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Theorem ivthdec 14798
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 8399 . . 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 2193 . . . . 5  |-  ( w  e.  D  |->  -u ( F `  w )
)  =  ( w  e.  D  |->  -u ( F `  w )
)
98negfcncf 14760 . . . 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 5554 . . . . . 6  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
1211negeqd 8214 . . . . 5  |-  ( w  =  x  ->  -u ( F `  w )  =  -u ( F `  x ) )
136sselda 3179 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  D )
14 ivth.8 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
1514renegcld 8399 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  -u ( F `
 x )  e.  RR )
168, 12, 13, 15fvmptd3 5651 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
w  e.  D  |->  -u ( F `  w ) ) `  x )  =  -u ( F `  x ) )
1716, 15eqeltrd 2270 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
w  e.  D  |->  -u ( F `  w ) ) `  x )  e.  RR )
18 fveq2 5554 . . . . . . 7  |-  ( w  =  A  ->  ( F `  w )  =  ( F `  A ) )
1918negeqd 8214 . . . . . 6  |-  ( w  =  A  ->  -u ( F `  w )  =  -u ( F `  A ) )
201rexrd 8069 . . . . . . . 8  |-  ( ph  ->  A  e.  RR* )
212rexrd 8069 . . . . . . . 8  |-  ( ph  ->  B  e.  RR* )
221, 2, 5ltled 8138 . . . . . . . 8  |-  ( ph  ->  A  <_  B )
23 lbicc2 10050 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2420, 21, 22, 23syl3anc 1249 . . . . . . 7  |-  ( ph  ->  A  e.  ( A [,] B ) )
256, 24sseldd 3180 . . . . . 6  |-  ( ph  ->  A  e.  D )
26 fveq2 5554 . . . . . . . . 9  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
2726eleq1d 2262 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
2814ralrimiva 2567 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
2927, 28, 24rspcdva 2869 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
3029renegcld 8399 . . . . . 6  |-  ( ph  -> 
-u ( F `  A )  e.  RR )
318, 19, 25, 30fvmptd3 5651 . . . . 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 8542 . . . . . 6  |-  ( ph  ->  ( U  <  ( F `  A )  <->  -u ( F `  A
)  <  -u U ) )
3533, 34mpbid 147 . . . . 5  |-  ( ph  -> 
-u ( F `  A )  <  -u U
)
3631, 35eqbrtrd 4051 . . . 4  |-  ( ph  ->  ( ( w  e.  D  |->  -u ( F `  w ) ) `  A )  <  -u U
)
3732simpld 112 . . . . . 6  |-  ( ph  ->  ( F `  B
)  <  U )
38 fveq2 5554 . . . . . . . . 9  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
3938eleq1d 2262 . . . . . . . 8  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
40 ubicc2 10051 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
4120, 21, 22, 40syl3anc 1249 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
4239, 28, 41rspcdva 2869 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  RR )
4342, 3ltnegd 8542 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  <  U  <->  -u U  <  -u ( F `  B )
) )
4437, 43mpbid 147 . . . . 5  |-  ( ph  -> 
-u U  <  -u ( F `  B )
)
45 fveq2 5554 . . . . . . 7  |-  ( w  =  B  ->  ( F `  w )  =  ( F `  B ) )
4645negeqd 8214 . . . . . 6  |-  ( w  =  B  ->  -u ( F `  w )  =  -u ( F `  B ) )
476, 41sseldd 3180 . . . . . 6  |-  ( ph  ->  B  e.  D )
4842renegcld 8399 . . . . . 6  |-  ( ph  -> 
-u ( F `  B )  e.  RR )
498, 46, 47, 48fvmptd3 5651 . . . . 5  |-  ( ph  ->  ( ( w  e.  D  |->  -u ( F `  w ) ) `  B )  =  -u ( F `  B ) )
5044, 49breqtrrd 4057 . . . 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 5554 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
5453eleq1d 2262 . . . . . . 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 2869 . . . . . 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 8542 . . . . 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 5651 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( ( w  e.  D  |->  -u ( F `  w )
) `  x )  =  -u ( F `  x ) )
65 fveq2 5554 . . . . . 6  |-  ( w  =  y  ->  ( F `  w )  =  ( F `  y ) )
6665negeqd 8214 . . . . 5  |-  ( w  =  y  ->  -u ( F `  w )  =  -u ( F `  y ) )
676sseld 3178 . . . . . 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 8399 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  -u ( F `  y )  e.  RR )
708, 66, 68, 69fvmptd3 5651 . . . 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 4061 . . 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 14797 . 2  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( ( w  e.  D  |->  -u ( F `  w ) ) `  c )  =  -u U )
73 fveq2 5554 . . . . . . 7  |-  ( w  =  c  ->  ( F `  w )  =  ( F `  c ) )
7473negeqd 8214 . . . . . 6  |-  ( w  =  c  ->  -u ( F `  w )  =  -u ( F `  c ) )
75 ioossicc 10025 . . . . . . . 8  |-  ( A (,) B )  C_  ( A [,] B )
7675, 6sstrid 3190 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  D )
7776sselda 3179 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  D )
78 fveq2 5554 . . . . . . . . 9  |-  ( x  =  c  ->  ( F `  x )  =  ( F `  c ) )
7978eleq1d 2262 . . . . . . . 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 3175 . . . . . . . . 9  |-  ( c  e.  ( A (,) B )  ->  c  e.  ( A [,] B
) )
8281adantl 277 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  ( A [,] B ) )
8379, 80, 82rspcdva 2869 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  RR )
8483renegcld 8399 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  -u ( F `
 c )  e.  RR )
858, 74, 77, 84fvmptd3 5651 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
w  e.  D  |->  -u ( F `  w ) ) `  c )  =  -u ( F `  c ) )
8685eqeq1d 2202 . . . 4  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
( w  e.  D  |-> 
-u ( F `  w ) ) `  c )  =  -u U 
<-> 
-u ( F `  c )  =  -u U ) )
87 cncff 14732 . . . . . . . 8  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
887, 87syl 14 . . . . . . 7  |-  ( ph  ->  F : D --> CC )
8988ffvelcdmda 5693 . . . . . 6  |-  ( (
ph  /\  c  e.  D )  ->  ( F `  c )  e.  CC )
9077, 89syldan 282 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  CC )
913recnd 8048 . . . . . 6  |-  ( ph  ->  U  e.  CC )
9291adantr 276 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  U  e.  CC )
9390, 92neg11ad 8326 . . . 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 2491 . 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 1364    e. wcel 2164   A.wral 2472   E.wrex 2473    C_ wss 3153   class class class wbr 4029    |-> cmpt 4090   -->wf 5250   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   RR*cxr 8053    < clt 8054    <_ cle 8055   -ucneg 8191   (,)cioo 9954   [,]cicc 9957   -cn->ccncf 14725
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992  ax-pre-suploc 7993
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-map 6704  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-ioo 9958  df-icc 9961  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-cncf 14726
This theorem is referenced by:  cosz12  14915  ioocosf1o  14989
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