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Theorem ivthinc 14504
Description: The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
Hypotheses
Ref Expression
ivth.1  |-  ( ph  ->  A  e.  RR )
ivth.2  |-  ( ph  ->  B  e.  RR )
ivth.3  |-  ( ph  ->  U  e.  RR )
ivth.4  |-  ( ph  ->  A  <  B )
ivth.5  |-  ( ph  ->  ( A [,] B
)  C_  D )
ivth.7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
ivth.8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
ivth.9  |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
ivthinc.i  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  x )  <  ( F `  y )
)
Assertion
Ref Expression
ivthinc  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c
)  =  U )
Distinct variable groups:    A, c, x   
y, A, x    B, c, x    y, B    F, c, x    y, F    U, c, x    y, U    ph, c, x    ph, y
Allowed substitution hints:    D( x, y, c)

Proof of Theorem ivthinc
Dummy variables  p  r  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ivth.1 . . . 4  |-  ( ph  ->  A  e.  RR )
2 ivth.2 . . . 4  |-  ( ph  ->  B  e.  RR )
3 ivth.3 . . . 4  |-  ( ph  ->  U  e.  RR )
4 ivth.4 . . . 4  |-  ( ph  ->  A  <  B )
5 ivth.5 . . . 4  |-  ( ph  ->  ( A [,] B
)  C_  D )
6 ivth.7 . . . 4  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
7 ivth.8 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
8 ivth.9 . . . 4  |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
9 ivthinc.i . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  x )  <  ( F `  y )
)
10 eqid 2188 . . . 4  |-  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U }  =  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
11 eqid 2188 . . . 4  |-  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) }  =  {
w  e.  ( A [,] B )  |  U  <  ( F `
 w ) }
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11ivthinclemex 14503 . . 3  |-  ( ph  ->  E! c  e.  ( A (,) B ) ( A. p  e. 
{ w  e.  ( A [,] B )  |  ( F `  w )  <  U } p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )
13 reurex 2703 . . 3  |-  ( E! c  e.  ( A (,) B ) ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r )  ->  E. c  e.  ( A (,) B
) ( A. p  e.  { w  e.  ( A [,] B )  |  ( F `  w )  <  U } p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )
1412, 13syl 14 . 2  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( A. p  e. 
{ w  e.  ( A [,] B )  |  ( F `  w )  <  U } p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )
15 elioore 9929 . . . . . . . . . 10  |-  ( c  e.  ( A (,) B )  ->  c  e.  RR )
1615ad2antlr 489 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  c  e.  RR )
1716ltnrd 8086 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  -.  c  <  c )
18 breq1 4020 . . . . . . . . 9  |-  ( p  =  c  ->  (
p  <  c  <->  c  <  c ) )
19 simplrl 535 . . . . . . . . 9  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  ( F `  c )  <  U )  ->  A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c )
20 fveq2 5529 . . . . . . . . . . 11  |-  ( w  =  c  ->  ( F `  w )  =  ( F `  c ) )
2120breq1d 4027 . . . . . . . . . 10  |-  ( w  =  c  ->  (
( F `  w
)  <  U  <->  ( F `  c )  <  U
) )
22 ioossicc 9976 . . . . . . . . . . . . 13  |-  ( A (,) B )  C_  ( A [,] B )
2322sseli 3165 . . . . . . . . . . . 12  |-  ( c  e.  ( A (,) B )  ->  c  e.  ( A [,] B
) )
2423adantl 277 . . . . . . . . . . 11  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  ( A [,] B ) )
2524ad2antrr 488 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  ( F `  c )  <  U )  -> 
c  e.  ( A [,] B ) )
26 simpr 110 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  ( F `  c )  <  U )  -> 
( F `  c
)  <  U )
2721, 25, 26elrabd 2909 . . . . . . . . 9  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  ( F `  c )  <  U )  -> 
c  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } )
2818, 19, 27rspcdva 2860 . . . . . . . 8  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  ( F `  c )  <  U )  -> 
c  <  c )
2917, 28mtand 666 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  -.  ( F `  c )  <  U )
30 breq2 4021 . . . . . . . . 9  |-  ( r  =  c  ->  (
c  <  r  <->  c  <  c ) )
31 simplrr 536 . . . . . . . . 9  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  U  <  ( F `  c ) )  ->  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r )
