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Theorem ivth2 18815
Description: The intermediate value theorem, decreasing case. (Contributed by Paul Chapman, 22-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)
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 )
ivth2.9  |-  ( ph  ->  ( ( F `  B )  <  U  /\  U  <  ( F `
 A ) ) )
Assertion
Ref Expression
ivth2  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c
)  =  U )
Distinct variable groups:    x, c, B    D, c, x    F, c, x    ph, c, x    A, c, x    U, c, x

Proof of Theorem ivth2
Dummy variable  y 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 9210 . . 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 2283 . . . . 5  |-  ( y  e.  D  |->  -u ( F `  y )
)  =  ( y  e.  D  |->  -u ( F `  y )
)
98negfcncf 18422 . . . 4  |-  ( F  e.  ( D -cn-> CC )  ->  ( y  e.  D  |->  -u ( F `  y )
)  e.  ( D
-cn-> CC ) )
107, 9syl 15 . . 3  |-  ( ph  ->  ( y  e.  D  |-> 
-u ( F `  y ) )  e.  ( D -cn-> CC ) )
116sselda 3180 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  D )
12 fveq2 5525 . . . . . . 7  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
1312negeqd 9046 . . . . . 6  |-  ( y  =  x  ->  -u ( F `  y )  =  -u ( F `  x ) )
14 negex 9050 . . . . . 6  |-  -u ( F `  x )  e.  _V
1513, 8, 14fvmpt 5602 . . . . 5  |-  ( x  e.  D  ->  (
( y  e.  D  |-> 
-u ( F `  y ) ) `  x )  =  -u ( F `  x ) )
1611, 15syl 15 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
y  e.  D  |->  -u ( F `  y ) ) `  x )  =  -u ( F `  x ) )
17 ivth.8 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
1817renegcld 9210 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  -u ( F `
 x )  e.  RR )
1916, 18eqeltrd 2357 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
y  e.  D  |->  -u ( F `  y ) ) `  x )  e.  RR )
201rexrd 8881 . . . . . . . 8  |-  ( ph  ->  A  e.  RR* )
212rexrd 8881 . . . . . . . 8  |-  ( ph  ->  B  e.  RR* )
221, 2, 5ltled 8967 . . . . . . . 8  |-  ( ph  ->  A  <_  B )
23 lbicc2 10752 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2420, 21, 22, 23syl3anc 1182 . . . . . . 7  |-  ( ph  ->  A  e.  ( A [,] B ) )
256, 24sseldd 3181 . . . . . 6  |-  ( ph  ->  A  e.  D )
26 fveq2 5525 . . . . . . . 8  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
2726negeqd 9046 . . . . . . 7  |-  ( y  =  A  ->  -u ( F `  y )  =  -u ( F `  A ) )
28 negex 9050 . . . . . . 7  |-  -u ( F `  A )  e.  _V
2927, 8, 28fvmpt 5602 . . . . . 6  |-  ( A  e.  D  ->  (
( y  e.  D  |-> 
-u ( F `  y ) ) `  A )  =  -u ( F `  A ) )
3025, 29syl 15 . . . . 5  |-  ( ph  ->  ( ( y  e.  D  |->  -u ( F `  y ) ) `  A )  =  -u ( F `  A ) )
31 ivth2.9 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  <  U  /\  U  <  ( F `
 A ) ) )
3231simprd 449 . . . . . 6  |-  ( ph  ->  U  <  ( F `
 A ) )
3317ralrimiva 2626 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
34 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
3534eleq1d 2349 . . . . . . . . 9  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
3635rspcv 2880 . . . . . . . 8  |-  ( A  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  A )  e.  RR ) )
3724, 33, 36sylc 56 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
383, 37ltnegd 9350 . . . . . 6  |-  ( ph  ->  ( U  <  ( F `  A )  <->  -u ( F `  A
)  <  -u U ) )
3932, 38mpbid 201 . . . . 5  |-  ( ph  -> 
-u ( F `  A )  <  -u U
)
4030, 39eqbrtrd 4043 . . . 4  |-  ( ph  ->  ( ( y  e.  D  |->  -u ( F `  y ) ) `  A )  <  -u U
)
4131simpld 445 . . . . . 6  |-  ( ph  ->  ( F `  B
)  <  U )
42 ubicc2 10753 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
4320, 21, 22, 42syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
44 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
