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Theorem ftc1lem5 19403
Description: Lemma for ftc1 19405. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
Hypotheses
Ref Expression
ftc1.g  |-  G  =  ( x  e.  ( A [,] B ) 
|->  S. ( A (,) x ) ( F `
 t )  _d t )
ftc1.a  |-  ( ph  ->  A  e.  RR )
ftc1.b  |-  ( ph  ->  B  e.  RR )
ftc1.le  |-  ( ph  ->  A  <_  B )
ftc1.s  |-  ( ph  ->  ( A (,) B
)  C_  D )
ftc1.d  |-  ( ph  ->  D  C_  RR )
ftc1.i  |-  ( ph  ->  F  e.  L ^1 )
ftc1.c  |-  ( ph  ->  C  e.  ( A (,) B ) )
ftc1.f  |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `
 C ) )
ftc1.j  |-  J  =  ( Lt  RR )
ftc1.k  |-  K  =  ( Lt  D )
ftc1.l  |-  L  =  ( TopOpen ` fld )
ftc1.h  |-  H  =  ( z  e.  ( ( A [,] B
)  \  { C } )  |->  ( ( ( G `  z
)  -  ( G `
 C ) )  /  ( z  -  C ) ) )
ftc1.e  |-  ( ph  ->  E  e.  RR+ )
ftc1.r  |-  ( ph  ->  R  e.  RR+ )
ftc1.fc  |-  ( (
ph  /\  y  e.  D )  ->  (
( abs `  (
y  -  C ) )  <  R  -> 
( abs `  (
( F `  y
)  -  ( F `
 C ) ) )  <  E ) )
ftc1.x1  |-  ( ph  ->  X  e.  ( A [,] B ) )
ftc1.x2  |-  ( ph  ->  ( abs `  ( X  -  C )
)  <  R )
Assertion
Ref Expression
ftc1lem5  |-  ( (
ph  /\  X  =/=  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
Distinct variable groups:    x, t,
y, z, C    t, D, x, y, z    y, G, z    t, A, x, y, z    t, B, x, y, z    t, X, x, z    t, E, y    y, H    ph, t, x, y, z    t, F, x, y, z    x, L, y, z    y, R
Allowed substitution hints:    R( x, z, t)    E( x, z)    G( x, t)    H( x, z, t)    J( x, y, z, t)    K( x, y, z, t)    L( t)    X( y)

Proof of Theorem ftc1lem5
StepHypRef Expression
1 ftc1.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
2 ftc1.b . . . . . 6  |-  ( ph  ->  B  e.  RR )
3 iccssre 10747 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3syl2anc 642 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
5 ftc1.x1 . . . . 5  |-  ( ph  ->  X  e.  ( A [,] B ) )
64, 5sseldd 3194 . . . 4  |-  ( ph  ->  X  e.  RR )
7 ioossicc 10751 . . . . . 6  |-  ( A (,) B )  C_  ( A [,] B )
8 ftc1.c . . . . . 6  |-  ( ph  ->  C  e.  ( A (,) B ) )
97, 8sseldi 3191 . . . . 5  |-  ( ph  ->  C  e.  ( A [,] B ) )
104, 9sseldd 3194 . . . 4  |-  ( ph  ->  C  e.  RR )
116, 10lttri2d 8974 . . 3  |-  ( ph  ->  ( X  =/=  C  <->  ( X  <  C  \/  C  <  X ) ) )
1211biimpa 470 . 2  |-  ( (
ph  /\  X  =/=  C )  ->  ( X  <  C  \/  C  < 
X ) )
135adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  X  <  C )  ->  X  e.  ( A [,] B ) )
146adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  X  <  C )  ->  X  e.  RR )
15 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  X  <  C )  ->  X  <  C )
1614, 15ltned 8971 . . . . . . . . 9  |-  ( (
ph  /\  X  <  C )  ->  X  =/=  C )
17 eldifsn 3762 . . . . . . . . 9  |-  ( X  e.  ( ( A [,] B )  \  { C } )  <->  ( X  e.  ( A [,] B
)  /\  X  =/=  C ) )
1813, 16, 17sylanbrc 645 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  X  e.  ( ( A [,] B )  \  { C } ) )
19 fveq2 5541 . . . . . . . . . . 11  |-  ( z  =  X  ->  ( G `  z )  =  ( G `  X ) )
2019oveq1d 5889 . . . . . . . . . 10  |-  ( z  =  X  ->  (
( G `  z
)  -  ( G `
 C ) )  =  ( ( G `
 X )  -  ( G `  C ) ) )
21 oveq1 5881 . . . . . . . . . 10  |-  ( z  =  X  ->  (
z  -  C )  =  ( X  -  C ) )
2220, 21oveq12d 5892 . . . . . . . . 9  |-  ( z  =  X  ->  (
( ( G `  z )  -  ( G `  C )
)  /  ( z  -  C ) )  =  ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) ) )
23 ftc1.h . . . . . . . . 9  |-  H  =  ( z  e.  ( ( A [,] B
)  \  { C } )  |->  ( ( ( G `  z
)  -  ( G `
 C ) )  /  ( z  -  C ) ) )
24 ovex 5899 . . . . . . . . 9  |-  ( ( ( G `  X
)  -  ( G `
 C ) )  /  ( X  -  C ) )  e. 
