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Theorem ivthdec 13064
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 8256 . . 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 2157 . . . . 5  |-  ( w  e.  D  |->  -u ( F `  w )
)  =  ( w  e.  D  |->  -u ( F `  w )
)
98negfcncf 13031 . . . 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 5469 . . . . . 6  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
1211negeqd 8071 . . . . 5  |-  ( w  =  x  ->  -u ( F `  w )  =  -u ( F `  x ) )
136sselda 3128 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  D )
14 ivth.8 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
1514renegcld 8256 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  -u ( F `
 x )  e.  RR )
168, 12, 13, 15fvmptd3 5562 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
w  e.  D  |->  -u ( F `  w ) ) `  x )  =  -u ( F `  x ) )
1716, 15eqeltrd 2234 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
w  e.  D  |->  -u ( F `  w ) ) `  x )  e.  RR )
18 fveq2 5469 . . . . . . 7  |-  ( w  =  A  ->  ( F `  w )  =  ( F `  A ) )
1918negeqd 8071 . . . . . 6  |-  ( w  =  A  ->  -u ( F `  w )  =  -u ( F `  A ) )
201rexrd 7928 . . . . . . . 8  |-  ( ph  ->  A  e.  RR* )
212rexrd 7928 . . . . . . . 8  |-  ( ph  ->  B  e.  RR* )
221, 2, 5ltled 7995 . . . . . . . 8  |-  ( ph  ->  A  <_  B )
23 lbicc2 9889 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2420, 21, 22, 23syl3anc 1220 . . . . . . 7  |-  ( ph  ->  A  e.  ( A [,] B ) )
256, 24sseldd 3129 . . . . . 6  |-  ( ph  ->  A  e.  D )
26 fveq2 5469 . . . . . . . . 9  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
2726eleq1d 2226 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
2814ralrimiva 2530 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
2927, 28, 24rspcdva 2821 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
3029renegcld 8256 . . . . . 6  |-  ( ph  -> 
-u ( F `  A )  e.  RR )
318, 19, 25, 30fvmptd3 5562 . . . . 5  |-  ( ph  ->  ( ( w  e.  D  |->  -u ( F `  w ) ) `  A )  =  -u ( F `  A ) )
32 ivthdec.9 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  <  U  /\  U  <  ( F `
 A ) ) )
3332simprd 113 . . . . . 6  |-  ( ph  ->  U  <  ( F `
 A ) )
343, 29ltnegd 8399 . . . . . 6  |-  ( ph  ->  ( U  <  ( F `  A )  <->  -u ( F `  A
)  <  -u U ) )
3533, 34mpbid 146 . . . . 5  |-  ( ph  -> 
-u ( F `  A )  <  -u U
)
3631, 35eqbrtrd 3987 . . . 4  |-  ( ph  ->  ( ( w  e.  D  |->  -u ( F `  w ) ) `  A )  <  -u U
)
3732simpld 111 . . . . . 6  |-  ( ph  ->  ( F `  B
)  <  U )
38 fveq2 5469 . . . . . . . . 9  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
3938eleq1d 2226 . . . . . . . 8  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
40 ubicc2 9890 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
4120, 21, 22, 40syl3anc 1220 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
4239, 28, 41rspcdva 2821 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  RR )
4342, 3ltnegd 8399 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  <  U  <->  -u U  <  -u ( F `  B )
) )
4437, 43mpbid 146 . . . . 5  |-  ( ph  -> 
-u U  <  -u ( F `  B )
)
45 fveq2 5469 . . . . . . 7  |-  ( w  =  B  ->  ( F `  w )  =  ( F `  B ) )
4645negeqd 8071 . . . . . 6  |-  ( w  =  B  ->  -u ( F `  w )  =  -u ( F `  B ) )
476, 41sseldd 3129 . . . . . 6  |-  ( ph  ->  B  e.  D )
4842renegcld 8256 . . . . . 6  |-  ( ph  -> 
-u ( F `  B )  e.  RR )
498, 46, 47, 48fvmptd3 5562 . . . . 5  |-  ( ph  ->  ( ( w  e.  D  |->  -u ( F `  w ) ) `  B )  =  -u ( F `  B ) )
5044, 49breqtrrd 3993 . . . 4  |-  ( ph  -> 
-u U  <  (
( w  e.  D  |-> 
-u ( F `  w ) ) `  B ) )
5136, 50jca 304 . . 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 5469 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
5453eleq1d 2226 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  e.  RR  <->  ( F `  y )  e.  RR ) )
55 simpll 519 . . . . . . . 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 521 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  y  e.  ( A [,] B ) )
5854, 56, 57rspcdva 2821 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  y )  e.  RR )
5914adantr 274 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  x )  e.  RR )
6058, 59ltnegd 8399 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( ( F `
 y )  < 
( F `  x
)  <->  -u ( F `  x )  <  -u ( F `  y )
