Theorem infssuzcldc 10599

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 10597	. . 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 3325	. . . . . . 7 |- { n e. ( ZZ>= ` M ) | ps } C_ ( ZZ>= ` M )
7	2, 6	eqsstri 3272	. . . . . 6 |- S C_ ( ZZ>= ` M )
8		uzssz 9877	. . . . . 6 |- ( ZZ>= ` M ) C_ ZZ
9	7, 8	sstri 3249	. . . . 5 |- S C_ ZZ
10		zssre 9586	. . . . 5 |- ZZ C_ RR
11	9, 10	sstri 3249	. . . 4 |- S C_ RR
12	11	a1i 9	. . 3 |- ( ph -> S C_ RR )
13	5, 12	infrenegsupex 9929	. 2 |- ( ph -> inf ( S , RR , < ) = -u sup ( { w e. RR | -u w e. S } , RR , < ) )
14	1, 2, 3, 4	infssuzex 10597	. . . . . 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 9931	. . . . 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 8468	. . . . . . . . . 10 |- ( w = u -> -u w = -u u )
17	16	eleq1d 2303	. . . . . . . . 9 |- ( w = u -> ( -u w e. S <-> -u u e. S ) )
18	17	elrab 2975	. . . . . . . 8 |- ( u e. { w e. RR | -u w e. S } <-> ( u e. RR /\ -u u e. S ) )
19	9	sseli 3236	. . . . . . . . . 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 8304	. . . . . . . . . 10 |- ( ( u e. RR /\ -u u e. S ) -> u e. CC )
23		znegclb 9612	. . . . . . . . . 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 3244	. . . . . 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 9679	. . . 4 |- ( ph -> sup ( { w e. RR | -u w e. S } , RR , < ) e. { w e. RR | -u w e. S } )
30		nfrab1 2726	. . . . . 6 |- F/_ w { w e. RR | -u w e. S }
31		nfcv 2386	. . . . . 6 |- F/_ w RR
32		nfcv 2386	. . . . . 6 |- F/_ w <
33	30, 31, 32	nfsup 7285	. . . . 5 |- F/_ w sup ( { w e. RR | -u w e. S } , RR , < )
34	33	nfneg 8472	. . . . . 6 |- F/_ w -u sup ( { w e. RR | -u w e. S } , RR , < )
35	34	nfel1 2397	. . . . 5 |- F/ w -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S
36		negeq 8468	. . . . . 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 2303	. . . . 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 2973	. . . 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 2311	1 |- ( ph -> inf ( S , RR , < ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2205   {crab 2526   C_ wss 3213   ` cfv 5354  (class class class)co 6052   supcsup 7275  infcinf 7276   CCcc 8127   RRcr 8128   < clt 8310   -ucneg 8447   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345

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 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-sup 7277  df-inf 7278  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346  df-fzo 10481

This theorem is referenced by:  zsupssdc  10602  bitsfzolem  12644  nnmindc  12734  nninfctlemfo  12740  lcmval  12764  lcmcllem  12768  odzcllem  12944  4sqlem13m  13105  4sqlem14  13106  4sqlem17  13109  4sqlem18  13110