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Theorem ivthle 18832
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 10751 . . . . 5  |-  ( A (,) B )  C_  ( A [,] B )
2 ivth.1 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
32adantr 451 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  A  e.  RR )
4 ivth.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
54adantr 451 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  B  e.  RR )
6 ivth.3 . . . . . . 7  |-  ( ph  ->  U  e.  RR )
76adantr 451 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  U  e.  RR )
8 ivth.4 . . . . . . 7  |-  ( ph  ->  A  <  B )
98adantr 451 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  A  <  B )
10 ivth.5 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  D )
1110adantr 451 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  -> 
( A [,] B
)  C_  D )
12 ivth.7 . . . . . . 7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
1312adantr 451 . . . . . 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 695 . . . . . 6  |-  ( ( ( ph  /\  (
( F `  A
)  <  U  /\  U  <  ( F `  B ) ) )  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
16 simpr 447 . . . . . 6  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  -> 
( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
173, 5, 7, 9, 11, 13, 15, 16ivth 18830 . . . . 5  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
18 ssrexv 3251 . . . . 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 59 . . . 4  |-  ( (
ph  /\  ( ( F `  A )  <  U  /\  U  < 
( F `  B
) ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
2019anassrs 629 . . 3  |-  ( ( ( ph  /\  ( F `  A )  <  U )  /\  U  <  ( F `  B
) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
212rexrd 8897 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
224rexrd 8897 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
232, 4, 8ltled 8983 . . . . . 6  |-  ( ph  ->  A  <_  B )
24 ubicc2 10769 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
2521, 22, 23, 24syl3anc 1182 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
26 eqcom 2298 . . . . . . 7  |-  ( ( F `  c )  =  U  <->  U  =  ( F `  c ) )
27 fveq2 5541 . . . . . . . 8  |-  ( c  =  B  ->  ( F `  c )  =  ( F `  B ) )
2827eqeq2d 2307 . . . . . . 7  |-  ( c  =  B  ->  ( U  =  ( F `  c )  <->  U  =  ( F `  B ) ) )
2926, 28syl5bb 248 . . . . . 6  |-  ( c  =  B  ->  (
( F `  c
)  =  U  <->  U  =  ( F `  B ) ) )
3029rspcev 2897 . . . . 5  |-  ( ( B  e.  ( A [,] B )  /\  U  =  ( F `  B ) )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
3125, 30sylan 457 . . . 4  |-  ( (
ph  /\  U  =  ( F `  B ) )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
3231adantlr 695 . . 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 449 . . . . 5  |-  ( ph  ->  U  <_  ( F `  B ) )
3514ralrimiva 2639 . . . . . . 7  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
36 fveq2 5541 . . . . . . . . 9  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
3736eleq1d 2362 . . . . . . . 8  |-  ( x  =  B  ->  (
( F `  x
)  e.  RR  <->  ( F `  B )  e.  RR ) )
3837rspcv 2893 . . . . . . 7  |-  ( B  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  B )  e.  RR ) )
3925, 35, 38sylc 56 . . . . . 6  |-  ( ph  ->  ( F `  B
)  e.  RR )
406, 39leloed 8978 . . . . 5  |-  ( ph  ->  ( U  <_  ( F `  B )  <->  ( U  <  ( F `
 B )  \/  U  =  ( F `
 B ) ) ) )
4134, 40mpbid 201 . . . 4  |-  ( ph  ->  ( U  <  ( F `  B )  \/  U  =  ( F `  B )
) )
4241adantr 451 . . 3  |-  ( (
ph  /\  ( F `  A )  <  U
)  ->  ( U  <  ( F `  B
)  \/  U  =  ( F `  B
) ) )
4320, 32, 42mpjaodan 761 . 2  |-  ( (
ph  /\  ( F `  A )  <  U
)  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
44 lbicc2 10768 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
4521, 22, 23, 44syl3anc 1182 . . 3  |-  ( ph  ->  A  e.  ( A [,] B ) )
46 fveq2 5541 . . . . 5  |-  ( c  =  A  ->  ( F `  c )  =  ( F `  A ) )
4746eqeq1d 2304 . . . 4  |-  ( c  =  A  ->  (
( F `  c
)  =  U  <->  ( F `  A )  =  U ) )
4847rspcev 2897 . . 3  |-  ( ( A  e.  ( A [,] B )  /\  ( F `  A )  =  U )  ->  E. c  e.  ( A [,] B ) ( F `  c )  =  U )
4945, 48sylan 457 . 2  |-  ( (
ph  /\  ( F `  A )  =  U )  ->  E. c  e.  ( A [,] B
) ( F `  c )  =  U )
5033simpld 445 . . 3  |-  ( ph  ->  ( F `  A
)  <_  U )
51 fveq2 5541 . . . . . . 7  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
5251eleq1d 2362 . . . . . 6  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
5352rspcv 2893 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  A )  e.  RR ) )
5445, 35, 53sylc 56 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  RR )
5554, 6leloed 8978 . . 3  |-  ( ph  ->  ( ( F `  A )  <_  U  <->  ( ( F `  A
)  <  U  \/  ( F `  A )  =  U ) ) )
5650, 55mpbid 201 . 2  |-  ( ph  ->  ( ( F `  A )  <  U  \/  ( F `  A
)  =  U ) )
5743, 49, 56mpjaodan 761 1  |-  ( ph  ->  E. c  e.  ( A [,] B ) ( F `  c
)  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   RR*cxr 8882    < clt 8883    <_ cle 8884   (,)cioo 10672   [,]cicc 10675   -cn->ccncf 18396
This theorem is referenced by:  ivthicc  18834  volivth  18978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ioo 10676  df-icc 10679  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-cncf 18398
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