4544eleq1d 2349 . . . . . . . . 9  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
4645rspcv 2880 . . . . . . . 8  |-  ( B  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  B )  e.  RR ) )
4743, 33, 46sylc 56 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  RR )
4847, 3ltnegd 9350 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  <  U  <->  -u U  <  -u ( F `  B )
) )
4941, 48mpbid 201 . . . . 5  |-  ( ph  -> 
-u U  <  -u ( F `  B )
)
506, 43sseldd 3181 . . . . . 6  |-  ( ph  ->  B  e.  D )
51 fveq2 5525 . . . . . . . 8  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
5251negeqd 9046 . . . . . . 7  |-  ( y  =  B  ->  -u ( F `  y )  =  -u ( F `  B ) )
53 negex 9050 . . . . . . 7  |-  -u ( F `  B )  e.  _V
5452, 8, 53fvmpt 5602 . . . . . 6  |-  ( B  e.  D  ->  (
( y  e.  D  |-> 
-u ( F `  y ) ) `  B )  =  -u ( F `  B ) )
5550, 54syl 15 . . . . 5  |-  ( ph  ->  ( ( y  e.  D  |->  -u ( F `  y ) ) `  B )  =  -u ( F `  B ) )
5649, 55breqtrrd 4049 . . . 4  |-  ( ph  -> 
-u U  <  (
( y  e.  D  |-> 
-u ( F `  y ) ) `  B ) )
5740, 56jca 518 . . 3  |-  ( ph  ->  ( ( ( y  e.  D  |->  -u ( F `  y )
) `  A )  <  -u U  /\  -u U  <  ( ( y  e.  D  |->  -u ( F `  y ) ) `  B ) ) )
581, 2, 4, 5, 6, 10, 19, 57ivth 18814 . 2  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( ( y  e.  D  |->  -u ( F `  y ) ) `  c )  =  -u U )
59 ioossicc 10735 . . . . . . . 8  |-  ( A (,) B )  C_  ( A [,] B )
6059, 6syl5ss 3190 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  D )
6160sselda 3180 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  D )
62 fveq2 5525 . . . . . . . 8  |-  ( y  =  c  ->  ( F `  y )  =  ( F `  c ) )
6362negeqd 9046 . . . . . . 7  |-  ( y  =  c  ->  -u ( F `  y )  =  -u ( F `  c ) )
64 negex 9050 . . . . . . 7  |-  -u ( F `  c )  e.  _V
6563, 8, 64fvmpt 5602 . . . . . 6  |-  ( c  e.  D  ->  (
( y  e.  D  |-> 
-u ( F `  y ) ) `  c )  =  -u ( F `  c ) )
6661, 65syl 15 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
y  e.  D  |->  -u ( F `  y ) ) `  c )  =  -u ( F `  c ) )
6766eqeq1d 2291 . . . 4  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
( y  e.  D  |-> 
-u ( F `  y ) ) `  c )  =  -u U 
<-> 
-u ( F `  c )  =  -u U ) )
68 cncff 18397 . . . . . . . 8  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
697, 68syl 15 . . . . . . 7  |-  ( ph  ->  F : D --> CC )
70 ffvelrn 5663 . . . . . . 7  |-  ( ( F : D --> CC  /\  c  e.  D )  ->  ( F `  c
)  e.  CC )
7169, 70sylan 457 . . . . . 6  |-  ( (
ph  /\  c  e.  D )  ->  ( F `  c )  e.  CC )
7261, 71syldan 456 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( F `  c )  e.  CC )
733recnd 8861 . . . . . 6  |-  ( ph  ->  U  e.  CC )
7473adantr 451 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  U  e.  CC )
75 neg11 9098 . . . . 5  |-  ( ( ( F `  c
)  e.  CC  /\  U  e.  CC )  ->  ( -u ( F `
 c )  = 
-u U  <->  ( F `  c )  =  U ) )
7672, 74, 75syl2anc 642 . . . 4  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( -u ( F `  c )  =  -u U  <->  ( F `  c )  =  U ) )
7767, 76bitrd 244 . . 3  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
( y  e.  D  |-> 
-u ( F `  y ) ) `  c )  =  -u U 
<->  ( F `  c
)  =  U ) )
7877rexbidva 2560 . 2  |-  ( ph  ->  ( E. c  e.  ( A (,) B
) ( ( y  e.  D  |->  -u ( F `  y )
) `  c )  =  -u U  <->  E. c  e.  ( A (,) B
) ( F `  c )  =  U ) )
7958, 78mpbid 201 1  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c
)  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   RR*cxr 8866    < clt 8867    <_ cle 8868   -ucneg 9038   (,)cioo 10656   [,]cicc 10659   -cn->ccncf 18380
This theorem is referenced by:  ivthle2  18817  pilem3  19829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382
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