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Theorem ivthle2 19223
Description: The intermediate value theorem with weak inequality, decreasing case. (Contributed by Mario Carneiro, 12-May-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 )
ivthle2.9  |-  ( ph  ->  ( ( F `  B )  <_  U  /\  U  <_  ( F `
 A ) ) )
Assertion
Ref Expression
ivthle2  |-  ( 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 ivthle2
StepHypRef Expression
1 ioossicc 10930 . . . . 5  |-  ( A (,) B )  C_  ( A [,] B )
2 ivth.1 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
32adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  A  e.  RR )
4 ivth.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
54adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  B  e.  RR )
6 ivth.3 . . . . . . 7  |-  ( ph  ->  U  e.  RR )
76adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  U  e.  RR )
8 ivth.4 . . . . . . 7  |-  ( ph  ->  A  <  B )
98adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  A  <  B )
10 ivth.5 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  D )
1110adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  -> 
( A [,] B
)  C_  D )
12 ivth.7 . . . . . . 7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
1312adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  F  e.  ( D -cn->
CC ) )
14 ivth.8 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
1514adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  (
( F `  B
)  <  U  /\  U  <  ( F `  A ) ) )  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
16 simpr 448 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  -> 
( ( F `  B )  <  U  /\  U  <  ( F `
 A ) ) )
173, 5, 7, 9, 11, 13, 15, 16ivth2 19221 . . . . 5  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
18 ssrexv 3353 . . . . 5  |-  ( ( A (,) B ) 
C_  ( A [,] B )  ->  ( E. c  e.  ( A (,) B ) ( F `  c )  =  U  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U ) )
191, 17, 18mpsyl 61 . . . 4  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
2019anassrs 630 . . 3  |-  ( ( ( ph  /\  ( F `  B )  <  U )  /\  U  <  ( F `  A
) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
212rexrd 9069 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
224rexrd 9069 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
232, 4, 8ltled 9155 . . . . . 6  |-  ( ph  ->  A  <_  B )
24 lbicc2 10947 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2521, 22, 23, 24syl3anc 1184 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
26 eqcom 2391 . . . . . . 7  |-  ( ( F `  c )  =  U  <->  U  =  ( F `  c ) )
27 fveq2 5670 . . . . . . . 8  |-  ( c  =  A  ->  ( F `  c )  =  ( F `  A ) )
2827eqeq2d 2400 . . . . . . 7  |-  ( c  =  A  ->  ( U  =  ( F `  c )  <->  U  =  ( F `  A ) ) )
2926, 28syl5bb 249 . . . . . 6  |-  ( c  =  A  ->  (
( F `  c
)  =  U  <->  U  =  ( F `  A ) ) )
3029rspcev 2997 . . . . 5  |-  ( ( A  e.  ( A [,] B )  /\  U  =  ( F `  A ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
3125, 30sylan 458 . . . 4  |-  ( (
ph  /\  U  =  ( F `  A ) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
3231adantlr 696 . . 3  |-  ( ( ( ph  /\  ( F `  B )  <  U )  /\  U  =  ( F `  A ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
33 ivthle2.9 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  <_  U  /\  U  <_  ( F `
 A ) ) )
3433simprd 450 . . . . 5  |-  ( ph  ->  U  <_  ( F `  A ) )
3514ralrimiva 2734 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
36 fveq2 5670 . . . . . . . . 9  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
3736eleq1d 2455 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
3837rspcv 2993 . . . . . . 7  |-  ( A  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  A )  e.  RR ) )
3925, 35, 38sylc 58 . . . . . 6  |-  ( ph  ->  ( F `  A
)  e.  RR )
406, 39leloed 9150 . . . . 5  |-  ( ph  ->  ( U  <_  ( F `  A )  <->  ( U  <  ( F `
 A )  \/  U  =  ( F `
 A ) ) ) )
4134, 40mpbid 202 . . . 4  |-  ( ph  ->  ( U  <  ( F `  A )  \/  U  =  ( F `  A )
) )
4241adantr 452 . . 3  |-  ( (
ph  /\  ( F `  B )  <  U
)  ->  ( U  <  ( F `  A
)  \/  U  =  ( F `  A
) ) )
4320, 32, 42mpjaodan 762 . 2  |-  ( (
ph  /\  ( F `  B )  <  U
)  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
44 ubicc2 10948 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
4521, 22, 23, 44syl3anc 1184 . . 3  |-  ( ph  ->  B  e.  ( A [,] B ) )
46 fveq2 5670 . . . . 5  |-  ( c  =  B  ->  ( F `  c )  =  ( F `  B ) )
4746eqeq1d 2397 . . . 4  |-  ( c  =  B  ->  (
( F `  c
)  =  U  <->  ( F `  B )  =  U ) )
4847rspcev 2997 . . 3  |-  ( ( B  e.  ( A [,] B )  /\  ( F `  B )  =  U )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
4945, 48sylan 458 . 2  |-  ( (
ph  /\  ( F `  B )  =  U )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
5033simpld 446 . . 3  |-  ( ph  ->  ( F `  B
)  <_  U )
51 fveq2 5670 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
5251eleq1d 2455 . . . . . 6  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
5352rspcv 2993 . . . . 5  |-  ( B  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  B )  e.  RR ) )
5445, 35, 53sylc 58 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  RR )
5554, 6leloed 9150 . . 3  |-  ( ph  ->  ( ( F `  B )  <_  U  <->  ( ( F `  B
)  <  U  \/  ( F `  B )  =  U ) ) )
5650, 55mpbid 202 . 2  |-  ( ph  ->  ( ( F `  B )  <  U  \/  ( F `  B
)  =  U ) )
5743, 49, 56mpjaodan 762 1  |-  ( ph  ->  E. c  e.  ( A [,] B ) ( F `  c
)  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652    C_ wss 3265   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   CCcc 8923   RRcr 8924   RR*cxr 9054    < clt 9055    <_ cle 9056   (,)cioo 10850   [,]cicc 10853   -cn->ccncf 18779
This theorem is referenced by:  ivthicc  19224  recosf1o  20306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-mulf 9005
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-oi 7414  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-ioo 10854  df-icc 10857  df-fz 10978  df-fzo 11068  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-hom 13482  df-cco 13483  df-rest 13579  df-topn 13580  df-topgen 13596  df-pt 13597  df-prds 13600  df-xrs 13655  df-0g 13656  df-gsum 13657  df-qtop 13662  df-imas 13663  df-xps 13665  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-submnd 14668  df-mulg 14744  df-cntz 15045  df-cmn 15343  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cn 17215  df-cnp 17216  df-tx 17517  df-hmeo 17710  df-xms 18261  df-ms 18262  df-tms 18263  df-cncf 18781
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