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Theorem ivthlem1 18811
Description: Lemma for ivth 18814. 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 8881 . . . 4  |-  ( ph  ->  A  e.  RR* )
3 ivth.2 . . . . 5  |-  ( ph  ->  B  e.  RR )
43rexrd 8881 . . . 4  |-  ( ph  ->  B  e.  RR* )
5 ivth.4 . . . . 5  |-  ( ph  ->  A  <  B )
61, 3, 5ltled 8967 . . . 4  |-  ( ph  ->  A  <_  B )
7 lbicc2 10752 . . . 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 2626 . . . . 5  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
11 fveq2 5525 . . . . . . 7  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
1211eleq1d 2349 . . . . . 6  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
1312rspcv 2880 . . . . 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 8967 . . 3  |-  ( ph  ->  ( F `  A
)  <_  U )
1911breq1d 4033 . . . 4  |-  ( x  =  A  ->  (
( F `  x
)  <_  U  <->  ( F `  A )  <_  U
) )
20 ivth.10 . . . 4  |-  S  =  { x  e.  ( A [,] B )  |  ( F `  x )  <_  U }
2119, 20elrab2 2925 . . 3  |-  ( A  e.  S  <->  ( A  e.  ( A [,] B
)  /\  ( F `  A )  <_  U
) )
228, 18, 21sylanbrc 645 . 2  |-  ( ph  ->  A  e.  S )
23 ssrab2 3258 . . . . . 6  |-  { x  e.  ( A [,] B
)  |  ( F `
 x )  <_  U }  C_  ( A [,] B )
2420, 23eqsstri 3208 . . . . 5  |-  S  C_  ( A [,] B )
2524sseli 3176 . . . 4  |-  ( z  e.  S  ->  z  e.  ( A [,] B
) )
26 iccleub 10707 . . . . . 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 2625 . 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 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   RR*cxr 8866    < clt 8867    <_ cle 8868   [,]cicc 10659   -cn->ccncf 18380
This theorem is referenced by:  ivthlem2  18812  ivthlem3  18813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663
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