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Theorem ivthlem1 19340
Description: Lemma for ivth 19343. 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 9126 . . . 4  |-  ( ph  ->  A  e.  RR* )
3 ivth.2 . . . . 5  |-  ( ph  ->  B  e.  RR )
43rexrd 9126 . . . 4  |-  ( ph  ->  B  e.  RR* )
5 ivth.4 . . . . 5  |-  ( ph  ->  A  <  B )
61, 3, 5ltled 9213 . . . 4  |-  ( ph  ->  A  <_  B )
7 lbicc2 11005 . . . 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 2781 . . . . 5  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
11 fveq2 5720 . . . . . . 7  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
1211eleq1d 2501 . . . . . 6  |-  ( x  =  A  ->  (
( F `  x
)  e.  RR  <->  ( F `  A )  e.  RR ) )
1312rspcv 3040 . . . . 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 9213 . . 3  |-  ( ph  ->  ( F `  A
)  <_  U )
1911breq1d 4214 . . . 4  |-  ( x  =  A  ->  (
( F `  x
)  <_  U  <->  ( F `  A )  <_  U
) )
20 ivth.10 . . . 4  |-  S  =  { x  e.  ( A [,] B )  |  ( F `  x )  <_  U }
2119, 20elrab2 3086 . . 3  |-  ( A  e.  S  <->  ( A  e.  ( A [,] B
)  /\  ( F `  A )  <_  U
) )
228, 18, 21sylanbrc 646 . 2  |-  ( ph  ->  A  e.  S )
23 ssrab2 3420 . . . . . 6  |-  { x  e.  ( A [,] B
)  |  ( F `
 x )  <_  U }  C_  ( A [,] B )
2420, 23eqsstri 3370 . . . . 5  |-  S  C_  ( A [,] B )
2524sseli 3336 . . . 4  |-  ( z  e.  S  ->  z  e.  ( A [,] B
) )
26 iccleub 10959 . . . . . 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 2780 . 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 1652    e. wcel 1725   A.wral 2697   {crab 2701    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   RR*cxr 9111    < clt 9112    <_ cle 9113   [,]cicc 10911   -cn->ccncf 18898
This theorem is referenced by:  ivthlem2  19341  ivthlem3  19342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-icc 10915
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