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Theorem ftc1lem5 22866
Description: Lemma for ftc1 22868. (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 11705 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3syl2anc 665 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
5 ftc1.x1 . . . . 5  |-  ( ph  ->  X  e.  ( A [,] B ) )
64, 5sseldd 3462 . . . 4  |-  ( ph  ->  X  e.  RR )
7 ioossicc 11709 . . . . . 6  |-  ( A (,) B )  C_  ( A [,] B )
8 ftc1.c . . . . . 6  |-  ( ph  ->  C  e.  ( A (,) B ) )
97, 8sseldi 3459 . . . . 5  |-  ( ph  ->  C  e.  ( A [,] B ) )
104, 9sseldd 3462 . . . 4  |-  ( ph  ->  C  e.  RR )
116, 10lttri2d 9763 . . 3  |-  ( ph  ->  ( X  =/=  C  <->  ( X  <  C  \/  C  <  X ) ) )
1211biimpa 486 . 2  |-  ( (
ph  /\  X  =/=  C )  ->  ( X  <  C  \/  C  < 
X ) )
135adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  X  <  C )  ->  X  e.  ( A [,] B ) )
146adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  X  <  C )  ->  X  e.  RR )
15 simpr 462 . . . . . . . . . 10  |-  ( (
ph  /\  X  <  C )  ->  X  <  C )
1614, 15ltned 9760 . . . . . . . . 9  |-  ( (
ph  /\  X  <  C )  ->  X  =/=  C )
17 eldifsn 4119 . . . . . . . . 9  |-  ( X  e.  ( ( A [,] B )  \  { C } )  <->  ( X  e.  ( A [,] B
)  /\  X  =/=  C ) )
1813, 16, 17sylanbrc 668 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  X  e.  ( ( A [,] B )  \  { C } ) )
19 fveq2 5872 . . . . . . . . . . 11  |-  ( z  =  X  ->  ( G `  z )  =  ( G `  X ) )
2019oveq1d 6311 . . . . . . . . . 10  |-  ( z  =  X  ->  (
( G `  z
)  -  ( G `
 C ) )  =  ( ( G `
 X )  -  ( G `  C ) ) )
21 oveq1 6303 . . . . . . . . . 10  |-  ( z  =  X  ->  (
z  -  C )  =  ( X  -  C ) )
2220, 21oveq12d 6314 . . . . . . . . 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 6324 . . . . . . . . 9  |-  ( ( ( G `  X
)  -  ( G `
 C ) )  /  ( X  -  C ) )  e. 
_V
2522, 23, 24fvmpt 5955 . . . . . . . 8  |-  ( X  e.  ( ( A [,] B )  \  { C } )  -> 
( H `  X
)  =  ( ( ( G `  X
)  -  ( G `
 C ) )  /  ( X  -  C ) ) )
2618, 25syl 17 . . . . . . 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 22864 . . . . . . . . . . . 12  |-  ( ph  ->  F : D --> CC )
3727, 1, 2, 28, 29, 30, 31, 36ftc1lem2 22862 . . . . . . . . . . 11  |-  ( ph  ->  G : ( A [,] B ) --> CC )
3837, 5ffvelrnd 6029 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  e.  CC )
3937, 9ffvelrnd 6029 . . . . . . . . . 10  |-  ( ph  ->  ( G `  C
)  e.  CC )
4038, 39subcld 9975 . . . . . . . . 9  |-  ( ph  ->  ( ( G `  X )  -  ( G `  C )
)  e.  CC )
4140adantr 466 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( ( G `  X )  -  ( G `  C ) )  e.  CC )
426recnd 9658 . . . . . . . . . 10  |-  ( ph  ->  X  e.  CC )
4310recnd 9658 . . . . . . . . . 10  |-  ( ph  ->  C  e.  CC )
4442, 43subcld 9975 . . . . . . . . 9  |-  ( ph  ->  ( X  -  C
)  e.  CC )
4544adantr 466 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( X  -  C )  e.  CC )
4642, 43subeq0ad 9985 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X  -  C )  =  0  <-> 
X  =  C ) )
4746necon3bid 2680 . . . . . . . . . 10  |-  ( ph  ->  ( ( X  -  C )  =/=  0  <->  X  =/=  C ) )
4847biimpar 487 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  C )  ->  ( X  -  C )  =/=  0
)
4916, 48syldan 472 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( X  -  C )  =/=  0
)
5041, 45, 49div2negd 10387 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( -u (
( G `  X
)  -  ( G `
 C ) )  /  -u ( X  -  C ) )  =  ( ( ( G `
 X )  -  ( G `  C ) )  /  ( X  -  C ) ) )
5138, 39negsubdi2d 9991 . . . . . . . . 9  |-  ( ph  -> 
-u ( ( G `
 X )  -  ( G `  C ) )  =  ( ( G `  C )  -  ( G `  X ) ) )
5242, 43negsubdi2d 9991 . . . . . . . . 9  |-  ( ph  -> 
-u ( X  -  C )  =  ( C  -  X ) )
5351, 52oveq12d 6314 . . . . . . . 8  |-  ( ph  ->  ( -u ( ( G `  X )  -  ( G `  C ) )  /  -u ( X  -  C
) )  =  ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) ) )
5453adantr 466 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( -u (
