Theorem hovera 15499

Description: A point at which the hover function is less than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)

Hypothesis

Ref	Expression
hover.f	|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )

Assertion

Ref	Expression
hovera	|- ( Z e. RR -> ( F ` ( Z - 1 ) ) < Z )

Distinct variable group:   x, Z

Allowed substitution hint:   F( x)

Proof of Theorem hovera

Step	Hyp	Ref	Expression
1		hover.f	. . 3 |- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )
2		preq1 3767	. . . . . 6 |- ( x = ( Z - 1 ) -> { x , 0 } = { ( Z - 1 ) , 0 } )
3	2	infeq1d 7302	. . . . 5 |- ( x = ( Z - 1 ) -> inf ( { x , 0 } , RR , < ) = inf ( { ( Z - 1 ) , 0 } , RR , < ) )
4		oveq1 6056	. . . . 5 |- ( x = ( Z - 1 ) -> ( x - 1 ) = ( ( Z - 1 ) - 1 ) )
5	3, 4	preq12d 3775	. . . 4 |- ( x = ( Z - 1 ) -> {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } = {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } )
6	5	supeq1d 7277	. . 3 |- ( x = ( Z - 1 ) -> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) = sup ( {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } , RR , < ) )
7		peano2rem 8536	. . 3 |- ( Z e. RR -> ( Z - 1 ) e. RR )
8		0red 8271	. . . . 5 |- ( Z e. RR -> 0 e. RR )
9		mincl 11909	. . . . 5 |- ( ( ( Z - 1 ) e. RR /\ 0 e. RR ) -> inf ( { ( Z - 1 ) , 0 } , RR , < ) e. RR )
10	7, 8, 9	syl2anc 411	. . . 4 |- ( Z e. RR -> inf ( { ( Z - 1 ) , 0 } , RR , < ) e. RR )
11		peano2rem 8536	. . . . 5 |- ( ( Z - 1 ) e. RR -> ( ( Z - 1 ) - 1 ) e. RR )
12	7, 11	syl 14	. . . 4 |- ( Z e. RR -> ( ( Z - 1 ) - 1 ) e. RR )
13		maxcl 11888	. . . 4 |- ( (inf ( { ( Z - 1 ) , 0 } , RR , < ) e. RR /\ ( ( Z - 1 ) - 1 ) e. RR ) -> sup ( {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } , RR , < ) e. RR )
14	10, 12, 13	syl2anc 411	. . 3 |- ( Z e. RR -> sup ( {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } , RR , < ) e. RR )
15	1, 6, 7, 14	fvmptd3 5770	. 2 |- ( Z e. RR -> ( F ` ( Z - 1 ) ) = sup ( {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } , RR , < ) )
16		id 19	. . . 4 |- ( Z e. RR -> Z e. RR )
17		0re 8270	. . . . 5 |- 0 e. RR
18		min1inf 11910	. . . . 5 |- ( ( ( Z - 1 ) e. RR /\ 0 e. RR ) -> inf ( { ( Z - 1 ) , 0 } , RR , < ) <_ ( Z - 1 ) )
19	7, 17, 18	sylancl 413	. . . 4 |- ( Z e. RR -> inf ( { ( Z - 1 ) , 0 } , RR , < ) <_ ( Z - 1 ) )
20		ltm1 9116	. . . 4 |- ( Z e. RR -> ( Z - 1 ) < Z )
21	10, 7, 16, 19, 20	lelttrd 8394	. . 3 |- ( Z e. RR -> inf ( { ( Z - 1 ) , 0 } , RR , < ) < Z )
22	7	ltm1d 9202	. . . 4 |- ( Z e. RR -> ( ( Z - 1 ) - 1 ) < ( Z - 1 ) )
23	12, 7, 16, 22, 20	lttrd 8395	. . 3 |- ( Z e. RR -> ( ( Z - 1 ) - 1 ) < Z )
24		maxltsup 11896	. . . 4 |- ( (inf ( { ( Z - 1 ) , 0 } , RR , < ) e. RR /\ ( ( Z - 1 ) - 1 ) e. RR /\ Z e. RR ) -> ( sup ( {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } , RR , < ) < Z <-> (inf ( { ( Z - 1 ) , 0 } , RR , < ) < Z /\ ( ( Z - 1 ) - 1 ) < Z ) ) )
25	10, 12, 16, 24	syl3anc 1274	. . 3 |- ( Z e. RR -> ( sup ( {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } , RR , < ) < Z <-> (inf ( { ( Z - 1 ) , 0 } , RR , < ) < Z /\ ( ( Z - 1 ) - 1 ) < Z ) ) )
26	21, 23, 25	mpbir2and 953	. 2 |- ( Z e. RR -> sup ( {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } , RR , < ) < Z )
27	15, 26	eqbrtrd 4130	1 |- ( Z e. RR -> ( F ` ( Z - 1 ) ) < Z )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   {cpr 3689   class class class wbr 4108   |-> cmpt 4170   ` cfv 5351  (class class class)co 6049   supcsup 7272  infcinf 7273   RRcr 8122   0cc0 8123   1c1 8124   < clt 8304   <_ cle 8305   - cmin 8440

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242  ax-caucvg 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-sup 7274  df-inf 7275  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-n0 9493  df-z 9574  df-uz 9850  df-rp 9983  df-seqfrec 10806  df-exp 10897  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677

This theorem is referenced by:  ivthdichlem  15503