Theorem dveq0 23763

Description: If a continuous function has zero derivative at all points on the interior of a closed interval, then it must be a constant function. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by Mario Carneiro, 3-Mar-2015.)

Hypotheses

Ref	Expression
dveq0.a	|- ( ph -> A e. RR )
dveq0.b	|- ( ph -> B e. RR )
dveq0.c	|- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) )
dveq0.d	|- ( ph -> ( RR _D F ) = ( ( A (,) B ) X. { 0 } ) )

Assertion

Ref	Expression
dveq0	|- ( ph -> F = ( ( A [,] B ) X. { ( F ` A ) } ) )

Proof of Theorem dveq0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dveq0.c	. . . 4 |- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) )
2		cncff 22696	. . . 4 |- ( F e. ( ( A [,] B ) -cn-> CC ) -> F : ( A [,] B ) --> CC )
3	1, 2	syl 17	. . 3 |- ( ph -> F : ( A [,] B ) --> CC )
4		ffn 6045	. . 3 |- ( F : ( A [,] B ) --> CC -> F Fn ( A [,] B ) )
5	3, 4	syl 17	. 2 |- ( ph -> F Fn ( A [,] B ) )
6		fvex 6201	. . 3 |- ( F ` A ) e. _V
7		fnconstg 6093	. . 3 |- ( ( F ` A ) e. _V -> ( ( A [,] B ) X. { ( F ` A ) } ) Fn ( A [,] B ) )
8	6, 7	mp1i 13	. 2 |- ( ph -> ( ( A [,] B ) X. { ( F ` A ) } ) Fn ( A [,] B ) )
9	6	fvconst2 6469	. . . 4 |- ( x e. ( A [,] B ) -> ( ( ( A [,] B ) X. { ( F ` A ) } ) ` x ) = ( F ` A ) )
10	9	adantl 482	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( ( A [,] B ) X. { ( F ` A ) } ) ` x ) = ( F ` A ) )
11	3	adantr 481	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> F : ( A [,] B ) --> CC )
12		dveq0.a	. . . . . . . 8 |- ( ph -> A e. RR )
13	12	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR )
14	13	rexrd 10089	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR* )
15		dveq0.b	. . . . . . . 8 |- ( ph -> B e. RR )
16	15	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR )
17	16	rexrd 10089	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR* )
18		elicc2 12238	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
19	12, 15, 18	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
20	19	biimpa 501	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x e. RR /\ A <_ x /\ x <_ B ) )
21	20	simp1d 1073	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
22	20	simp2d 1074	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> A <_ x )
23	20	simp3d 1075	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ B )
24	13, 21, 16, 22, 23	letrd 10194	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> A <_ B )
25		lbicc2 12288	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
26	14, 17, 24, 25	syl3anc 1326	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. ( A [,] B ) )
27	11, 26	ffvelrnd 6360	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` A ) e. CC )
28	3	ffvelrnda 6359	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
29	27, 28	subcld 10392	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( F ` A ) - ( F ` x ) ) e. CC )
30		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) )
31	26, 30	jca 554	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( A e. ( A [,] B ) /\ x e. ( A [,] B ) ) )
32		dveq0.d	. . . . . . . . . . 11 |- ( ph -> ( RR _D F ) = ( ( A (,) B ) X. { 0 } ) )
33	32	dmeqd 5326	. . . . . . . . . 10 |- ( ph -> dom ( RR _D F ) = dom ( ( A (,) B ) X. { 0 } ) )
34		c0ex 10034	. . . . . . . . . . . 12 |- 0 e. _V
35	34	snnz 4309	. . . . . . . . . . 11 |- { 0 } =/= (/)
36		dmxp 5344	. . . . . . . . . . 11 |- ( { 0 } =/= (/) -> dom ( ( A (,) B ) X. { 0 } ) = ( A (,) B ) )
37	35, 36	ax-mp 5	. . . . . . . . . 10 |- dom ( ( A (,) B ) X. { 0 } ) = ( A (,) B )
38	33, 37	syl6eq 2672	. . . . . . . . 9 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
39		0red 10041	. . . . . . . . 9 |- ( ph -> 0 e. RR )
