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Theorem ivthlem1 19215
Description: Lemma for ivth 19218. The set  S of all 
x values with  ( F `  x ) less than  U is lower bounded by  A and upper bounded by  B. (Contributed by Mario Carneiro, 17-Jun-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 )
ivth.9  |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
ivth.10  |-  S  =  { x  e.  ( A [,] B )  |  ( F `  x )  <_  U }
Assertion
Ref Expression
ivthlem1  |-  ( ph  ->  ( A  e.  S  /\  A. z  e.  S  z  <_  B ) )
Distinct variable groups:    x, z, B    x, D, z    x, F, z    ph, x, z   
x, A    x, S, z    x, U, z
Allowed substitution hint:    A( z)

Proof of Theorem ivthlem1
StepHypRef Expression
1 ivth.1 . . . . 5  |-  ( ph  ->  A  e.  RR )
21rexrd 9067 . . . 4  |-  ( ph  ->  A  e.  RR* )
3 ivth.2 . . . . 5  |-  ( ph  ->  B  e.  RR )
43rexrd 9067 . . . 4  |-  ( ph  ->  B  e.  RR* )
5 ivth.4 . . . . 5  |-  ( ph  ->  A  <  B )
61, 3, 5ltled 9153 . . . 4  |-  ( ph  ->  A  <_  B )
7 lbicc2 10945 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
82, 4, 6, 7syl3anc 1184 . . 3  |-  ( ph  ->  A  e.  ( A [,] B ) )
9 ivth.8 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
109ralrimiva 2732 . . . . 5  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
11 fveq2 5668 . . . . . . 7  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
1211eleq1d 2453 . . . . . 6  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
1312rspcv 2991 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  A )  e.  RR ) )
148, 10, 13sylc 58 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  RR )
15 ivth.3 . . . 4  |-  ( ph  ->  U  e.  RR )
16 ivth.9 . . . . 5  |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
1716simpld 446 . . . 4  |-  ( ph  ->  ( F `  A
)  <  U )
1814, 15, 17ltled 9153 . . 3  |-  ( ph  ->  ( F `  A
)  <_  U )
1911breq1d 4163 . . . 4  |-  ( x  =  A  ->  (
( F `  x
)  <_  U  <->  ( F `  A )  <_  U
) )
20 ivth.10 . . . 4  |-  S  =  { x  e.  ( A [,] B )  |  ( F `  x )  <_  U }
2119, 20elrab2 3037 . . 3  |-  ( A  e.  S  <->  ( A  e.  ( A [,] B
)  /\  ( F `  A )  <_  U
) )
228, 18, 21sylanbrc 646 . 2  |-  ( ph  ->  A  e.  S )
23 ssrab2 3371 . . . . . 6  |-  { x  e.  ( A [,] B
)  |  ( F `
 x )  <_  U }  C_  ( A [,] B )
2420, 23eqsstri 3321 . . . . 5  |-  S  C_  ( A [,] B )
2524sseli 3287 . . . 4  |-  ( z  e.  S  ->  z  e.  ( A [,] B
) )
26 iccleub 10899 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  z  e.  ( A [,] B
) )  ->  z  <_  B )
27263expia 1155 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
z  e.  ( A [,] B )  -> 
z  <_  B )
)
282, 4, 27syl2anc 643 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B )  ->  z  <_  B
) )
2925, 28syl5 30 . . 3  |-  ( ph  ->  ( z  e.  S  ->  z  <_  B )
)
3029ralrimiv 2731 . 2  |-  ( ph  ->  A. z  e.  S  z  <_  B )
3122, 30jca 519 1  |-  ( ph  ->  ( A  e.  S  /\  A. z  e.  S  z  <_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653    C_ wss 3263   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   RR*cxr 9052    < clt 9053    <_ cle 9054   [,]cicc 10851   -cn->ccncf 18777
This theorem is referenced by:  ivthlem2  19216  ivthlem3  19217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-pre-lttri 8997  ax-pre-lttrn 8998
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-icc 10855
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