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Theorem ivthinc 14309
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 2177 . . . 4  |-  { w  e.  ( A [,] B
)  |  ( F `
 w )  < 
U }  =  {
w  e.  ( A [,] B )  |  ( F `  w
)  <  U }
11 eqid 2177 . . . 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 14308 . . 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 2691 . . 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 9915 . . . . . . . . . 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 8072 . . . . . . . 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 4008 . . . . . . . . 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 5517 . . . . . . . . . . 11  |-  ( w  =  c  ->  ( F `  w )  =  ( F `  c ) )
2120breq1d 4015 . . . . . . . . . 10  |-  ( w  =  c  ->  (
( F `  w
)  <  U  <->  ( F `  c )  <  U
) )
22 ioossicc 9962 . . . . . . . . . . . . 13  |-  ( A (,) B )  C_  ( A [,] B )
2322sseli 3153 . . . . . . . . . . . 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 2897 . . . . . . . . 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 2848 . . . . . . . 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 665 . . . . . . 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 4009 . . . . . . . . 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 4017 . . . . . . . . . 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 2897 . . . . . . . . 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 2848 . . . . . . . 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 665 . . . . . . 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 752 . . . . . . 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 5517 . . . . . . . . . 10  |-  ( x  =  c  ->  ( F `  x )  =  ( F `  c ) )
4140eleq1d 2246 . . . . . . . . 9  |-  ( x  =  c  ->  (
( F `  x
)  e.  RR  <->  ( F `  c )  e.  RR ) )
427ralrimiva 2550 . . . . . . . . . 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 2848 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  RR )
453adantr 276 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  U  e.  RR )
46 reaplt 8548 . . . . . . . 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 673 . . . . 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 7989 . . . . . . 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 7989 . . . . . . 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 8582 . . . . . 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 2579 . 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 708    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   E!wreu 2457   {crab 2459    C_ wss 3131   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   RRcr 7813    < clt 7995   # cap 8541   (,)cioo 9891   [,]cicc 9894   -cn->ccncf 14245
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934  ax-pre-suploc 7935
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-map 6653  df-sup 6986  df-inf 6987  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-rp 9657  df-ioo 9895  df-icc 9898  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-cncf 14246
This theorem is referenced by:  ivthdec  14310  reeff1olem  14380
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