3220breq2d 4029 . . . . . . . . . 10  |-  ( w  =  c  ->  ( U  <  ( F `  w )  <->  U  <  ( F `  c ) ) )
3324ad2antrr 488 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  U  <  ( F `  c ) )  -> 
c  e.  ( A [,] B ) )
34 simpr 110 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  U  <  ( F `  c ) )  ->  U  <  ( F `  c ) )
3532, 33, 34elrabd 2909 . . . . . . . . 9  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  U  <  ( F `  c ) )  -> 
c  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } )
3630, 31, 35rspcdva 2860 . . . . . . . 8  |-  ( ( ( ( ph  /\  c  e.  ( A (,) B ) )  /\  ( A. p  e.  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r ) )  /\  U  <  ( F `  c ) )  -> 
c  <  c )
3717, 36mtand 666 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  -.  U  <  ( F `  c ) )
38 ioran 753 . . . . . . 7  |-  ( -.  ( ( F `  c )  <  U  \/  U  <  ( F `
 c ) )  <-> 
( -.  ( F `
 c )  < 
U  /\  -.  U  <  ( F `  c
) ) )
3929, 37, 38sylanbrc 417 . . . . . 6  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  -.  ( ( F `  c )  <  U  \/  U  <  ( F `
 c ) ) )
40 fveq2 5529 . . . . . . . . . 10  |-  ( x  =  c  ->  ( F `  x )  =  ( F `  c ) )
4140eleq1d 2257 . . . . . . . . 9  |-  ( x  =  c  ->  (
( F `  x
)  e.  RR  <->  ( F `  c )  e.  RR ) )
427ralrimiva 2562 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
4342adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  A. x  e.  ( A [,] B
) ( F `  x )  e.  RR )
4441, 43, 24rspcdva 2860 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  RR )
453adantr 276 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  U  e.  RR )
46 reaplt 8562 . . . . . . . 8  |-  ( ( ( F `  c
)  e.  RR  /\  U  e.  RR )  ->  ( ( F `  c ) #  U  <->  ( ( F `  c )  <  U  \/  U  < 
( F `  c
) ) ) )
4744, 45, 46syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( ( F `  c ) #  U 
<->  ( ( F `  c )  <  U  \/  U  <  ( F `
 c ) ) ) )
4847adantr 276 . . . . . 6  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  (
( F `  c
) #  U  <->  ( ( F `  c )  <  U  \/  U  < 
( F `  c
) ) ) )
4939, 48mtbird 674 . . . . 5  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  -.  ( F `  c ) #  U )
5044recnd 8003 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  CC )
5150adantr 276 . . . . . 6  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  ( F `  c )  e.  CC )
523recnd 8003 . . . . . . 7  |-  ( ph  ->  U  e.  CC )
5352ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  U  e.  CC )
54 apti 8596 . . . . . 6  |-  ( ( ( F `  c
)  e.  CC  /\  U  e.  CC )  ->  ( ( F `  c )  =  U  <->  -.  ( F `  c
) #  U ) )
5551, 53, 54syl2anc 411 . . . . 5  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  (
( F `  c
)  =  U  <->  -.  ( F `  c ) #  U ) )
5649, 55mpbird 167 . . . 4  |-  ( ( ( ph  /\  c  e.  ( A (,) B
) )  /\  ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
) )  ->  ( F `  c )  =  U )
5756ex 115 . . 3  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( ( A. p  e.  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U } p  < 
c  /\  A. r  e.  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) } c  <  r
)  ->  ( F `  c )  =  U ) )
5857reximdva 2591 . 2  |-  ( ph  ->  ( E. c  e.  ( A (,) B
) ( A. p  e.  { w  e.  ( A [,] B )  |  ( F `  w )  <  U } p  <  c  /\  A. r  e.  { w  e.  ( A [,] B
)  |  U  < 
( F `  w
) } c  < 
r )  ->  E. c  e.  ( A (,) B
) ( F `  c )  =  U ) )
5914, 58mpd 13 1  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c
)  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1363    e. wcel 2159   A.wral 2467   E.wrex 2468   E!wreu 2469   {crab 2471    C_ wss 3143   class class class wbr 4017   ` cfv 5230  (class class class)co 5890   CCcc 7826   RRcr 7827    < clt 8009   # cap 8555   (,)cioo 9905   [,]cicc 9908   -cn->ccncf 14440
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948  ax-pre-suploc 7949
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-isom 5239  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-map 6667  df-sup 7000  df-inf 7001  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-rp 9671  df-ioo 9909  df-icc 9912  df-seqfrec 10463  df-exp 10537  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-cncf 14441
This theorem is referenced by:  ivthdec  14505  reeff1olem  14575
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