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Theorem ivthinc 13032
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 2157 . . . 4  |-  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U }  =  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
11 eqid 2157 . . . 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 13031 . . 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 2670 . . 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 9816 . . . . . . . . . 10  |-  ( c  e.  ( A (,) B )  ->  c  e.  RR )
1615ad2antlr 481 . . . . . . . . 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 7988 . . . . . . . 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 3968 . . . . . . . . 9  |-  ( p  =  c  ->  (
p  <  c  <->  c  <  c ) )
19 simplrl 525 . . . . . . . . 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 5468 . . . . . . . . . . 11  |-  ( w  =  c  ->  ( F `  w )  =  ( F `  c ) )
2120breq1d 3975 . . . . . . . . . 10  |-  ( w  =  c  ->  (
( F `  w
)  <  U  <->  ( F `  c )  <  U
) )
22 ioossicc 9863 . . . . . . . . . . . . 13  |-  ( A (,) B )  C_  ( A [,] B )
2322sseli 3124 . . . . . . . . . . . 12  |-  ( c  e.  ( A (,) B )  ->  c  e.  ( A [,] B
) )
2423adantl 275 . . . . . . . . . . 11  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  ( A [,] B ) )
2524ad2antrr 480 . . . . . . . . . 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 109 . . . . . . . . . 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 2870 . . . . . . . . 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 2821 . . . . . . . 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 655 . . . . . . 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 3969 . . . . . . . . 9  |-  ( r  =  c  ->  (
c  <  r  <->  c  <  c ) )
31 simplrr 526 . . . . . . . . 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 3977 . . . . . . . . . 10  |-  ( w  =  c  ->  ( U  <  ( F `  w )  <->  U  <  ( F `  c ) ) )
3324ad2antrr 480 . . . . . . . . . 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 109 . . . . . . . . . 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 2870 . . . . . . . . 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 2821 . . . . . . . 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 655 . . . . . . 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 742 . . . . . . 7  |-  ( -.  ( ( F `  c )  <  U  \/  U  <  ( F `
 c ) )  <-> 
( -.  ( F `
 c )  < 
U  /\  -.  U  <  ( F `  c
) ) )
3929, 37, 38sylanbrc 414 . . . . . 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 5468 . . . . . . . . . 10  |-  ( x  =  c  ->  ( F `  x )  =  ( F `  c ) )
4140eleq1d 2226 . . . . . . . . 9  |-  ( x  =  c  ->  (
( F `  x
)  e.  RR  <->  ( F `  c )  e.  RR ) )
427ralrimiva 2530 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
4342adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  A. x  e.  ( A [,] B
) ( F `  x )  e.  RR )
4441, 43, 24rspcdva 2821 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  RR )
453adantr 274 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  U  e.  RR )
46 reaplt 8463 . . . . . . . 8  |-  ( ( ( F `  c
)  e.  RR  /\  U  e.  RR )  ->  ( ( F `  c ) #  U  <->  ( ( F `  c )  <  U  \/  U  < 
( F `  c
) ) ) )
4744, 45, 46syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( ( F `  c ) #  U 
<->  ( ( F `  c )  <  U  \/  U  <  ( F `
 c ) ) ) )
4847adantr 274 . . . . . 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 663 . . . . 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 7906 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  CC )
5150adantr 274 . . . . . 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 7906 . . . . . . 7  |-  ( ph  ->  U  e.  CC )
5352ad2antrr 480 . . . . . 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 8497 . . . . . 6  |-  ( ( ( F `  c
)  e.  CC  /\  U  e.  CC )  ->  ( ( F `  c )  =  U  <->  -.  ( F `  c
) #  U ) )
5551, 53, 54syl2anc 409 . . . . 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 166 . . . 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 114 . . 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 2559 . 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 103    <-> wb 104    \/ wo 698    = wceq 1335    e. wcel 2128   A.wral 2435   E.wrex 2436   E!wreu 2437   {crab 2439    C_ wss 3102   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   RRcr 7731    < clt 7912   # cap 8456   (,)cioo 9792   [,]cicc 9795   -cn->ccncf 12968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852  ax-pre-suploc 7853
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-map 6595  df-sup 6928  df-inf 6929  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-rp 9561  df-ioo 9796  df-icc 9799  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-cncf 12969
This theorem is referenced by:  ivthdec  13033  reeff1olem  13103
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