Theorem infssuzcldc 11954

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 11952	. . 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 3242	. . . . . . 7 |- { n e. ( ZZ>= ` M ) | ps } C_ ( ZZ>= ` M )
7	2, 6	eqsstri 3189	. . . . . 6 |- S C_ ( ZZ>= ` M )
8		uzssz 9549	. . . . . 6 |- ( ZZ>= ` M ) C_ ZZ
9	7, 8	sstri 3166	. . . . 5 |- S C_ ZZ
10		zssre 9262	. . . . 5 |- ZZ C_ RR
11	9, 10	sstri 3166	. . . 4 |- S C_ RR
12	11	a1i 9	. . 3 |- ( ph -> S C_ RR )
13	5, 12	infrenegsupex 9596	. 2 |- ( ph -> inf ( S , RR , < ) = -u sup ( { w e. RR | -u w e. S } , RR , < ) )
14	1, 2, 3, 4	infssuzex 11952	. . . . . 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 9598	. . . . 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 8152	. . . . . . . . . 10 |- ( w = u -> -u w = -u u )
17	16	eleq1d 2246	. . . . . . . . 9 |- ( w = u -> ( -u w e. S <-> -u u e. S ) )
18	17	elrab 2895	. . . . . . . 8 |- ( u e. { w e. RR | -u w e. S } <-> ( u e. RR /\ -u u e. S ) )
19	9	sseli 3153	. . . . . . . . . 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 7988	. . . . . . . . . 10 |- ( ( u e. RR /\ -u u e. S ) -> u e. CC )
23		znegclb 9288	. . . . . . . . . 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 3161	. . . . . 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 9353	. . . 4 |- ( ph -> sup ( { w e. RR | -u w e. S } , RR , < ) e. { w e. RR | -u w e. S } )
30		nfrab1 2657	. . . . . 6 |- F/_ w { w e. RR | -u w e. S }
31		nfcv 2319	. . . . . 6 |- F/_ w RR
32		nfcv 2319	. . . . . 6 |- F/_ w <
33	30, 31, 32	nfsup 6993	. . . . 5 |- F/_ w sup ( { w e. RR | -u w e. S } , RR , < )
34	33	nfneg 8156	. . . . . 6 |- F/_ w -u sup ( { w e. RR | -u w e. S } , RR , < )
35	34	nfel1 2330	. . . . 5 |- F/ w -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S
36		negeq 8152	. . . . . 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 2246	. . . . 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 2893	. . . 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 2254	1 |- ( ph -> inf ( S , RR , < ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 834   = wceq 1353   e. wcel 2148   {crab 2459   C_ wss 3131   ` cfv 5218  (class class class)co 5877   supcsup 6983  infcinf 6984   CCcc 7811   RRcr 7812   < clt 7994   -ucneg 8131   ZZcz 9255   ZZ>=cuz 9530   ...cfz 10010

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011  df-fzo 10145

This theorem is referenced by:  zsupssdc  11957  nnmindc  12037  lcmval  12065  lcmcllem  12069  odzcllem  12244