ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ivthdec Unicode version

Theorem ivthdec 14092
Description: The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-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 )
ivthdec.9  |-  ( ph  ->  ( ( F `  B )  <  U  /\  U  <  ( F `
 A ) ) )
ivthdec.i  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  y )  <  ( F `  x )
)
Assertion
Ref Expression
ivthdec  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c
)  =  U )
Distinct variable groups:    A, c, x   
y, A, x    B, c, x    y, B    D, c, x    y, D    F, c, x    y, F    U, c, x    y, U    ph, c, x    ph, y

Proof of Theorem ivthdec
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ivth.1 . . 3  |-  ( ph  ->  A  e.  RR )
2 ivth.2 . . 3  |-  ( ph  ->  B  e.  RR )
3 ivth.3 . . . 4  |-  ( ph  ->  U  e.  RR )
43renegcld 8336 . . 3  |-  ( ph  -> 
-u U  e.  RR )
5 ivth.4 . . 3  |-  ( ph  ->  A  <  B )
6 ivth.5 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  D )
7 ivth.7 . . . 4  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
8 eqid 2177 . . . . 5  |-  ( w  e.  D  |->  -u ( F `  w )
)  =  ( w  e.  D  |->  -u ( F `  w )
)
98negfcncf 14059 . . . 4  |-  ( F  e.  ( D -cn-> CC )  ->  ( w  e.  D  |->  -u ( F `  w )
)  e.  ( D
-cn-> CC ) )
107, 9syl 14 . . 3  |-  ( ph  ->  ( w  e.  D  |-> 
-u ( F `  w ) )  e.  ( D -cn-> CC ) )
11 fveq2 5515 . . . . . 6  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
1211negeqd 8151 . . . . 5  |-  ( w  =  x  ->  -u ( F `  w )  =  -u ( F `  x ) )
136sselda 3155 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  D )
14 ivth.8 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
1514renegcld 8336 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  -u ( F `
 x )  e.  RR )
168, 12, 13, 15fvmptd3 5609 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
w  e.  D  |->  -u ( F `  w ) ) `  x )  =  -u ( F `  x ) )
1716, 15eqeltrd 2254 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
w  e.  D  |->  -u ( F `  w ) ) `  x )  e.  RR )
18 fveq2 5515 . . . . . . 7  |-  ( w  =  A  ->  ( F `  w )  =  ( F `  A ) )
1918negeqd 8151 . . . . . 6  |-  ( w  =  A  ->  -u ( F `  w )  =  -u ( F `  A ) )
201rexrd 8006 . . . . . . . 8  |-  ( ph  ->  A  e.  RR* )
212rexrd 8006 . . . . . . . 8  |-  ( ph  ->  B  e.  RR* )
221, 2, 5ltled 8075 . . . . . . . 8  |-  ( ph  ->  A  <_  B )
23 lbicc2 9983 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2420, 21, 22, 23syl3anc 1238 . . . . . . 7  |-  ( ph  ->  A  e.  ( A [,] B ) )
256, 24sseldd 3156 . . . . . 6  |-  ( ph  ->  A  e.  D )
26 fveq2 5515 . . . . . . . . 9  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
2726eleq1d 2246 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
2814ralrimiva 2550 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
2927, 28, 24rspcdva 2846 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
3029renegcld 8336 . . . . . 6  |-  ( ph  -> 
-u ( F `  A )  e.  RR )
318, 19, 25, 30fvmptd3 5609 . . . . 5  |-  ( ph  ->  ( ( w  e.  D  |->  -u ( F `  w ) ) `  A )  =  -u ( F `  A ) )
32 ivthdec.9 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  <  U  /\  U  <  ( F `
 A ) ) )
3332simprd 114 . . . . . 6  |-  ( ph  ->  U  <  ( F `
 A ) )
343, 29ltnegd 8479 . . . . . 6  |-  ( ph  ->  ( U  <  ( F `  A )  <->  -u ( F `  A
)  <  -u U ) )
3533, 34mpbid 147 . . . . 5  |-  ( ph  -> 
-u ( F `  A )  <  -u U
)
3631, 35eqbrtrd 4025 . . . 4  |-  ( ph  ->  ( ( w  e.  D  |->  -u ( F `  w ) ) `  A )  <  -u U
)
3732simpld 112 . . . . . 6  |-  ( ph  ->  ( F `  B
)  <  U )
38 fveq2 5515 . . . . . . . . 9  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
3938eleq1d 2246 . . . . . . . 8  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
40 ubicc2 9984 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
4120, 21, 22, 40syl3anc 1238 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
4239, 28, 41rspcdva 2846 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  RR )
4342, 3ltnegd 8479 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  <  U  <->  -u U  <  -u ( F `  B )
) )
4437, 43mpbid 147 . . . . 5  |-  ( ph  -> 
-u U  <  -u ( F `  B )
)
45 fveq2 5515 . . . . . . 7  |-  ( w  =  B  ->  ( F `  w )  =  ( F `  B ) )
4645negeqd 8151 . . . . . 6  |-  ( w  =  B  ->  -u ( F `  w )  =  -u ( F `  B ) )
476, 41sseldd 3156 . . . . . 6  |-  ( ph  ->  B  e.  D )
4842renegcld 8336 . . . . . 6  |-  ( ph  -> 
-u ( F `  B )  e.  RR )
498, 46, 47, 48fvmptd3 5609 . . . . 5  |-  ( ph  ->  ( ( w  e.  D  |->  -u ( F `  w ) ) `  B )  =  -u ( F `  B ) )
5044, 49breqtrrd 4031 . . . 4  |-  ( ph  -> 
-u U  <  (
( w  e.  D  |-> 
-u ( F `  w ) ) `  B ) )
5136, 50jca 306 . . 3  |-  ( ph  ->  ( ( ( w  e.  D  |->  -u ( F `  w )
) `  A )  <  -u U  /\  -u U  <  ( ( w  e.  D  |->  -u ( F `  w ) ) `  B ) ) )
52 ivthdec.i . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  y )  <  ( F `  x )
)
53 fveq2 5515 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
