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Theorem ftc1lem5 22204
Description: Lemma for ftc1 22206. (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 11606 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3syl2anc 661 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
5 ftc1.x1 . . . . 5  |-  ( ph  ->  X  e.  ( A [,] B ) )
64, 5sseldd 3505 . . . 4  |-  ( ph  ->  X  e.  RR )
7 ioossicc 11610 . . . . . 6  |-  ( A (,) B )  C_  ( A [,] B )
8 ftc1.c . . . . . 6  |-  ( ph  ->  C  e.  ( A (,) B ) )
97, 8sseldi 3502 . . . . 5  |-  ( ph  ->  C  e.  ( A [,] B ) )
104, 9sseldd 3505 . . . 4  |-  ( ph  ->  C  e.  RR )
116, 10lttri2d 9723 . . 3  |-  ( ph  ->  ( X  =/=  C  <->  ( X  <  C  \/  C  <  X ) ) )
1211biimpa 484 . 2  |-  ( (
ph  /\  X  =/=  C )  ->  ( X  <  C  \/  C  < 
X ) )
135adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  X  <  C )  ->  X  e.  ( A [,] B ) )
146adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  X  <  C )  ->  X  e.  RR )
15 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  X  <  C )  ->  X  <  C )
1614, 15ltned 9720 . . . . . . . . 9  |-  ( (
ph  /\  X  <  C )  ->  X  =/=  C )
17 eldifsn 4152 . . . . . . . . 9  |-  ( X  e.  ( ( A [,] B )  \  { C } )  <->  ( X  e.  ( A [,] B
)  /\  X  =/=  C ) )
1813, 16, 17sylanbrc 664 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  X  e.  ( ( A [,] B )  \  { C } ) )
19 fveq2 5866 . . . . . . . . . . 11  |-  ( z  =  X  ->  ( G `  z )  =  ( G `  X ) )
2019oveq1d 6299 . . . . . . . . . 10  |-  ( z  =  X  ->  (
( G `  z
)  -  ( G `
 C ) )  =  ( ( G `
 X )  -  ( G `  C ) ) )
21 oveq1 6291 . . . . . . . . . 10  |-  ( z  =  X  ->  (
z  -  C )  =  ( X  -  C ) )
2220, 21oveq12d 6302 . . . . . . . . 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 6309 . . . . . . . . 9  |-  ( ( ( G `  X
)  -  ( G `
 C ) )  /  ( X  -  C ) )  e. 
_V
2522, 23, 24fvmpt 5950 . . . . . . . 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 22202 . . . . . . . . . . . 12  |-  ( ph  ->  F : D --> CC )
3727, 1, 2, 28, 29, 30, 31, 36ftc1lem2 22200 . . . . . . . . . . 11  |-  ( ph  ->  G : ( A [,] B ) --> CC )
3837, 5ffvelrnd 6022 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  e.  CC )
3937, 9ffvelrnd 6022 . . . . . . . . . 10  |-  ( ph  ->  ( G `  C
)  e.  CC )
4038, 39subcld 9930 . . . . . . . . 9  |-  ( ph  ->  ( ( G `  X )  -  ( G `  C )
)  e.  CC )
4140adantr 465 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( ( G `  X )  -  ( G `  C ) )  e.  CC )
426recnd 9622 . . . . . . . . . 10  |-  ( ph  ->  X  e.  CC )
4310recnd 9622 . . . . . . . . . 10  |-  ( ph  ->  C  e.  CC )
4442, 43subcld 9930 . . . . . . . . 9  |-  ( ph  ->  ( X  -  C
)  e.  CC )
4544adantr 465 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( X  -  C )  e.  CC )
4642, 43subeq0ad 9940 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X  -  C )  =  0  <-> 
X  =  C ) )
4746necon3bid 2725 . . . . . . . . . 10  |-  ( ph  ->  ( ( X  -  C )  =/=  0  <->  X  =/=  C ) )
4847biimpar 485 . . . . . . . . 9  |-  ( (
ph  /\  X  =/=  C )  ->  ( X  -  C )  =/=  0
)
4916, 48syldan 470 . . . . . . . 8  |-  ( (
ph  /\  X  <  C )  ->  ( X  -  C )  =/=  0
)
5041, 45, 49div2negd 10335 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( -u (
( G `  X
)  -  ( G `
 C ) )  /  -u ( X  -  C ) )  =  ( ( ( G `
 X )  -  ( G `  C ) )  /  ( X  -  C ) ) )
5138, 39negsubdi2d 9946 . . . . . . . . 9  |-  ( ph  -> 