) )
6152, 60mpbid 146 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  -u ( F `  x )  <  -u ( F `  y )
)
6213adantr 274 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  x  e.  D
)
6315adantr 274 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  -u ( F `  x )  e.  RR )
648, 12, 62, 63fvmptd3 5562 . . . 4  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( ( w  e.  D  |->  -u ( F `  w )
) `  x )  =  -u ( F `  x ) )
65 fveq2 5469 . . . . . 6  |-  ( w  =  y  ->  ( F `  w )  =  ( F `  y ) )
6665negeqd 8071 . . . . 5  |-  ( w  =  y  ->  -u ( F `  w )  =  -u ( F `  y ) )
676sseld 3127 . . . . . 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 8256 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  -u ( F `  y )  e.  RR )
708, 66, 68, 69fvmptd3 5562 . . . 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 3997 . . 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 13063 . 2  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( ( w  e.  D  |->  -u ( F `  w ) ) `  c )  =  -u U )
73 fveq2 5469 . . . . . . 7  |-  ( w  =  c  ->  ( F `  w )  =  ( F `  c ) )
7473negeqd 8071 . . . . . 6  |-  ( w  =  c  ->  -u ( F `  w )  =  -u ( F `  c ) )
75 ioossicc 9864 . . . . . . . 8  |-  ( A (,) B )  C_  ( A [,] B )
7675, 6sstrid 3139 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  D )
7776sselda 3128 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  D )
78 fveq2 5469 . . . . . . . . 9  |-  ( x  =  c  ->  ( F `  x )  =  ( F `  c ) )
7978eleq1d 2226 . . . . . . . 8  |-  ( x  =  c  ->  (
( F `  x
)  e.  RR  <->  ( F `  c )  e.  RR ) )
8028adantr 274 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  A. x  e.  ( A [,] B
) ( F `  x )  e.  RR )
8175sseli 3124 . . . . . . . . 9  |-  ( c  e.  ( A (,) B )  ->  c  e.  ( A [,] B
) )
8281adantl 275 . . . . . . . 8  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  ( A [,] B ) )
8379, 80, 82rspcdva 2821 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  RR )
8483renegcld 8256 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  -u ( F `
 c )  e.  RR )
858, 74, 77, 84fvmptd3 5562 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
w  e.  D  |->  -u ( F `  w ) ) `  c )  =  -u ( F `  c ) )
8685eqeq1d 2166 . . . 4  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
( w  e.  D  |-> 
-u ( F `  w ) ) `  c )  =  -u U 
<-> 
-u ( F `  c )  =  -u U ) )
87 cncff 13006 . . . . . . . 8  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
887, 87syl 14 . . . . . . 7  |-  ( ph  ->  F : D --> CC )
8988ffvelrnda 5603 . . . . . 6  |-  ( (
ph  /\  c  e.  D )  ->  ( F `  c )  e.  CC )
9077, 89syldan 280 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  CC )
913recnd 7907 . . . . . 6  |-  ( ph  ->  U  e.  CC )
9291adantr 274 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  U  e.  CC )
9390, 92neg11ad 8183 . . . 4  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( -u ( F `  c )  =  -u U  <->  ( F `  c )  =  U ) )
9486, 93bitrd 187 . . 3  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
( w  e.  D  |-> 
-u ( F `  w ) ) `  c )  =  -u U 
<->  ( F `  c
)  =  U ) )
9594rexbidva 2454 . 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 146 1  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c
)  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   A.wral 2435   E.wrex 2436    C_ wss 3102   class class class wbr 3966    |-> cmpt 4026   -->wf 5167   ` cfv 5171  (class class class)co 5825   CCcc 7731   RRcr 7732   RR*cxr 7912    < clt 7913    <_ cle 7914   -ucneg 8048   (,)cioo 9793   [,]cicc 9796   -cn->ccncf 12999
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 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851  ax-arch 7852  ax-caucvg 7853  ax-pre-suploc 7854
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 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-isom 5180  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-frec 6339  df-map 6596  df-sup 6929  df-inf 6930  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-n0 9092  df-z 9169  df-uz 9441  df-rp 9562  df-ioo 9797  df-icc 9800  df-seqfrec 10349  df-exp 10423  df-cj 10746  df-re 10747  df-im 10748  df-rsqrt 10902  df-abs 10903  df-cncf 13000
This theorem is referenced by:  cosz12  13143  ioocosf1o  13217
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