Theorem ftc1lem5 23803

Description: Lemma for ftc1 23805. (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 = ( L ↾t RR )
ftc1.k	|- K = ( L ↾t D )
ftc1.l	|- L = ( TopOpen ` ℂfld )
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

Step	Hyp	Ref	Expression
1		ftc1.a	. . . . . 6 |- ( ph -> A e. RR )
2		ftc1.b	. . . . . 6 |- ( ph -> B e. RR )
3		iccssre 12255	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	syl2anc 693	. . . . 5 |- ( ph -> ( A [,] B ) C_ RR )
5		ftc1.x1	. . . . 5 |- ( ph -> X e. ( A [,] B ) )
6	4, 5	sseldd 3604	. . . 4 |- ( ph -> X e. RR )
7		ioossicc 12259	. . . . . 6 |- ( A (,) B ) C_ ( A [,] B )
8		ftc1.c	. . . . . 6 |- ( ph -> C e. ( A (,) B ) )
9	7, 8	sseldi 3601	. . . . 5 |- ( ph -> C e. ( A [,] B ) )
10	4, 9	sseldd 3604	. . . 4 |- ( ph -> C e. RR )
11	6, 10	lttri2d 10176	. . 3 |- ( ph -> ( X =/= C <-> ( X < C \/ C < X ) ) )
12	11	biimpa 501	. 2 |- ( ( ph /\ X =/= C ) -> ( X < C \/ C < X ) )
13	5	adantr 481	. . . . . . . . 9 |- ( ( ph /\ X < C ) -> X e. ( A [,] B ) )
14	6	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ X < C ) -> X e. RR )
15		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ X < C ) -> X < C )
16	14, 15	ltned 10173	. . . . . . . . 9 |- ( ( ph /\ X < C ) -> X =/= C )
17		eldifsn 4317	. . . . . . . . 9 |- ( X e. ( ( A [,] B ) \ { C } ) <-> ( X e. ( A [,] B ) /\ X =/= C ) )
18	13, 16, 17	sylanbrc 698	. . . . . . . 8 |- ( ( ph /\ X < C ) -> X e. ( ( A [,] B ) \ { C } ) )
19		fveq2 6191	. . . . . . . . . . 11 |- ( z = X -> ( G ` z ) = ( G ` X ) )
20	19	oveq1d 6665	. . . . . . . . . 10 |- ( z = X -> ( ( G ` z ) - ( G ` C ) ) = ( ( G ` X ) - ( G ` C ) ) )
21		oveq1 6657	. . . . . . . . . 10 |- ( z = X -> ( z - C ) = ( X - C ) )
22	20, 21	oveq12d 6668	. . . . . . . . 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 6678	. . . . . . . . 9 |- ( ( ( G ` X ) - ( G ` C ) ) / ( X - C ) ) e. _V
25	22, 23, 24	fvmpt 6282	. . . . . . . 8 |- ( X e. ( ( A [,] B ) \ { C } ) -> ( H ` X ) = ( ( ( G ` X ) - ( G ` C ) ) / ( X - C ) ) )
26	18, 25	syl 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 = ( L ↾t RR )
34		ftc1.k	. . . . . . . . . . . . 13 |- K = ( L ↾t D )
35		ftc1.l	. . . . . . . . . . . . 13 |- L = ( TopOpen ` ℂfld )
36	27, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35	ftc1lem3 23801	. . . . . . . . . . . 12 |- ( ph -> F : D --> CC )
37	27, 1, 2, 28, 29, 30, 31, 36	ftc1lem2 23799	. . . . . . . . . . 11 |- ( ph -> G : ( A [,] B ) --> CC )
38	37, 5	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( G ` X ) e. CC )
39	37, 9	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( G ` C ) e. CC )
40	38, 39	subcld 10392	. . . . . . . . 9 |- ( ph -> ( ( G ` X ) - ( G ` C ) ) e. CC )
41	40	adantr 481	. . . . . . . 8 |- ( ( ph /\ X < C ) -> ( ( G ` X ) - ( G ` C ) ) e. CC )
42	6	recnd 10068	. . . . . . . . . 10 |- ( ph -> X e. CC )
43	10	recnd 10068	. . . . . . . . . 10 |- ( ph -> C e. CC )
44	42, 43	subcld 10392	. . . . . . . . 9 |- ( ph -> ( X - C ) e. CC )
45	44	adantr 481	. . . . . . . 8 |- ( ( ph /\ X < C ) -> ( X - C ) e. CC )
46	42, 43	subeq0ad 10402	. . . . . . . . . . 11 |- ( ph -> ( ( X - C ) = 0 <-> X = C ) )
47	46	necon3bid 2838	. . . . . . . . . 10 |- ( ph -> ( ( X - C ) =/= 0 <-> X =/= C ) )
