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Theorem ivthle 19353
Description: The intermediate value theorem with weak inequality, increasing case. (Contributed by Mario Carneiro, 12-Aug-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 )
ivthle.9  |-  ( ph  ->  ( ( F `  A )  <_  U  /\  U  <_  ( F `
 B ) ) )
Assertion
Ref Expression
ivthle  |-  ( 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 ivthle
StepHypRef Expression
1 ioossicc 10996 . . . . 5  |-  ( A (,) B )  C_  ( A [,] B )
2 ivth.1 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
32adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  A  e.  RR )
4 ivth.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
54adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  B  e.  RR )
6 ivth.3 . . . . . . 7  |-  ( ph  ->  U  e.  RR )
76adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  U  e.  RR )
8 ivth.4 . . . . . . 7  |-  ( ph  ->  A  <  B )
98adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  A  <  B )
10 ivth.5 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  D )
1110adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  -> 
( A [,] B
)  C_  D )
12 ivth.7 . . . . . . 7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
1312adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  F  e.  ( D -cn->
CC ) )
14 ivth.8 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
1514adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  (
( F `  A
)  <  U  /\  U  <  ( F `  B ) ) )  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
16 simpr 448 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  -> 
( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
173, 5, 7, 9, 11, 13, 15, 16ivth 19351 . . . . 5  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
18 ssrexv 3408 . . . . 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 `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
2019anassrs 630 . . 3  |-  ( ( ( ph  /\  ( F `  A )  <  U )  /\  U  <  ( F `  B
) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
212rexrd 9134 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
224rexrd 9134 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
232, 4, 8ltled 9221 . . . . . 6  |-  ( ph  ->  A  <_  B )
24 ubicc2 11014 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
2521, 22, 23, 24syl3anc 1184 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
26 eqcom 2438 . . . . . . 7  |-  ( ( F `  c )  =  U  <->  U  =  ( F `  c ) )
27 fveq2 5728 . . . . . . . 8  |-  ( c  =  B  ->  ( F `  c )  =  ( F `  B ) )
2827eqeq2d 2447 . . . . . . 7  |-  ( c  =  B  ->  ( U  =  ( F `  c )  <->  U  =  ( F `  B ) ) )
2926, 28syl5bb 249 . . . . . 6  |-  ( c  =  B  ->  (
( F `  c
)  =  U  <->  U  =  ( F `  B ) ) )
3029rspcev 3052 . . . . 5  |-  ( ( B  e.  ( A [,] B )  /\  U  =  ( F `  B ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
3125, 30sylan 458 . . . 4  |-  ( (
ph  /\  U  =  ( F `  B ) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
3231adantlr 696 . . 3  |-  ( ( ( ph  /\  ( F `  A )  <  U )  /\  U  =  ( F `  B ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
33 ivthle.9 . . . . . 6  |-  ( ph  ->  ( ( F `  A )  <_  U  /\  U  <_  ( F `
 B ) ) )
3433simprd 450 . . . . 5  |-  ( ph  ->  U  <_  ( F `  B ) )
3514ralrimiva 2789 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
36 fveq2 5728 . . . . . . . . 9  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
3736eleq1d 2502 . . . . . . . 8  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
3837rspcv 3048 . . . . . . 7  |-  ( B  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  B )  e.  RR ) )
3925, 35, 38sylc 58 . . . . . 6  |-  ( ph  ->  ( F `  B
)  e.  RR )
406, 39leloed 9216 . . . . 5  |-  ( ph  ->  ( U  <_  ( F `  B )  <->  ( U  <  ( F `
 B )  \/  U  =  ( F `
 B ) ) ) )
4134, 40mpbid 202 . . . 4  |-  ( ph  ->  ( U  <  ( F `  B )  \/  U  =  ( F `  B )
) )
4241adantr 452 . . 3  |-  ( (
ph  /\  ( F `  A )  <  U
)  ->  ( U  <  ( F `  B
)  \/  U  =  ( F `  B
) ) )
4320, 32, 42mpjaodan 762 . 2  |-  ( (
ph  /\  ( F `  A )  <  U
)  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
44 lbicc2 11013 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
4521, 22, 23, 44syl3anc 1184 . . 3  |-  ( ph  ->  A  e.  ( A [,] B ) )
46 fveq2 5728 . . . . 5  |-  ( c  =  A  ->  ( F `  c )  =  ( F `  A ) )
4746eqeq1d 2444 . . . 4  |-  ( c  =  A  ->  (
( F `  c
)  =  U  <->  ( F `  A )  =  U ) )
4847rspcev 3052 . . 3  |-  ( ( A  e.  ( A [,] B )  /\  ( F `  A )  =  U )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
4945, 48sylan 458 . 2  |-  ( (
ph  /\  ( F `  A )  =  U )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
5033simpld 446 . . 3  |-  ( ph  ->  ( F `  A
)  <_  U )
51 fveq2 5728 . . . . . . 7  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
5251eleq1d 2502 . . . . . 6  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
5352rspcv 3048 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  A )  e.  RR ) )
5445, 35, 53sylc 58 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  RR )
5554, 6leloed 9216 . . 3  |-  ( ph  ->  ( ( F `  A )  <_  U  <->  ( ( F `  A
)  <  U  \/  ( F `  A )  =  U ) ) )
5650, 55mpbid 202 . 2  |-  ( ph  ->  ( ( F `  A )  <  U  \/  ( F `  A
)  =  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 2705   E.wrex 2706    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   RR*cxr 9119    < clt 9120    <_ cle 9121   (,)cioo 10916   [,]cicc 10919   -cn->ccncf 18906
This theorem is referenced by:  ivthicc  19355  volivth  19499
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ioo 10920  df-icc 10923  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-cncf 18908
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