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Theorem ivthle2 19344
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 10986 . . . . 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 19342 . . . . 5  |-  ( (
ph  /\  ( ( F `  B )  <  U  /\  U  < 
( F `  A
) ) )  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
18 ssrexv 3400 . . . . 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 9124 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
224rexrd 9124 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
232, 4, 8ltled 9211 . . . . . 6  |-  ( ph  ->  A  <_  B )
24 lbicc2 11003 . . . . . 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 2437 . . . . . . 7  |-  ( ( F `  c )  =  U  <->  U  =  ( F `  c ) )
27 fveq2 5720 . . . . . . . 8  |-  ( c  =  A  ->  ( F `  c )  =  ( F `  A ) )
2827eqeq2d 2446 . . . . . . 7  |-  ( c  =  A  ->  ( U  =  ( F `  c )  <->  U  =  ( F `  A ) ) )
2926, 28syl5bb 249 . . . . . 6  |-  ( c  =  A  ->  (
( F `  c
)  =  U  <->  U  =  ( F `  A ) ) )
3029rspcev 3044 . . . . 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 2781 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
36 fveq2 5720 . . . . . . . . 9  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
3736eleq1d 2501 . . . . . . . 8  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
3837rspcv 3040 . . . . . . 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 9206 . . . . 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 11004 . . . 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 5720 . . . . 5  |-  ( c  =  B  ->  ( F `  c )  =  ( F `  B ) )
4746eqeq1d 2443 . . . 4  |-  ( c  =  B  ->  (
( F `  c
)  =  U  <->  ( F `  B )  =  U ) )
4847rspcev 3044 . . 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 5720 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
5251eleq1d 2501 . . . . . 6  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
5352rspcv 3040 . . . . 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 9206 . . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8978   RRcr 8979   RR*cxr 9109    < clt 9110    <_ cle 9111   (,)cioo 10906   [,]cicc 10909   -cn->ccncf 18896
This theorem is referenced by:  ivthicc  19345  recosf1o  20427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-mulf 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7469  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-ioo 10910  df-icc 10913  df-fz 11034  df-fzo 11126  df-seq 11314  df-exp 11373  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-starv 13534  df-sca 13535  df-vsca 13536  df-tset 13538  df-ple 13539  df-ds 13541  df-unif 13542  df-hom 13543  df-cco 13544  df-rest 13640  df-topn 13641  df-topgen 13657  df-pt 13658  df-prds 13661  df-xrs 13716  df-0g 13717  df-gsum 13718  df-qtop 13723  df-imas 13724  df-xps 13726  df-mre 13801  df-mrc 13802  df-acs 13804  df-mnd 14680  df-submnd 14729  df-mulg 14805  df-cntz 15106  df-cmn 15404  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-cnfld 16694  df-top 16953  df-bases 16955  df-topon 16956  df-topsp 16957  df-cn 17281  df-cnp 17282  df-tx 17584  df-hmeo 17777  df-xms 18340  df-ms 18341  df-tms 18342  df-cncf 18898
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