( G `  X
)  -  ( G `
 C ) )  /  -u ( X  -  C ) )  =  ( ( ( G `
 C )  -  ( G `  X ) )  /  ( C  -  X ) ) )
5526, 50, 543eqtr2d 2467 . . . . . 6  |-  ( (
ph  /\  X  <  C )  ->  ( H `  X )  =  ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) ) )
5655oveq1d 6311 . . . . 5  |-  ( (
ph  /\  X  <  C )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )
5756fveq2d 5876 . . . 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 9963 . . . . . . 7  |-  ( ph  ->  ( C  -  C
)  =  0 )
6362abs00bd 13322 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  C )
)  =  0 )
6459rpgt0d 11333 . . . . . 6  |-  ( ph  ->  0  <  R )
6563, 64eqbrtrd 4437 . . . . 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 22865 . . . 4  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )  <  E
)
6757, 66eqbrtrd 4437 . . 3  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
685adantr 466 . . . . . . . 8  |-  ( (
ph  /\  C  <  X )  ->  X  e.  ( A [,] B ) )
6910adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  C  <  X )  ->  C  e.  RR )
70 simpr 462 . . . . . . . . 9  |-  ( (
ph  /\  C  <  X )  ->  C  <  X )
7169, 70gtned 9759 . . . . . . . 8  |-  ( (
ph  /\  C  <  X )  ->  X  =/=  C )
7268, 71, 17sylanbrc 668 . . . . . . 7  |-  ( (
ph  /\  C  <  X )  ->  X  e.  ( ( A [,] B )  \  { C } ) )
7372, 25syl 17 . . . . . 6  |-  ( (
ph  /\  C  <  X )  ->  ( H `  X )  =  ( ( ( G `  X )  -  ( G `  C )
)  /  ( X  -  C ) ) )
7473oveq1d 6311 . . . . 5  |-  ( (
ph  /\  C  <  X )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )
7574fveq2d 5876 . . . 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 22865 . . . 4  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )  <  E
)
7775, 76eqbrtrd 4437 . . 3  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
7867, 77jaodan 792 . 2  |-  ( (
ph  /\  ( X  <  C  \/  C  < 
X ) )  -> 
( abs `  (
( H `  X
)  -  ( F `
 C ) ) )  <  E )
7912, 78syldan 472 1  |-  ( (
ph  /\  X  =/=  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616    \ cdif 3430    C_ wss 3433   {csn 3993   class class class wbr 4417    |-> cmpt 4475   ` cfv 5592  (class class class)co 6296   CCcc 9526   RRcr 9527   0cc0 9528    < clt 9664    <_ cle 9665    - cmin 9849   -ucneg 9850    / cdiv 10258   RR+crp 11291   (,)cioo 11624   [,]cicc 11627   abscabs 13265   ↾t crest 15271   TopOpenctopn 15272  ℂfldccnfld 18898    CnP ccnp 20165   L^1cibl 22449   S.citg 22450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cc 8854  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-disj 4389  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-ofr 6537  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-omul 7186  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-acn 8366  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-ioo 11628  df-ioc 11629  df-ico 11630  df-icc 11631  df-fz 11772  df-fzo 11903  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-rlim 13520  df-sum 13720  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-starv 15157  df-sca 15158  df-vsca 15159  df-ip 15160  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-hom 15166  df-cco 15167  df-rest 15273  df-topn 15274  df-0g 15292  df-gsum 15293  df-topgen 15294  df-pt 15295  df-prds 15298  df-xrs 15352  df-qtop 15357  df-imas 15358  df-xps 15360  df-mre 15436  df-mrc 15437  df-acs 15439  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-submnd 16527  df-mulg 16620  df-cntz 16915  df-cmn 17360  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-cnfld 18899  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-cn 20167  df-cnp 20168  df-cmp 20326  df-tx 20501  df-hmeo 20694  df-xms 21259  df-ms 21260  df-tms 21261  df-cncf 21799  df-ovol 22290  df-vol 22292  df-mbf 22451  df-itg1 22452  df-itg2 22453  df-ibl 22454  df-itg 22455  df-0p 22502
This theorem is referenced by:  ftc1lem6  22867
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