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Theorem ivthle 20952
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 11393 . . . . 5  |-  ( A (,) B )  C_  ( A [,] B )
2 ivth.1 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
32adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  A  e.  RR )
4 ivth.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  B  e.  RR )
6 ivth.3 . . . . . . 7  |-  ( ph  ->  U  e.  RR )
76adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  U  e.  RR )
8 ivth.4 . . . . . . 7  |-  ( ph  ->  A  <  B )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  A  <  B )
10 ivth.5 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  D )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  -> 
( A [,] B
)  C_  D )
12 ivth.7 . . . . . . 7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
1312adantr 465 . . . . . 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 714 . . . . . 6  |-  ( ( ( ph  /\  (
( F `  A
)  <  U  /\  U  <  ( F `  B ) ) )  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
16 simpr 461 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  -> 
( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
173, 5, 7, 9, 11, 13, 15, 16ivth 20950 . . . . 5  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
18 ssrexv 3429 . . . . 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 `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
2019anassrs 648 . . 3  |-  ( ( ( ph  /\  ( F `  A )  <  U )  /\  U  <  ( F `  B
) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
212rexrd 9445 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
224rexrd 9445 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
232, 4, 8ltled 9534 . . . . . 6  |-  ( ph  ->  A  <_  B )
24 ubicc2 11414 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
2521, 22, 23, 24syl3anc 1218 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
26 eqcom 2445 . . . . . . 7  |-  ( ( F `  c )  =  U  <->  U  =  ( F `  c ) )
27 fveq2 5703 . . . . . . . 8  |-  ( c  =  B  ->  ( F `  c )  =  ( F `  B ) )
2827eqeq2d 2454 . . . . . . 7  |-  ( c  =  B  ->  ( U  =  ( F `  c )  <->  U  =  ( F `  B ) ) )
2926, 28syl5bb 257 . . . . . 6  |-  ( c  =  B  ->  (
( F `  c
)  =  U  <->  U  =  ( F `  B ) ) )
3029rspcev 3085 . . . . 5  |-  ( ( B  e.  ( A [,] B )  /\  U  =  ( F `  B ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
3125, 30sylan 471 . . . 4  |-  ( (
ph  /\  U  =  ( F `  B ) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
3231adantlr 714 . . 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 463 . . . . 5  |-  ( ph  ->  U  <_  ( F `  B ) )
3514ralrimiva 2811 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
36 fveq2 5703 . . . . . . . . 9  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
3736eleq1d 2509 . . . . . . . 8  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
3837rspcv 3081 . . . . . . 7  |-  ( B  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  B )  e.  RR ) )
3925, 35, 38sylc 60 . . . . . 6  |-  ( ph  ->  ( F `  B
)  e.  RR )
406, 39leloed 9529 . . . . 5  |-  ( ph  ->  ( U  <_  ( F `  B )  <->  ( U  <  ( F `
 B )  \/  U  =  ( F `
 B ) ) ) )
4134, 40mpbid 210 . . . 4  |-  ( ph  ->  ( U  <  ( F `  B )  \/  U  =  ( F `  B )
) )
4241adantr 465 . . 3  |-  ( (
ph  /\  ( F `  A )  <  U
)  ->  ( U  <  ( F `  B
)  \/  U  =  ( F `  B
) ) )
4320, 32, 42mpjaodan 784 . 2  |-  ( (
ph  /\  ( F `  A )  <  U
)  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
44 lbicc2 11413 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
4521, 22, 23, 44syl3anc 1218 . . 3  |-  ( ph  ->  A  e.  ( A [,] B ) )
46 fveq2 5703 . . . . 5  |-  ( c  =  A  ->  ( F `  c )  =  ( F `  A ) )
4746eqeq1d 2451 . . . 4  |-  ( c  =  A  ->  (
( F `  c
)  =  U  <->  ( F `  A )  =  U ) )
4847rspcev 3085 . . 3  |-  ( ( A  e.  ( A [,] B )  /\  ( F `  A )  =  U )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
4945, 48sylan 471 . 2  |-  ( (
ph  /\  ( F `  A )  =  U )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
5033simpld 459 . . 3  |-  ( ph  ->  ( F `  A
)  <_  U )
51 fveq2 5703 . . . . . . 7  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
5251eleq1d 2509 . . . . . 6  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
5352rspcv 3081 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  A )  e.  RR ) )
5445, 35, 53sylc 60 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  RR )
5554, 6leloed 9529 . . 3  |-  ( ph  ->  ( ( F `  A )  <_  U  <->  ( ( F `  A
)  <  U  \/  ( F `  A )  =  U ) ) )
5650, 55mpbid 210 . 2  |-  ( ph  ->  ( ( F `  A )  <  U  \/  ( F `  A
)  =  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 2727   E.wrex 2728    C_ wss 3340   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   CCcc 9292   RRcr 9293   RR*cxr 9429    < clt 9430    <_ cle 9431   (,)cioo 11312   [,]cicc 11315   -cn->ccncf 20464
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 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-ioo 11316  df-icc 11319  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-cncf 20466
This theorem is referenced by:  ivthicc  20954  volivth  21099
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