-u ( ( G `
 X )  -  ( G `  C ) )  =  ( ( G `  C )  -  ( G `  X ) ) )
5242, 43negsubdi2d 9946 . . . . . . . . 9  |-  ( ph  -> 
-u ( X  -  C )  =  ( C  -  X ) )
5351, 52oveq12d 6302 . . . . . . . 8  |-  ( ph  ->  ( -u ( ( G `  X )  -  ( G `  C ) )  /  -u ( X  -  C
) )  =  ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) ) )
5453adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  <  C )  ->  ( -u (
( G `  X
)  -  ( G `
 C ) )  /  -u ( X  -  C ) )  =  ( ( ( G `
 C )  -  ( G `  X ) )  /  ( C  -  X ) ) )
5526, 50, 543eqtr2d 2514 . . . . . 6  |-  ( (
ph  /\  X  <  C )  ->  ( H `  X )  =  ( ( ( G `  C )  -  ( G `  X )
)  /  ( C  -  X ) ) )
5655oveq1d 6299 . . . . 5  |-  ( (
ph  /\  X  <  C )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )
5756fveq2d 5870 . . . 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 9918 . . . . . . 7  |-  ( ph  ->  ( C  -  C
)  =  0 )
6362abs00bd 13087 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  C )
)  =  0 )
6459rpgt0d 11259 . . . . . 6  |-  ( ph  ->  0  <  R )
6563, 64eqbrtrd 4467 . . . . 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 22203 . . . 4  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( ( ( G `  C )  -  ( G `  X ) )  / 
( C  -  X
) )  -  ( F `  C )
) )  <  E
)
6757, 66eqbrtrd 4467 . . 3  |-  ( (
ph  /\  X  <  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
685adantr 465 . . . . . . . 8  |-  ( (
ph  /\  C  <  X )  ->  X  e.  ( A [,] B ) )
6910adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  C  <  X )  ->  C  e.  RR )
70 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  C  <  X )  ->  C  <  X )
7169, 70gtned 9719 . . . . . . . 8  |-  ( (
ph  /\  C  <  X )  ->  X  =/=  C )
7268, 71, 17sylanbrc 664 . . . . . . 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 6299 . . . . 5  |-  ( (
ph  /\  C  <  X )  ->  ( ( H `  X )  -  ( F `  C ) )  =  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )
7574fveq2d 5870 . . . 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 22203 . . . 4  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( ( ( G `  X )  -  ( G `  C ) )  / 
( X  -  C
) )  -  ( F `  C )
) )  <  E
)
7775, 76eqbrtrd 4467 . . 3  |-  ( (
ph  /\  C  <  X )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
7867, 77jaodan 783 . 2  |-  ( (
ph  /\  ( X  <  C  \/  C  < 
X ) )  -> 
( abs `  (
( H `  X
)  -  ( F `
 C ) ) )  <  E )
7912, 78syldan 470 1  |-  ( (
ph  /\  X  =/=  C )  ->  ( abs `  ( ( H `  X )  -  ( F `  C )
) )  <  E
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    C_ wss 3476   {csn 4027   class class class wbr 4447    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492    < clt 9628    <_ cle 9629    - cmin 9805   -ucneg 9806    / cdiv 10206   RR+crp 11220   (,)cioo 11529   [,]cicc 11532   abscabs 13030   ↾t crest 14676   TopOpenctopn 14677  ℂfldccnfld 18219    CnP ccnp 19520   L^1cibl 21789   S.citg 21790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cc 8815  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-ofr 6525  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cn 19522  df-cnp 19523  df-cmp 19681  df-tx 19826  df-hmeo 20019  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-ovol 21639  df-vol 21640  df-mbf 21791  df-itg1 21792  df-itg2 21793  df-ibl 21794  df-itg 21795  df-0p 21840
This theorem is referenced by:  ftc1lem6  22205
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