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Theorem ftc1lem5 19926
Description: Lemma for ftc1 19928. (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 10994 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3syl2anc 644 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
5 ftc1.x1 . . . . 5  |-  ( ph  ->  X  e.  ( A [,] B ) )
64, 5sseldd 3351 . . . 4  |-  ( ph  ->  X  e.  RR )
7 ioossicc 10998 . . . . . 6  |-  ( A (,) B )  C_  ( A [,] B )
8 ftc1.c . . . . . 6  |-  ( ph  ->  C  e.  ( A (,) B ) )
97, 8sseldi 3348 . . . . 5  |-  ( ph  ->  C  e.  ( A [,] B ) )
104, 9sseldd 3351 . . . 4  |-  ( ph  ->  C  e.  RR )
116, 10lttri2d 9214 . . 3  |-  ( ph  ->  ( X  =/=  C  <->  ( X  <  C  \/  C  <  X ) ) )
1211biimpa 472 . 2  |-  ( (
ph  /\  X  =/=  C )  ->  ( X  <  C  \/  C  < 
X ) )
135adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  X  <  C )  ->  X  e.  ( A [,] B ) )
146adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  X  <  C )  ->  X  e.  RR )
15 simpr 449 . . . . . . . . . 10  |-  ( (
ph  /\  X  <  C )  ->  X  <  C )
1614, 15ltned 9211 . . . . . . . . 9  |-  ( (
ph  /\  X  <  C )  ->  X  =/=  C )
17 eldifsn 3929 . . . . . . . . 9  |-  ( X  e.  ( ( A [,] B )  \  { C } )  <->  ( X  e.  ( A [,] B
)  /\  X  =/=  C ) )
1813, 16, 17sylanbrc 647 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  X  e.  ( ( A [,] B )  \  { C } ) )
19 fveq2 5730 . . . . . . . . . . 11  |-  ( z  =  X  ->  ( G `  z )  =  ( G `  X ) )
2019oveq1d 6098 . . . . . . . . . 10  |-  ( z  =  X  ->  (
( G `  z
)  -  ( G `
 C ) )  =  ( ( G `
 X )  -  ( G `  C ) ) )
21 oveq1 6090 . . . . . . . . . 10  |-  ( z  =  X  ->  (
z  -  C )  =  ( X  -  C ) )
2220, 21oveq12d 6101 . . . . . . . . 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 6108 . . . . . . . . 9  |-  ( ( ( G `  X
)  -  ( G `
 C ) )  /  ( X  -  C ) )  e. 
_V
2522, 23, 24fvmpt 5808 . . . . . . . 8  |-  ( X  e.  ( ( A [,] B )  \  { C } )  -> 
( H `  X
)  =  ( ( ( G `  X
)  -  ( G `
 C ) )  /  ( X  -  C ) ) )
2618, 25syl 16 . . . . . . 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 19924 . . . . . . . . . . . 12  |-  ( ph  ->  F : D --> CC )
3727, 1, 2, 28, 29, 30, 31, 36ftc1lem2 19922 . . . . . . . . . . 11  |-  ( ph  ->  G : ( A [,] B ) --> CC )
3837, 5ffvelrnd 5873 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  e.  CC )
3937, 9ffvelrnd 5873 . . . . . . . . . 10  |-  ( ph  ->  ( G `  C
)  e.  CC )
4038, 39subcld 9413 . . . . . . . . 9  |-  ( ph  ->  ( ( G `  X )  -  ( G `  C )
)  e.  CC )
4140adantr 453 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( ( G `  X )  -  ( G `  C ) )  e.  CC )
426recnd 9116 . . . . . . . . . 10  |-  ( ph  ->  X  e.  CC )
4310recnd 9116 . . . . . . . . . 10  |-  ( ph  ->  C  e.  CC )
4442, 43subcld 9413 . . . . . . . . 9  |-  ( ph  ->  ( X  -  C
)  e.  CC )
4544adantr 453 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( X  -  C )  e.  CC )
4642, 43subeq0ad 9423 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X  -  C )  =  0  <-> 
X  =  C ) )
4746necon3bid 2638 . . . . . . . . . 10  |-  ( ph  ->  ( ( X  -  C )  =/=  0  <->  X  =/=  C ) )
4847biimpar 473 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  C )  ->  ( X  -  C )  =/=  0
)
4916, 48syldan 458 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( X  -  C )  =/=  0
)
5041, 45, 49div2negd 9807 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( -u (
( G `  X
)  -  ( G `
 C ) )  /  -u ( X  -  C ) )  =  ( ( ( G `
 X )  -  ( G `  C ) )  /  ( X  -  C ) ) )
5138, 39negsubdi2d 9429 . . . . . . . . 9  |-  ( ph  -> 
-u ( ( G `
 X )  -  ( G `  C ) )  =  ( ( G `  C )  -  ( G `  X ) ) )
5242, 43negsubdi2d 9429 . . . . . . . . 9  |-  ( ph  -> 
-u ( X  -  C )  =  ( C  -  X ) )
5351, 52oveq12d 6101 . . . . . . . 8  |-  ( ph  ->  ( -u ( ( G `  X )  -  ( G `  C ) )  /  -u ( X  -  C