_V
2522, 23, 24fvmpt 5618 . . . . . . . 8  |-  ( X  e.  ( ( A [,] B )  \  { C } )  -> 
( H `  X
)  =  ( ( ( G `  X
)  -  ( G `
 C ) )  /  ( X  -  C ) ) )
2618, 25syl 15 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( H `  X )  =  ( ( ( G `  X )  -  ( G `  C )
)  /  ( X  -  C ) ) )
27 ftc1.g . . . . . . . . . . . 12  |-  G  =  ( x  e.  ( A [,] B ) 
|->  S. ( A (,) x ) ( F `
 t )  _d t )
28 ftc1.le . . . . . . . . . . . 12  |-  ( ph  ->  A  <_  B )
29 ftc1.s . . . . . . . . . . . 12  |-  ( ph  ->  ( A (,) B
)  C_  D )
30 ftc1.d . . . . . . . . . . . 12  |-  ( ph  ->  D  C_  RR )
31 ftc1.i . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  L ^1 )
32 ftc1.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `
 C ) )
33 ftc1.j . . . . . . . . . . . . 13  |-  J  =  ( Lt  RR )
34 ftc1.k . . . . . . . . . . . . 13  |-  K  =  ( Lt  D )
35 ftc1.l . . . . . . . . . . . . 13  |-  L  =  ( TopOpen ` fld )
3627, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35ftc1lem3 19401 . . . . . . . . . . . 12  |-  ( ph  ->  F : D --> CC )
3727, 1, 2, 28, 29, 30, 31, 36ftc1lem2 19399 . . . . . . . . . . 11  |-  ( ph  ->  G : ( A [,] B ) --> CC )
3837, 5ffvelrnd 5682 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  e.  CC )
3937, 9ffvelrnd 5682 . . . . . . . . . 10  |-  ( ph  ->  ( G `  C
)  e.  CC )
4038, 39subcld 9173 . . . . . . . . 9  |-  ( ph  ->  ( ( G `  X )  -  ( G `  C )
)  e.  CC )
4140adantr 451 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( ( G `  X )  -  ( G `  C ) )  e.  CC )
426recnd 8877 . . . . . . . . . 10  |-  ( ph  ->  X  e.  CC )
4310recnd 8877 . . . . . . . . . 10  |-  ( ph  ->  C  e.  CC )
4442, 43subcld 9173 . . . . . . . . 9  |-  ( ph  ->  ( X  -  C
)  e.  CC )
4544adantr 451 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( X  -  C )  e.  CC )
46 subeq0 9089 . . . . . . . . . . . 12  |-  ( ( X  e.  CC  /\  C  e.  CC )  ->  ( ( X  -  C )  =  0  <-> 
X  =  C ) )
4742, 43, 46syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X  -  C )  =  0  <-> 
X  =  C ) )
4847necon3bid 2494 . . . . . . . . . 10  |-  ( ph  ->  ( ( X  -  C )  =/=  0  <->  X  =/=  C ) )
4948biimpar 471 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  C )  ->  ( X  -  C )  =/=  0
)
5016, 49syldan 456 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( X  -  C )  =/=  0
)
5141, 45, 50div2negd 9567 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( -u (
( G `  X
)  -  ( G `
 C ) )  /  -u ( X  -  C ) )  =  ( ( ( G `
 X )  -  ( G `  C ) )  /  ( X  -  C ) ) )
5238, 39negsubdi2d 9189 . . . . . . . . 9  |-  ( ph  -> 
-u ( ( G `
 X )  -  ( G `  C ) )  =  ( ( G `  C )  -  ( G `  X ) ) )
5342, 43negsubdi2d 9189 . . . . . . . . 9  |-  ( ph  -> 
-u ( X  -  C )  =  ( C  -  X ) )
5452, 53oveq12d 5892 . . . . . . . 8  |-  ( ph  ->  ( -u ( ( G `  X )  -  ( G `  C ) )  /  -u ( X  -  C
) )  =  ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) ) )
5554adantr 451 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( -u (
( G `  X
)  -  ( G `
 C ) )  /  -u ( X  -  C ) )  =  ( ( ( G `
 C )  -  ( G `  X ) )  /  ( C  -  X ) ) )
5626, 51, 553eqtr2d 2334 . . . . . 6  |-  ( (
ph  /\  X  <  C )  ->  ( H `  X )  =  ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) ) )
