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Theorem ivthlem1 18827
Description: Lemma for ivth 18830. 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 8897 . . . 4  |-  ( ph  ->  A  e.  RR* )
3 ivth.2 . . . . 5  |-  ( ph  ->  B  e.  RR )
43rexrd 8897 . . . 4  |-  ( ph  ->  B  e.  RR* )
5 ivth.4 . . . . 5  |-  ( ph  ->  A  <  B )
61, 3, 5ltled 8983 . . . 4  |-  ( ph  ->  A  <_  B )
7 lbicc2 10768 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
82, 4, 6, 7syl3anc 1182 . . 3  |-  ( ph  ->  A  e.  ( A [,] B ) )
9 ivth.8 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
109ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
11 fveq2 5541 . . . . . . 7  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
1211eleq1d 2362 . . . . . 6  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
1312rspcv 2893 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  ( A. x  e.  ( A [,] B ) ( F `  x )  e.  RR  ->  ( F `  A )  e.  RR ) )
148, 10, 13sylc 56 . . . 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 445 . . . 4  |-  ( ph  ->  ( F `  A
)  <  U )
1814, 15, 17ltled 8983 . . 3  |-  ( ph  ->  ( F `  A
)  <_  U )
1911breq1d 4049 . . . 4  |-  ( x  =  A  ->  (
( F `  x
)  <_  U  <->  ( F `  A )  <_  U
) )
20 ivth.10 . . . 4  |-  S  =  { x  e.  ( A [,] B )  |  ( F `  x )  <_  U }
2119, 20elrab2 2938 . . 3  |-  ( A  e.  S  <->  ( A  e.  ( A [,] B
)  /\  ( F `  A )  <_  U
) )
228, 18, 21sylanbrc 645 . 2  |-  ( ph  ->  A  e.  S )
23 ssrab2 3271 . . . . . 6  |-  { x  e.  ( A [,] B
)  |  ( F `
 x )  <_  U }  C_  ( A [,] B )
2420, 23eqsstri 3221 . . . . 5  |-  S  C_  ( A [,] B )
2524sseli 3189 . . . 4  |-  ( z  e.  S  ->  z  e.  ( A [,] B
) )
26 iccleub 10723 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  z  e.  ( A [,] B
) )  ->  z  <_  B )
27263expia 1153 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
z  e.  ( A [,] B )  -> 
z  <_  B )
)
282, 4, 27syl2anc 642 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B )  ->  z  <_  B
) )
2925, 28syl5 28 . . 3  |-  ( ph  ->  ( z  e.  S  ->  z  <_  B )
)
3029ralrimiv 2638 . 2  |-  ( ph  ->  A. z  e.  S  z  <_  B )
3122, 30jca 518 1  |-  ( ph  ->  ( A  e.  S  /\  A. z  e.  S  z  <_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    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   [,]cicc 10675   -cn->ccncf 18396
This theorem is referenced by:  ivthlem2  18828  ivthlem3  18829
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-pre-lttri 8827  ax-pre-lttrn 8828
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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-icc 10679
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