) )  =  ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) ) )
5453adantr 453 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( -u (
( G `  X
)  -  ( G `
 C ) )  /  -u ( X  -  C ) )  =  ( ( ( G `
 C )  -  ( G `  X ) )  /  ( C  -  X ) ) )
5526, 50, 543eqtr2d 2476 . . . . . 6  |-  ( (
ph  /\  X  <  C )  ->  ( H `  X )  =  ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) ) )
5655oveq1d 6098 . . . . 5  |-  ( (
ph  /\  X  <  C )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )
5756fveq2d 5734 . . . 4  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  =  ( abs `  ( ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) )  -  ( F `  C ) ) ) )
58 ftc1.e . . . . 5  |-  ( ph  ->  E  e.  RR+ )
59 ftc1.r . . . . 5  |-  ( ph  ->  R  e.  RR+ )
60 ftc1.fc . . . . 5  |-  ( (
ph  /\  y  e.  D )  ->  (
( abs `  (
y  -  C ) )  <  R  -> 
( abs `  (
( F `  y
)  -  ( F `
 C ) ) )  <  E ) )
61 ftc1.x2 . . . . 5  |-  ( ph  ->  ( abs `  ( X  -  C )
)  <  R )
6243subidd 9401 . . . . . . 7  |-  ( ph  ->  ( C  -  C
)  =  0 )
6362abs00bd 12098 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  C )
)  =  0 )
6459rpgt0d 10653 . . . . . 6  |-  ( ph  ->  0  <  R )
6563, 64eqbrtrd 4234 . . . . 5  |-  ( ph  ->  ( abs `  ( C  -  C )
)  <  R )
6627, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35, 23, 58, 59, 60, 5, 61, 9, 65ftc1lem4 19925 . . . 4  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )  <  E
)
6757, 66eqbrtrd 4234 . . 3  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
685adantr 453 . . . . . . . 8  |-  ( (
ph  /\  C  <  X )  ->  X  e.  ( A [,] B ) )
6910adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  C  <  X )  ->  C  e.  RR )
70 simpr 449 . . . . . . . . 9  |-  ( (
ph  /\  C  <  X )  ->  C  <  X )
7169, 70gtned 9210 . . . . . . . 8  |-  ( (
ph  /\  C  <  X )  ->  X  =/=  C )
7268, 71, 17sylanbrc 647 . . . . . . 7  |-  ( (
ph  /\  C  <  X )  ->  X  e.  ( ( A [,] B )  \  { C } ) )
7372, 25syl 16 . . . . . 6  |-  ( (
ph  /\  C  <  X )  ->  ( H `  X )  =  ( ( ( G `  X )  -  ( G `  C )
)  /  ( X  -  C ) ) )
7473oveq1d 6098 . . . . 5  |-  ( (
ph  /\  C  <  X )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )
7574fveq2d 5734 . . . 4  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  =  ( abs `  ( ( ( ( G `  X )  -  ( G `  C )
)  /  ( X  -  C ) )  -  ( F `  C ) ) ) )
7627, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35, 23, 58, 59, 60, 9, 65, 5, 61ftc1lem4 19925 . . . 4  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )  <  E
)
7775, 76eqbrtrd 4234 . . 3  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
7867, 77jaodan 762 . 2  |-  ( (
ph  /\  ( X  <  C  \/  C  < 
X ) )  -> 
( abs `  (
( H `  X
)  -  ( F `
 C ) ) )  <  E )
7912, 78syldan 458 1  |-  ( (
ph  /\  X  =/=  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601    \ cdif 3319    C_ wss 3322   {csn 3816   class class class wbr 4214    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992    < clt 9122    <_ cle 9123    - cmin 9293   -ucneg 9294    / cdiv 9679   RR+crp 10614   (,)cioo 10918   [,]cicc 10921   abscabs 12041   ↾t crest 13650   TopOpenctopn 13651  ℂfldccnfld 16705    CnP ccnp 17291   L ^1cibl 19511   S.citg 19512
This theorem is referenced by:  ftc1lem6  19927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cc 8317  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-ofr 6308  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-acn 7831  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cn 17293  df-cnp 17294  df-cmp 17452  df-tx 17596  df-hmeo 17789  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-ovol 19363  df-vol 19364  df-mbf 19514  df-itg1 19515  df-itg2 19516  df-ibl 19517  df-itg 19518  df-0p 19564
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