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Theorem ivthle2 20939
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 11379 . . . . 5  |-  ( A (,) B )  C_  ( A [,] B )
2 ivth.1 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
32adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  A  e.  RR )
4 ivth.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  B  e.  RR )
6 ivth.3 . . . . . . 7  |-  ( ph  ->  U  e.  RR )
76adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  U  e.  RR )
8 ivth.4 . . . . . . 7  |-  ( ph  ->  A  <  B )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  A  <  B )
10 ivth.5 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  D )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  -> 
( A [,] B
)  C_  D )
12 ivth.7 . . . . . . 7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
1312adantr 465 . . . . . 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 714 . . . . . 6  |-  ( ( ( ph  /\  (
( F `  B
)  <  U  /\  U  <  ( F `  A ) ) )  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
16 simpr 461 . . . . . 6  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  -> 
( ( F `  B )  <  U  /\  U  <  ( F `
 A ) ) )
173, 5, 7, 9, 11, 13, 15, 16ivth2 20937 . . . . 5  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
18 ssrexv 3415 . . . . 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 63 . . . 4  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
2019anassrs 648 . . 3  |-  ( ( ( ph  /\  ( F `  B )  <  U )  /\  U  <  ( F `  A
) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
212rexrd 9431 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
224rexrd 9431 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
232, 4, 8ltled 9520 . . . . . 6  |-  ( ph  ->  A  <_  B )
24 lbicc2 11399 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2521, 22, 23, 24syl3anc 1218 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
26 eqcom 2443 . . . . . . 7  |-  ( ( F `  c )  =  U  <->  U  =  ( F `  c ) )
27 fveq2 5689 . . . . . . . 8  |-  ( c  =  A  ->  ( F `  c )  =  ( F `  A ) )
2827eqeq2d 2452 . . . . . . 7  |-  ( c  =  A  ->  ( U  =  ( F `  c )  <->  U  =  ( F `  A ) ) )
2926, 28syl5bb 257 . . . . . 6  |-  ( c  =  A  ->  (
( F `  c
)  =  U  <->  U  =  ( F `  A ) ) )
3029rspcev 3071 . . . . 5  |-  ( ( A  e.  ( A [,] B )  /\  U  =  ( F `  A ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
3125, 30sylan 471 . . . 4  |-  ( (
ph  /\  U  =  ( F `  A ) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
3231adantlr 714 . . 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 463 . . . . 5  |-  ( ph  ->  U  <_  ( F `  A ) )
3514ralrimiva 2797 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
36 fveq2 5689 . . . . . . . . 9  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
3736eleq1d 2507 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
3837rspcv 3067 . . . . . . 7  |-  ( A  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  A )  e.  RR ) )
3925, 35, 38sylc 60 . . . . . 6  |-  ( ph  ->  ( F `  A
)  e.  RR )
406, 39leloed 9515 . . . . 5  |-  ( ph  ->  ( U  <_  ( F `  A )  <->  ( U  <  ( F `
 A )  \/  U  =  ( F `
 A ) ) ) )
4134, 40mpbid 210 . . . 4  |-  ( ph  ->  ( U  <  ( F `  A )  \/  U  =  ( F `  A )
) )
4241adantr 465 . . 3  |-  ( (
ph  /\  ( F `  B )  <  U
)  ->  ( U  <  ( F `  A
)  \/  U  =  ( F `  A
) ) )
4320, 32, 42mpjaodan 784 . 2  |-  ( (
ph  /\  ( F `  B )  <  U
)  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
44 ubicc2 11400 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
4521, 22, 23, 44syl3anc 1218 . . 3  |-  ( ph  ->  B  e.  ( A [,] B ) )
46 fveq2 5689 . . . . 5  |-  ( c  =  B  ->  ( F `  c )  =  ( F `  B ) )
4746eqeq1d 2449 . . . 4  |-  ( c  =  B  ->  (
( F `  c
)  =  U  <->  ( F `  B )  =  U ) )
4847rspcev 3071 . . 3  |-  ( ( B  e.  ( A [,] B )  /\  ( F `  B )  =  U )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
4945, 48sylan 471 . 2  |-  ( (
ph  /\  ( F `  B )  =  U )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
5033simpld 459 . . 3  |-  ( ph  ->  ( F `  B
)  <_  U )
51 fveq2 5689 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
5251eleq1d 2507 . . . . . 6  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
5352rspcv 3067 . . . . 5  |-  ( B  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  B )  e.  RR ) )
5445, 35, 53sylc 60 . . . 4  |-  ( ph  ->  ( F `  B
)  e.  RR )
5554, 6leloed 9515 . . 3  |-  ( ph  ->  ( ( F `  B )  <_  U  <->  ( ( F `  B
)  <  U  \/  ( F `  B )  =  U ) ) )
5650, 55mpbid 210 . 2  |-  ( ph  ->  ( ( F `  B )  <  U  \/  ( F `  B
)  =  U ) )
5743, 49, 56mpjaodan 784 1  |-  ( ph  ->  E. c  e.  ( A [,] B ) ( F `  c
)  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714    C_ wss 3326   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   CCcc 9278   RRcr 9279   RR*cxr 9415    < clt 9416    <_ cle 9417   (,)cioo 11298   [,]cicc 11301   -cn->ccncf 20450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-fi 7659  df-sup 7689  df-oi 7722  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-q 10952  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-ioo 11302  df-icc 11305  df-fz 11436  df-fzo 11547  df-seq 11805  df-exp 11864  df-hash 12102  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-hom 14260  df-cco 14261  df-rest 14359  df-topn 14360  df-0g 14378  df-gsum 14379  df-topgen 14380  df-pt 14381  df-prds 14384  df-xrs 14438  df-qtop 14443  df-imas 14444  df-xps 14446  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-submnd 15463  df-mulg 15546  df-cntz 15833  df-cmn 16277  df-psmet 17807  df-xmet 17808  df-met 17809  df-bl 17810  df-mopn 17811  df-cnfld 17817  df-top 18501  df-bases 18503  df-topon 18504  df-topsp 18505  df-cn 18829  df-cnp 18830  df-tx 19133  df-hmeo 19326  df-xms 19893  df-ms 19894  df-tms 19895  df-cncf 20452
This theorem is referenced by:  ivthicc  20940  recosf1o  21989
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