40	32	fveq1d 6193	. . . . . . . . . . . 12 |- ( ph -> ( ( RR _D F ) ` y ) = ( ( ( A (,) B ) X. { 0 } ) ` y ) )
41	34	fvconst2 6469	. . . . . . . . . . . 12 |- ( y e. ( A (,) B ) -> ( ( ( A (,) B ) X. { 0 } ) ` y ) = 0 )
42	40, 41	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( A (,) B ) ) -> ( ( RR _D F ) ` y ) = 0 )
43	42	abs00bd 14031	. . . . . . . . . 10 |- ( ( ph /\ y e. ( A (,) B ) ) -> ( abs ` ( ( RR _D F ) ` y ) ) = 0 )
44		0le0 11110	. . . . . . . . . 10 |- 0 <_ 0
45	43, 44	syl6eqbr 4692	. . . . . . . . 9 |- ( ( ph /\ y e. ( A (,) B ) ) -> ( abs ` ( ( RR _D F ) ` y ) ) <_ 0 )
46	12, 15, 1, 38, 39, 45	dvlip 23756	. . . . . . . 8 |- ( ( ph /\ ( A e. ( A [,] B ) /\ x e. ( A [,] B ) ) ) -> ( abs ` ( ( F ` A ) - ( F ` x ) ) ) <_ ( 0 x. ( abs ` ( A - x ) ) ) )
47	31, 46	syldan 487	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( abs ` ( ( F ` A ) - ( F ` x ) ) ) <_ ( 0 x. ( abs ` ( A - x ) ) ) )
48	13	recnd 10068	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. CC )
49	21	recnd 10068	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
50	48, 49	subcld 10392	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( A - x ) e. CC )
51	50	abscld 14175	. . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( abs ` ( A - x ) ) e. RR )
52	51	recnd 10068	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( abs ` ( A - x ) ) e. CC )
53	52	mul02d 10234	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( 0 x. ( abs ` ( A - x ) ) ) = 0 )
54	47, 53	breqtrd 4679	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( abs ` ( ( F ` A ) - ( F ` x ) ) ) <_ 0 )
55	29	absge0d 14183	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> 0 <_ ( abs ` ( ( F ` A ) - ( F ` x ) ) ) )
56	29	abscld 14175	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( abs ` ( ( F ` A ) - ( F ` x ) ) ) e. RR )
57		0re 10040	. . . . . . 7 |- 0 e. RR
58		letri3 10123	. . . . . . 7 |- ( ( ( abs ` ( ( F ` A ) - ( F ` x ) ) ) e. RR /\ 0 e. RR ) -> ( ( abs ` ( ( F ` A ) - ( F ` x ) ) ) = 0 <-> ( ( abs ` ( ( F ` A ) - ( F ` x ) ) ) <_ 0 /\ 0 <_ ( abs ` ( ( F ` A ) - ( F ` x ) ) ) ) ) )
59	56, 57, 58	sylancl 694	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( abs ` ( ( F ` A ) - ( F ` x ) ) ) = 0 <-> ( ( abs ` ( ( F ` A ) - ( F ` x ) ) ) <_ 0 /\ 0 <_ ( abs ` ( ( F ` A ) - ( F ` x ) ) ) ) ) )
60	54, 55, 59	mpbir2and 957	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( abs ` ( ( F ` A ) - ( F ` x ) ) ) = 0 )
61	29, 60	abs00d 14185	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( F ` A ) - ( F ` x ) ) = 0 )
62	27, 28, 61	subeq0d 10400	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` A ) = ( F ` x ) )
63	10, 62	eqtr2d 2657	. 2 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) = ( ( ( A [,] B ) X. { ( F ` A ) } ) ` x ) )
64	5, 8, 63	eqfnfvd 6314	1 |- ( ph -> F = ( ( A [,] B ) X. { ( F ` A ) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   RR*cxr 10073   <_ cle 10075   - cmin 10266   (,)cioo 12175   [,]cicc 12178   abscabs 13974   -cn->ccncf 22679   _D cdv 23627

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-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-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-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-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-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-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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  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-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-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  ftc2  23807  ftc2nc  33494