Theorem infssfzledc 10619

Description: The infimum of a decidable inhabited subset of an integer range is a lower bound for that set. (Contributed by Jim Kingdon, 12-Jun-2026.)

Hypotheses

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

Assertion

Ref	Expression
infssfzledc	|- ( ph -> inf ( S , RR , < ) <_ A )

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

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

Proof of Theorem infssfzledc

Step	Hyp	Ref	Expression
1		infssfzledc.s	. . . 4 |- S = { n e. ( M ... N ) | ps }
2		elfzuz 10374	. . . . . . . 8 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
3	2	ad2antrl 490	. . . . . . 7 |- ( ( ph /\ ( n e. ( M ... N ) /\ ps ) ) -> n e. ( ZZ>= ` M ) )
4		elfzle2 10382	. . . . . . . 8 |- ( n e. ( M ... N ) -> n <_ N )
5	4	ad2antrl 490	. . . . . . 7 |- ( ( ph /\ ( n e. ( M ... N ) /\ ps ) ) -> n <_ N )
6		simprr 533	. . . . . . 7 |- ( ( ph /\ ( n e. ( M ... N ) /\ ps ) ) -> ps )
7	3, 5, 6	jca32 310	. . . . . 6 |- ( ( ph /\ ( n e. ( M ... N ) /\ ps ) ) -> ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) )
8		simprl 531	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) ) -> n e. ( ZZ>= ` M ) )
9		simprrl 541	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) ) -> n <_ N )
10		eluzelz 9881	. . . . . . . . . . 11 |- ( n e. ( ZZ>= ` M ) -> n e. ZZ )
11	10	adantr 276	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) -> n e. ZZ )
12		infssfzledc.a	. . . . . . . . . . . . 13 |- ( ph -> A e. S )
13	1	eleq2i 2301	. . . . . . . . . . . . . 14 |- ( A e. S <-> A e. { n e. ( M ... N ) | ps } )
14		nfcv 2386	. . . . . . . . . . . . . . 15 |- F/_ n ( M ... N )
15	14	elrabsf 3084	. . . . . . . . . . . . . 14 |- ( A e. { n e. ( M ... N ) | ps } <-> ( A e. ( M ... N ) /\ [. A / n ]. ps ) )
16	13, 15	bitri 184	. . . . . . . . . . . . 13 |- ( A e. S <-> ( A e. ( M ... N ) /\ [. A / n ]. ps ) )
17	12, 16	sylib 122	. . . . . . . . . . . 12 |- ( ph -> ( A e. ( M ... N ) /\ [. A / n ]. ps ) )
18	17	simpld 112	. . . . . . . . . . 11 |- ( ph -> A e. ( M ... N ) )
19		elfzel2 10376	. . . . . . . . . . 11 |- ( A e. ( M ... N ) -> N e. ZZ )
20	18, 19	syl 14	. . . . . . . . . 10 |- ( ph -> N e. ZZ )
21		eluz 9885	. . . . . . . . . 10 |- ( ( n e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` n ) <-> n <_ N ) )
22	11, 20, 21	syl2anr 290	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) ) -> ( N e. ( ZZ>= ` n ) <-> n <_ N ) )
23	9, 22	mpbird 167	. . . . . . . 8 |- ( ( ph /\ ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) ) -> N e. ( ZZ>= ` n ) )
24		elfzuzb 10372	. . . . . . . 8 |- ( n e. ( M ... N ) <-> ( n e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` n ) ) )
25	8, 23, 24	sylanbrc 417	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) ) -> n e. ( M ... N ) )
26		simprrr 542	. . . . . . 7 |- ( ( ph /\ ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) ) -> ps )
27	25, 26	jca 306	. . . . . 6 |- ( ( ph /\ ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) ) -> ( n e. ( M ... N ) /\ ps ) )
28	7, 27	impbida 600	. . . . 5 |- ( ph -> ( ( n e. ( M ... N ) /\ ps ) <-> ( n e. ( ZZ>= ` M ) /\ ( n <_ N /\ ps ) ) ) )
29	28	rabbidva2 2799	. . . 4 |- ( ph -> { n e. ( M ... N ) | ps } = { n e. ( ZZ>= ` M ) | ( n <_ N /\ ps ) } )
30	1, 29	eqtrid 2279	. . 3 |- ( ph -> S = { n e. ( ZZ>= ` M ) | ( n <_ N /\ ps ) } )
31	30	infeq1d 7316	. 2 |- ( ph -> inf ( S , RR , < ) = inf ( { n e. ( ZZ>= ` M ) | ( n <_ N /\ ps ) } , RR , < ) )
32		elfzel1 10377	. . . 4 |- ( A e. ( M ... N ) -> M e. ZZ )
33	18, 32	syl 14	. . 3 |- ( ph -> M e. ZZ )
34		eqid 2234	. . 3 |- { n e. ( ZZ>= ` M ) | ( n <_ N /\ ps ) } = { n e. ( ZZ>= ` M ) | ( n <_ N /\ ps ) }
35	12, 30	eleqtrd 2313	. . 3 |- ( ph -> A e. { n e. ( ZZ>= ` M ) | ( n <_ N /\ ps ) } )
36		elfzelz 10378	. . . . 5 |- ( n e. ( M ... A ) -> n e. ZZ )
37		zdcle 9671	. . . . 5 |- ( ( n e. ZZ /\ N e. ZZ ) -> DECID n <_ N )
38	36, 20, 37	syl2anr 290	. . . 4 |- ( ( ph /\ n e. ( M ... A ) ) -> DECID n <_ N )
39		infssfzledc.dc	. . . 4 |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )
40	38, 39	dcand 941	. . 3 |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ( n <_ N /\ ps ) )
41	33, 34, 35, 40	infssuzledc 10616	. 2 |- ( ph -> inf ( { n e. ( ZZ>= ` M ) | ( n <_ N /\ ps ) } , RR , < ) <_ A )
42	31, 41	eqbrtrd 4136	1 |- ( ph -> inf ( S , RR , < ) <_ A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2205   {crab 2526   [.wsbc 3045   class class class wbr 4114   ` cfv 5357  (class class class)co 6058  infcinf 7287   RRcr 8142   < clt 8324   <_ cle 8325   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499

This theorem is referenced by:  ballotfilemsle  13192