48	47	biimpar 502	. . . . . . . . 9 |- ( ( ph /\ X =/= C ) -> ( X - C ) =/= 0 )
49	16, 48	syldan 487	. . . . . . . 8 |- ( ( ph /\ X < C ) -> ( X - C ) =/= 0 )
50	41, 45, 49	div2negd 10816	. . . . . . 7 |- ( ( ph /\ X < C ) -> ( -u ( ( G ` X ) - ( G ` C ) ) / -u ( X - C ) ) = ( ( ( G ` X ) - ( G ` C ) ) / ( X - C ) ) )
51	38, 39	negsubdi2d 10408	. . . . . . . . 9 |- ( ph -> -u ( ( G ` X ) - ( G ` C ) ) = ( ( G ` C ) - ( G ` X ) ) )
52	42, 43	negsubdi2d 10408	. . . . . . . . 9 |- ( ph -> -u ( X - C ) = ( C - X ) )
53	51, 52	oveq12d 6668	. . . . . . . 8 |- ( ph -> ( -u ( ( G ` X ) - ( G ` C ) ) / -u ( X - C ) ) = ( ( ( G ` C ) - ( G ` X ) ) / ( C - X ) ) )
54	53	adantr 481	. . . . . . 7 |- ( ( ph /\ X < C ) -> ( -u ( ( G ` X ) - ( G ` C ) ) / -u ( X - C ) ) = ( ( ( G ` C ) - ( G ` X ) ) / ( C - X ) ) )
55	26, 50, 54	3eqtr2d 2662	. . . . . 6 |- ( ( ph /\ X < C ) -> ( H ` X ) = ( ( ( G ` C ) - ( G ` X ) ) / ( C - X ) ) )
56	55	oveq1d 6665	. . . . 5 |- ( ( ph /\ X < C ) -> ( ( H ` X ) - ( F ` C ) ) = ( ( ( ( G ` C ) - ( G ` X ) ) / ( C - X ) ) - ( F ` C ) ) )
57	56	fveq2d 6195	. . . 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 )
62	43	subidd 10380	. . . . . . 7 |- ( ph -> ( C - C ) = 0 )
63	62	abs00bd 14031	. . . . . 6 |- ( ph -> ( abs ` ( C - C ) ) = 0 )
64	59	rpgt0d 11875	. . . . . 6 |- ( ph -> 0 < R )
65	63, 64	eqbrtrd 4675	. . . . 5 |- ( ph -> ( abs ` ( C - C ) ) < R )
66	27, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35, 23, 58, 59, 60, 5, 61, 9, 65	ftc1lem4 23802	. . . 4 |- ( ( ph /\ X < C ) -> ( abs ` ( ( ( ( G ` C ) - ( G ` X ) ) / ( C - X ) ) - ( F ` C ) ) ) < E )
67	57, 66	eqbrtrd 4675	. . 3 |- ( ( ph /\ X < C ) -> ( abs ` ( ( H ` X ) - ( F ` C ) ) ) < E )
68	5	adantr 481	. . . . . . . 8 |- ( ( ph /\ C < X ) -> X e. ( A [,] B ) )
69	10	adantr 481	. . . . . . . . 9 |- ( ( ph /\ C < X ) -> C e. RR )
70		simpr 477	. . . . . . . . 9 |- ( ( ph /\ C < X ) -> C < X )
71	69, 70	gtned 10172	. . . . . . . 8 |- ( ( ph /\ C < X ) -> X =/= C )
72	68, 71, 17	sylanbrc 698	. . . . . . 7 |- ( ( ph /\ C < X ) -> X e. ( ( A [,] B ) \ { C } ) )
73	72, 25	syl 17	. . . . . 6 |- ( ( ph /\ C < X ) -> ( H ` X ) = ( ( ( G ` X ) - ( G ` C ) ) / ( X - C ) ) )
74	73	oveq1d 6665	. . . . 5 |- ( ( ph /\ C < X ) -> ( ( H ` X ) - ( F ` C ) ) = ( ( ( ( G ` X ) - ( G ` C ) ) / ( X - C ) ) - ( F ` C ) ) )
75	74	fveq2d 6195	. . . 4 |- ( ( ph /\ C < X ) -> ( abs ` ( ( H ` X ) - ( F ` C ) ) ) = ( abs ` ( ( ( ( G ` X ) - ( G ` C ) ) / ( X - C ) ) - ( F ` C ) ) ) )
76	27, 1, 2, 28, 29, 30, 31, 8, 32, 33, 34, 35, 23, 58, 59, 60, 9, 65, 5, 61	ftc1lem4 23802	. . . 4 |- ( ( ph /\ C < X ) -> ( abs ` ( ( ( ( G ` X ) - ( G ` C ) ) / ( X - C ) ) - ( F ` C ) ) ) < E )
77	75, 76	eqbrtrd 4675	. . 3 |- ( ( ph /\ C < X ) -> ( abs ` ( ( H ` X ) - ( F ` C ) ) ) < E )
78	67, 77	jaodan 826	. 2 |- ( ( ph /\ ( X < C \/ C < X ) ) -> ( abs ` ( ( H ` X ) - ( F ` C ) ) ) < E )
79	12, 78	syldan 487	1 |- ( ( ph /\ X =/= C ) -> ( abs ` ( ( H ` X ) - ( F ` C ) ) ) < E )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   RR+crp 11832   (,)cioo 12175   [,]cicc 12178   abscabs 13974   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   CnP ccnp 21029   L^1cibl 23386   S.citg 23387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437

This theorem is referenced by:  ftc1lem6  23804