Theorem hovera 15561

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 3770	. . . . . 6 |- ( x = ( Z - 1 ) -> { x , 0 } = { ( Z - 1 ) , 0 } )
3	2	infeq1d 7305	. . . . 5 |- ( x = ( Z - 1 ) -> inf ( { x , 0 } , RR , < ) = inf ( { ( Z - 1 ) , 0 } , RR , < ) )
4		oveq1 6059	. . . . 5 |- ( x = ( Z - 1 ) -> ( x - 1 ) = ( ( Z - 1 ) - 1 ) )
5	3, 4	preq12d 3778	. . . 4 |- ( x = ( Z - 1 ) -> {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } = {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } )
6	5	supeq1d 7280	. . 3 |- ( x = ( Z - 1 ) -> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) = sup ( {inf ( { ( Z - 1 ) , 0 } , RR , < ) , ( ( Z - 1 ) - 1 ) } , RR , < ) )
7		peano2rem 8545	. . 3 |- ( Z e. RR -> ( Z - 1 ) e. RR )
8		0red 8280	. . . . 5 |- ( Z e. RR -> 0 e. RR )
9		mincl 11924	. . . . 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 8545	. . . . 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 11903	. . . 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 5773	. 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 8279	. . . . 5 |- 0 e. RR
18		min1inf 11925	. . . . 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 9125	. . . 4 |- ( Z e. RR -> ( Z - 1 ) < Z )
21	10, 7, 16, 19, 20	lelttrd 8403	. . 3 |- ( Z e. RR -> inf ( { ( Z - 1 ) , 0 } , RR , < ) < Z )
22	7	ltm1d 9211	. . . 4 |- ( Z e. RR -> ( ( Z - 1 ) - 1 ) < ( Z - 1 ) )
23	12, 7, 16, 22, 20	lttrd 8404	. . 3 |- ( Z e. RR -> ( ( Z - 1 ) - 1 ) < Z )
24		maxltsup 11911	. . . 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 4133	1 |- ( Z e. RR -> ( F ` ( Z - 1 ) ) < Z )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {cpr 3692   class class class wbr 4111   |-> cmpt 4173   ` cfv 5354  (class class class)co 6052   supcsup 7275  infcinf 7276   RRcr 8131   0cc0 8132   1c1 8133   < clt 8313   <_ cle 8314   - cmin 8449

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-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

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-nul 3511  df-if 3623  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-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  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-recs 6538  df-frec 6624  df-sup 7277  df-inf 7278  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-rp 9993  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692

This theorem is referenced by:  ivthdichlem  15565