5453eleq1d 2246 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  e.  RR  <->  ( F `  y )  e.  RR ) )
55 simpll 527 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ph )
5655, 28syl 14 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
57 simprl 529 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  y  e.  ( A [,] B ) )
5854, 56, 57rspcdva 2846 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  y )  e.  RR )
5914adantr 276 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  x )  e.  RR )
6058, 59ltnegd 8479 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( ( F `
 y )  < 
( F `  x
)  <->  -u ( F `  x )  <  -u ( F `  y )
) )
6152, 60mpbid 147 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  -u ( F `  x )  <  -u ( F `  y )
)
6213adantr 276 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  x  e.  D
)
6315adantr 276 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  -u ( F `  x )  e.  RR )
648, 12, 62, 63fvmptd3 5609 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( ( w  e.  D  |->  -u ( F `  w )
) `  x )  =  -u ( F `  x ) )
65 fveq2 5515 . . . . . 6  |-  ( w  =  y  ->  ( F `  w )  =  ( F `  y ) )
6665negeqd 8151 . . . . 5  |-  ( w  =  y  ->  -u ( F `  w )  =  -u ( F `  y ) )
676sseld 3154 . . . . . 6  |-  ( ph  ->  ( y  e.  ( A [,] B )  ->  y  e.  D
) )
6855, 57, 67sylc 62 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  y  e.  D
)
6958renegcld 8336 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  -u ( F `  y )  e.  RR )
708, 66, 68, 69fvmptd3 5609 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( ( w  e.  D  |->  -u ( F `  w )
) `  y )  =  -u ( F `  y ) )
7161, 64, 703brtr4d 4035 . . 3  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( ( w  e.  D  |->  -u ( F `  w )
) `  x )  <  ( ( w  e.  D  |->  -u ( F `  w ) ) `  y ) )
721, 2, 4, 5, 6, 10, 17, 51, 71ivthinc 14091 . 2  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( ( w  e.  D  |->  -u ( F `  w ) ) `  c )  =  -u U )
73 fveq2 5515 . . . . . . 7  |-  ( w  =  c  ->  ( F `  w )  =  ( F `  c ) )
7473negeqd 8151 . . . . . 6  |-  ( w  =  c  ->  -u ( F `  w )  =  -u ( F `  c ) )
75 ioossicc 9958 . . . . . . . 8  |-  ( A (,) B )  C_  ( A [,] B )
7675, 6sstrid 3166 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  D )
7776sselda 3155 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  D )
78 fveq2 5515 . . . . . . . . 9  |-  ( x  =  c  ->  ( F `  x )  =  ( F `  c ) )
7978eleq1d 2246 . . . . . . . 8  |-  ( x  =  c  ->  (
( F `  x
)  e.  RR  <->  ( F `  c )  e.  RR ) )
8028adantr 276 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  A. x  e.  ( A [,] B
) ( F `  x )  e.  RR )
8175sseli 3151 . . . . . . . . 9  |-  ( c  e.  ( A (,) B )  ->  c  e.  ( A [,] B
) )
8281adantl 277 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  ( A [,] B ) )
8379, 80, 82rspcdva 2846 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  RR )
8483renegcld 8336 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  -u ( F `
 c )  e.  RR )
858, 74, 77, 84fvmptd3 5609 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
w  e.  D  |->  -u ( F `  w ) ) `  c )  =  -u ( F `  c ) )
8685eqeq1d 2186 . . . 4  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
( w  e.  D  |-> 
-u ( F `  w ) ) `  c )  =  -u U 
<-> 
-u ( F `  c )  =  -u U ) )
87 cncff 14034 . . . . . . . 8  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
887, 87syl 14 . . . . . . 7  |-  ( ph  ->  F : D --> CC )
8988ffvelcdmda 5651 . . . . . 6  |-  ( (
ph  /\  c  e.  D )  ->  ( F `  c )  e.  CC )
9077, 89syldan 282 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  CC )
913recnd 7985 . . . . . 6  |-  ( ph  ->  U  e.  CC )
9291adantr 276 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  U  e.  CC )
9390, 92neg11ad 8263 . . . 4  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( -u ( F `  c )  =  -u U  <->  ( F `  c )  =  U ) )
9486, 93bitrd 188 . . 3  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
( w  e.  D  |-> 
-u ( F `  w ) ) `  c )  =  -u U 
<->  ( F `  c
)  =  U ) )
9594rexbidva 2474 . 2  |-  ( ph  ->  ( E. c  e.  ( A (,) B
) ( ( w  e.  D  |->  -u ( F `  w )
) `  c )  =  -u U  <->  E. c  e.  ( A (,) B
) ( F `  c )  =  U ) )
9672, 95mpbid 147 1  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c
)  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456    C_ wss 3129   class class class wbr 4003    |-> cmpt 4064   -->wf 5212   ` cfv 5216  (class class class)co 5874   CCcc 7808   RRcr 7809   RR*cxr 7990    < clt 7991    <_ cle 7992   -ucneg 8128   (,)cioo 9887   [,]cicc 9890   -cn->ccncf 14027
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930  ax-pre-suploc 7931
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-map 6649  df-sup 6982  df-inf 6983  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-rp 9653  df-ioo 9891  df-icc 9894  df-seqfrec 10445  df-exp 10519  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-cncf 14028
This theorem is referenced by:  cosz12  14171  ioocosf1o  14245
  Copyright terms: Public domain W3C validator