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Theorem ivth2 19030
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 9357 . . 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 2366 . . . . 5  |-  ( y  e.  D  |->  -u ( F `  y )
)  =  ( y  e.  D  |->  -u ( F `  y )
)
98negfcncf 18637 . . . 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 3266 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  D )
12 fveq2 5632 . . . . . . 7  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
1312negeqd 9193 . . . . . 6  |-  ( y  =  x  ->  -u ( F `  y )  =  -u ( F `  x ) )
14 negex 9197 . . . . . 6  |-  -u ( F `  x )  e.  _V
1513, 8, 14fvmpt 5709 . . . . 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 9357 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  -u ( F `
 x )  e.  RR )
1916, 18eqeltrd 2440 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
y  e.  D  |->  -u ( F `  y ) ) `  x )  e.  RR )
201rexrd 9028 . . . . . . . 8  |-  ( ph  ->  A  e.  RR* )
212rexrd 9028 . . . . . . . 8  |-  ( ph  ->  B  e.  RR* )
221, 2, 5ltled 9114 . . . . . . . 8  |-  ( ph  ->  A  <_  B )
23 lbicc2 10905 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2420, 21, 22, 23syl3anc 1183 . . . . . . 7  |-  ( ph  ->  A  e.  ( A [,] B ) )
256, 24sseldd 3267 . . . . . 6  |-  ( ph  ->  A  e.  D )
26 fveq2 5632 . . . . . . . 8  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
2726negeqd 9193 . . . . . . 7  |-  ( y  =  A  ->  -u ( F `  y )  =  -u ( F `  A ) )
28 negex 9197 . . . . . . 7  |-  -u ( F `  A )  e.  _V
2927, 8, 28fvmpt 5709 . . . . . 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 2711 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
34 fveq2 5632 . . . . . . . . . 10  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
3534eleq1d 2432 . . . . . . . . 9  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
3635rspcv 2965 . . . . . . . 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 9497 . . . . . 6  |-  ( ph  ->  ( U  <  ( F `  A )  <->  -u ( F `  A
)  <  -u U ) )
3932, 38mpbid 201 . . . . 5  |-  ( ph  -> 
-u ( F `  A )  <  -u U
)
4030, 39eqbrtrd 4145 . . . 4  |-  ( ph  ->  ( ( y  e.  D  |->  -u ( F `  y ) ) `  A )  <  -u U
)
4131simpld 445 . . . . . 6  |-  ( ph  ->  ( F `  B
)  <  U )
42 ubicc2 10906 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
4320, 21, 22, 42syl3anc 1183 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
44 fveq2 5632 . . . . . . . . . 10  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
4544eleq1d 2432 . . . . . . . . 9  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
4645rspcv 2965 . . . . . . . 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 9497 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  <  U  <->  -u U  <  -u ( F `  B )
) )
4941, 48mpbid 201 . . . . 5  |-  ( ph  -> 
-u U  <  -u ( F `  B )
)
506, 43sseldd 3267 . . . . . 6  |-  ( ph  ->  B  e.  D )
51 fveq2 5632 . . . . . . . 8  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
5251negeqd 9193 . . . . . . 7  |-  ( y  =  B  ->  -u ( F `  y )  =  -u ( F `  B ) )
53 negex 9197 . . . . . . 7  |-  -u ( F `  B )  e.  _V
5452, 8, 53fvmpt 5709 . . . . . 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 4151 . . . 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 19029 . 2  |-  ( ph  ->  E. c  e.  ( A (,) B ) ( ( y  e.  D  |->  -u ( F `  y ) ) `  c )  =  -u U )
59 ioossicc 10888 . . . . . . . 8  |-  ( A (,) B )  C_  ( A [,] B )
6059, 6syl5ss 3276 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  D )
6160sselda 3266 . . . . . 6  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  c  e.  D )
62 fveq2 5632 . . . . . . . 8  |-  ( y  =  c  ->  ( F `  y )  =  ( F `  c ) )
6362negeqd 9193 . . . . . . 7  |-  ( y  =  c  ->  -u ( F `  y )  =  -u ( F `  c ) )
64 negex 9197 . . . . . . 7  |-  -u ( F `  c )  e.  _V
6563, 8, 64fvmpt 5709 . . . . . 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 2374 . . . 4  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  ( (
( y  e.  D  |-> 
-u ( F `  y ) ) `  c )  =  -u U 
<-> 
-u ( F `  c )  =  -u U ) )
68 cncff 18611 . . . . . . . 8  |-  ( F  e.  ( D -cn-> CC )  ->  F : D
--> CC )
697, 68syl 15 . . . . . . 7  |-  ( ph  ->  F : D --> CC )
70 ffvelrn 5770 . . . . . . 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 9008 . . . . . 6  |-  ( ph  ->  U  e.  CC )
7473adantr 451 . . . . 5  |-  ( (
ph  /\  c  e.  ( A (,) B ) )  ->  U  e.  CC )
75 neg11 9245 . . . . 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 2645 . 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 1647    e. wcel 1715   A.wral 2628   E.wrex 2629    C_ wss 3238   class class class wbr 4125    e. cmpt 4179   -->wf 5354   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   RR*cxr 9013    < clt 9014    <_ cle 9015   -ucneg 9185   (,)cioo 10809   [,]cicc 10812   -cn->ccncf 18594
This theorem is referenced by:  ivthle2  19032  pilem3  20047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-icc 10816  df-fz 10936  df-fzo 11026  df-seq 11211  df-exp 11270  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-hom 13440  df-cco 13441  df-rest 13537  df-topn 13538  df-topgen 13554  df-pt 13555  df-prds 13558  df-xrs 13613  df-0g 13614  df-gsum 13615  df-qtop 13620  df-imas 13621  df-xps 13623  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-submnd 14626  df-mulg 14702  df-cntz 15003  df-cmn 15301  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-cnfld 16594  df-top 16853  df-bases 16855  df-topon 16856  df-topsp 16857  df-cn 17174  df-cnp 17175  df-tx 17474  df-hmeo 17663  df-xms 18098  df-ms 18099  df-tms 18100  df-cncf 18596
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