -u w e. S } )
30		nfrab1 2677	. . . . . 6 |- F/_ w { w e. RR | -u w e. S }
31		nfcv 2339	. . . . . 6 |- F/_ w RR
32		nfcv 2339	. . . . . 6 |- F/_ w <
33	30, 31, 32	nfsup 7058	. . . . 5 |- F/_ w sup ( { w e. RR | -u w e. S } , RR , < )
34	33	nfneg 8223	. . . . . 6 |- F/_ w -u sup ( { w e. RR | -u w e. S } , RR , < )
35	34	nfel1 2350	. . . . 5 |- F/ w -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S
36		negeq 8219	. . . . . 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 2265	. . . . 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 2918	. . . 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 122	. . 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 114	. 2 |- ( ph -> -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S )
41	13, 40	eqeltrd 2273	1 |- ( ph -> inf ( S , RR , < ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   = wceq 1364   e. wcel 2167   {crab 2479   C_ wss 3157   ` cfv 5258  (class class class)co 5922   supcsup 7048  infcinf 7049   CCcc 7877   RRcr 7878   < clt 8061   -ucneg 8198   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218

This theorem is referenced by:  zsupssdc  10328  bitsfzolem  12118  nnmindc  12201  nninfctlemfo  12207  lcmval  12231  lcmcllem  12235  odzcllem  12411  4sqlem13m  12572  4sqlem14  12573  4sqlem17  12576  4sqlem18  12577