5756oveq1d 5889 . . . . 5  |-  ( (
ph  /\  X  <  C )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )
5857fveq2d 5545 . . . 4  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  =  ( abs `  ( ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) )  -  ( F `  C ) ) ) )
59 ftc1.e . . . . 5  |-  ( ph  ->  E  e.  RR+ )
60 ftc1.r . . . . 5  |-  ( ph  ->  R  e.  RR+ )
61 ftc1.fc . . . . 5  |-  ( (
ph  /\  y  e.  D )  ->  (
( abs `  (
y  -  C ) )  <  R  -> 
( abs `  (
( F `  y
)  -  ( F `
 C ) ) )  <  E ) )
62 ftc1.x2 . . . . 5  |-  ( ph  ->  ( abs `  ( X  -  C )
)  <  R )
6343subidd 9161 . . . . . . . 8  |-  ( ph  ->  ( C  -  C
)  =  0 )
6463fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( abs `  ( C  -  C )
)  =  ( abs `  0 ) )
65 abs0 11786 . . . . . . 7  |-  ( abs `  0 )  =  0
6664, 65syl6eq 2344 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  C )
)  =  0 )
6760rpgt0d 10409 . . . . . 6  |-  ( ph  ->  0  <  R )
6866, 67eqbrtrd 4059 . . . . 5  |-  ( ph  ->  ( abs `  ( C  -  C )
)  <  R )
6927, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35, 23, 59, 60, 61, 5, 62, 9, 68ftc1lem4 19402 . . . 4  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )  <  E
)
7058, 69eqbrtrd 4059 . . 3  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
715adantr 451 . . . . . . . 8  |-  ( (
ph  /\  C  <  X )  ->  X  e.  ( A [,] B ) )
7210adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  C  <  X )  ->  C  e.  RR )
73 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  C  <  X )  ->  C  <  X )
7472, 73gtned 8970 . . . . . . . 8  |-  ( (
ph  /\  C  <  X )  ->  X  =/=  C )
7571, 74, 17sylanbrc 645 . . . . . . 7  |-  ( (
ph  /\  C  <  X )  ->  X  e.  ( ( A [,] B )  \  { C } ) )
7675, 25syl 15 . . . . . 6  |-  ( (
ph  /\  C  <  X )  ->  ( H `  X )  =  ( ( ( G `  X )  -  ( G `  C )
)  /  ( X  -  C ) ) )
7776oveq1d 5889 . . . . 5  |-  ( (
ph  /\  C  <  X )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )
7877fveq2d 5545 . . . 4  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  =  ( abs `  ( ( ( ( G `  X )  -  ( G `  C )
)  /  ( X  -  C ) )  -  ( F `  C ) ) ) )
7927, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35, 23, 59, 60, 61, 9, 68, 5, 62ftc1lem4 19402 . . . 4  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )  <  E
)
8078, 79eqbrtrd 4059 . . 3  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
8170, 80jaodan 760 . 2  |-  ( (
ph  /\  ( X  <  C  \/  C  < 
X ) )  -> 
( abs `  (
( H `  X
)  -  ( F `
 C ) ) )  <  E )
8212, 81syldan 456 1  |-  ( (
ph  /\  X  =/=  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   abscabs 11735   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393    CnP ccnp 16971   L ^1cibl 18988   S.citg 18989
This theorem is referenced by:  ftc1lem6  19404
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  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  ax-addf 8832  ax-mulf 8833
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-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  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-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992  df-itg2 18993  df-ibl 18994  df-itg 18995  df-0p 19041
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