Theorem infssuzcldc 10685

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 10683	. . 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 3333	. . . . . . 7 |- { n e. ( ZZ>= ` M ) | ps } C_ ( ZZ>= ` M )
7	2, 6	eqsstri 3280	. . . . . 6 |- S C_ ( ZZ>= ` M )
8		uzssz 9958	. . . . . 6 |- ( ZZ>= ` M ) C_ ZZ
9	7, 8	sstri 3257	. . . . 5 |- S C_ ZZ
10		zssre 9662	. . . . 5 |- ZZ C_ RR
11	9, 10	sstri 3257	. . . 4 |- S C_ RR
12	11	a1i 9	. . 3 |- ( ph -> S C_ RR )
13	5, 12	infrenegsupex 10010	. 2 |- ( ph -> inf ( S , RR , < ) = -u sup ( { w e. RR | -u w e. S } , RR , < ) )
14	1, 2, 3, 4	infssuzex 10683	. . . . . 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 10012	. . . . 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 8526	. . . . . . . . . 10 |- ( w = u -> -u w = -u u )
17	16	eleq1d 2307	. . . . . . . . 9 |- ( w = u -> ( -u w e. S <-> -u u e. S ) )
18	17	elrab 2982	. . . . . . . 8 |- ( u e. { w e. RR | -u w e. S } <-> ( u e. RR /\ -u u e. S ) )
19	9	sseli 3244	. . . . . . . . . 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 8360	. . . . . . . . . 10 |- ( ( u e. RR /\ -u u e. S ) -> u e. CC )
23		znegclb 9688	. . . . . . . . . 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 3252	. . . . . 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 9755	. . . 4 |- ( ph -> sup ( { w e. RR | -u w e. S } , RR , < ) e. { w e. RR | -u w e. S } )
30		nfrab1 2732	. . . . . 6 |- F/_ w { w e. RR | -u w e. S }
31		nfcv 2392	. . . . . 6 |- F/_ w RR
32		nfcv 2392	. . . . . 6 |- F/_ w <
33	30, 31, 32	nfsup 7338	. . . . 5 |- F/_ w sup ( { w e. RR | -u w e. S } , RR , < )
34	33	nfneg 8530	. . . . . 6 |- F/_ w -u sup ( { w e. RR | -u w e. S } , RR , < )
35	34	nfel1 2403	. . . . 5 |- F/ w -u sup ( { w e. RR | -u w e. S } , RR , < ) e. S
36		negeq 8526	. . . . . 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 2307	. . . . 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 2980	. . . 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 2315	1 |- ( ph -> inf ( S , RR , < ) e. S )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220   ` cfv 5380  (class class class)co 6088   supcsup 7328  infcinf 7329   CCcc 8183   RRcr 8184   < clt 8366   -ucneg 8505   ZZcz 9655   ZZ>=cuz 9937   ...cfz 10428

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567

This theorem is used by:  infssfzcldc  10686  zsupssdc  10690  bitsfzolem  12746  nnmindc  12836  nninfctlemfo  12842  lcmval  12866  lcmcllem  12870  odzcllem  13050  4sqlem13m  13211  4sqlem14  13212  4sqlem17  13215  4sqlem18  13216