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Theorem ftc1lem5 19387
Description: Lemma for ftc1 19389. (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 10731 . . . . . 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 3181 . . . 4  |-  ( ph  ->  X  e.  RR )
7 ioossicc 10735 . . . . . 6  |-  ( A (,) B )  C_  ( A [,] B )
8 ftc1.c . . . . . 6  |-  ( ph  ->  C  e.  ( A (,) B ) )
97, 8sseldi 3178 . . . . 5  |-  ( ph  ->  C  e.  ( A [,] B ) )
104, 9sseldd 3181 . . . 4  |-  ( ph  ->  C  e.  RR )
116, 10lttri2d 8958 . . 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 8955 . . . . . . . . 9  |-  ( (
ph  /\  X  <  C )  ->  X  =/=  C )
17 eldifsn 3749 . . . . . . . . 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 5525 . . . . . . . . . . 11  |-  ( z  =  X  ->  ( G `  z )  =  ( G `  X ) )
2019oveq1d 5873 . . . . . . . . . 10  |-  ( z  =  X  ->  (
( G `  z
)  -  ( G `
 C ) )  =  ( ( G `
 X )  -  ( G `  C ) ) )
21 oveq1 5865 . . . . . . . . . 10  |-  ( z  =  X  ->  (
z  -  C )  =  ( X  -  C ) )
2220, 21oveq12d 5876 . . . . . . . . 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 5883 . . . . . . . . 9  |-  ( ( ( G `  X
)  -  ( G `
 C ) )  /  ( X  -  C ) )  e. 
_V
2522, 23, 24fvmpt 5602 . . . . . . . 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 19385 . . . . . . . . . . . 12  |-  ( ph  ->  F : D --> CC )
3727, 1, 2, 28, 29, 30, 31, 36ftc1lem2 19383 . . . . . . . . . . 11  |-  ( ph  ->  G : ( A [,] B ) --> CC )
3837, 5ffvelrnd 5666 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  e.  CC )
3937, 9ffvelrnd 5666 . . . . . . . . . 10  |-  ( ph  ->  ( G `  C
)  e.  CC )
4038, 39subcld 9157 . . . . . . . . 9  |-  ( ph  ->  ( ( G `  X )  -  ( G `  C )
)  e.  CC )
4140adantr 451 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( ( G `  X )  -  ( G `  C ) )  e.  CC )
426recnd 8861 . . . . . . . . . 10  |-  ( ph  ->  X  e.  CC )
4310recnd 8861 . . . . . . . . . 10  |-  ( ph  ->  C  e.  CC )
4442, 43subcld 9157 . . . . . . . . 9  |-  ( ph  ->  ( X  -  C
)  e.  CC )
4544adantr 451 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( X  -  C )  e.  CC )
46 subeq0 9073 . . . . . . . . . . . 12  |-  ( ( X  e.  CC  /\  C  e.  CC )  ->  ( ( X  -  C )  =  0  <-> 
X  =  C ) )
4742, 43, 46syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X  -  C )  =  0  <-> 
X  =  C ) )
4847necon3bid 2481 . . . . . . . . . 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 9551 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( -u (
( G `  X
)  -  ( G `
 C ) )  /  -u ( X  -  C ) )  =  ( ( ( G `
 X )  -  ( G `  C ) )  /  ( X  -  C ) ) )
5238, 39negsubdi2d 9173 . . . . . . . . 9  |-  ( ph  -> 
-u ( ( G `
 X )  -  ( G `  C ) )  =  ( ( G `  C )  -  ( G `  X ) ) )
5342, 43negsubdi2d 9173 . . . . . . . . 9  |-  ( ph  -> 
-u ( X  -  C )  =  ( C  -  X ) )
5452, 53oveq12d 5876 . . . . . . . 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 2321 . . . . . 6  |-  ( (
ph  /\  X  <  C )  ->  ( H `  X )  =  ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) ) )
5756oveq1d 5873 . . . . 5  |-  ( (
ph  /\  X  <  C )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )
5857fveq2d 5529 . . . 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 9145 . . . . . . . 8  |-  ( ph  ->  ( C  -  C
)  =  0 )
6463fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( abs `  ( C  -  C )
)  =  ( abs `  0 ) )
65 abs0 11770 . . . . . . 7  |-  ( abs `  0 )  =  0
6664, 65syl6eq 2331 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  C )
)  =  0 )
6760rpgt0d 10393 . . . . . 6  |-  ( ph  ->  0  <  R )
6866, 67eqbrtrd 4043 . . . . 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 19386 . . . 4  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )  <  E
)
7058, 69eqbrtrd 4043 . . 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 8954 . . . . . . . 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 5873 . . . . 5  |-  ( (
ph  /\  C  <  X )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )
7877fveq2d 5529 . . . 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 19386 . . . 4  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )  <  E
)
8078, 79eqbrtrd 4043 . . 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 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   RR+crp 10354   (,)cioo 10656   [,]cicc 10659   abscabs 11719   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377    CnP ccnp 16955   L ^1cibl 18972   S.citg 18973
This theorem is referenced by:  ftc1lem6  19388
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976  df-itg2 18977  df-ibl 18978  df-itg 18979  df-0p 19025
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