Theorem dvmptresicc 40134

Description: Derivative of a function restricted to a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
dvmptresicc.f	|- F = ( x e. CC |-> A )
dvmptresicc.a	|- ( ( ph /\ x e. CC ) -> A e. CC )
dvmptresicc.fdv	|- ( ph -> ( CC _D F ) = ( x e. CC |-> B ) )
dvmptresicc.b	|- ( ( ph /\ x e. CC ) -> B e. CC )
dvmptresicc.c	|- ( ph -> C e. RR )
dvmptresicc.d	|- ( ph -> D e. RR )

Assertion

Ref	Expression
dvmptresicc	|- ( ph -> ( RR _D ( x e. ( C [,] D ) |-> A ) ) = ( x e. ( C (,) D ) |-> B ) )

Distinct variable groups:   x, C   x, D   ph, x

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem dvmptresicc

Step	Hyp	Ref	Expression
1		dvmptresicc.f	. . . . 5 |- F = ( x e. CC |-> A )
2	1	reseq1i 5392	. . . 4 |- ( F |` ( C [,] D ) ) = ( ( x e. CC |-> A ) |` ( C [,] D ) )
3		dvmptresicc.c	. . . . . . 7 |- ( ph -> C e. RR )
4		dvmptresicc.d	. . . . . . 7 |- ( ph -> D e. RR )
5	3, 4	iccssred 39727	. . . . . 6 |- ( ph -> ( C [,] D ) C_ RR )
6		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
7	6	a1i 11	. . . . . 6 |- ( ph -> RR C_ CC )
8	5, 7	sstrd 3613	. . . . 5 |- ( ph -> ( C [,] D ) C_ CC )
9	8	resmptd 5452	. . . 4 |- ( ph -> ( ( x e. CC |-> A ) |` ( C [,] D ) ) = ( x e. ( C [,] D ) |-> A ) )
10	2, 9	syl5eq 2668	. . 3 |- ( ph -> ( F |` ( C [,] D ) ) = ( x e. ( C [,] D ) |-> A ) )
11	10	oveq2d 6666	. 2 |- ( ph -> ( RR _D ( F |` ( C [,] D ) ) ) = ( RR _D ( x e. ( C [,] D ) |-> A ) ) )
12	5	resabs1d 5428	. . . . 5 |- ( ph -> ( ( F |` RR ) |` ( C [,] D ) ) = ( F |` ( C [,] D ) ) )
13	12	eqcomd 2628	. . . 4 |- ( ph -> ( F |` ( C [,] D ) ) = ( ( F |` RR ) |` ( C [,] D ) ) )
14	13	oveq2d 6666	. . 3 |- ( ph -> ( RR _D ( F |` ( C [,] D ) ) ) = ( RR _D ( ( F |` RR ) |` ( C [,] D ) ) ) )
15		dvmptresicc.a	. . . . . 6 |- ( ( ph /\ x e. CC ) -> A e. CC )
16	15, 1	fmptd 6385	. . . . 5 |- ( ph -> F : CC --> CC )
17	16, 7	fssresd 6071	. . . 4 |- ( ph -> ( F |` RR ) : RR --> CC )
18		ssid 3624	. . . . 5 |- RR C_ RR
19	18	a1i 11	. . . 4 |- ( ph -> RR C_ RR )
20		eqid 2622	. . . . 5 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 22606	. . . . 5 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22	20, 21	dvres 23675	. . . 4 |- ( ( ( RR C_ CC /\ ( F |` RR ) : RR --> CC ) /\ ( RR C_ RR /\ ( C [,] D ) C_ RR ) ) -> ( RR _D ( ( F |` RR ) |` ( C [,] D ) ) ) = ( ( RR _D ( F |` RR ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) ) )
23	7, 17, 19, 5, 22	syl22anc 1327	. . 3 |- ( ph -> ( RR _D ( ( F |` RR ) |` ( C [,] D ) ) ) = ( ( RR _D ( F |` RR ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) ) )
24		reelprrecn 10028	. . . . . . 7 |- RR e. { RR , CC }
25	24	a1i 11	. . . . . 6 |- ( ph -> RR e. { RR , CC } )
26		ssid 3624	. . . . . . 7 |- CC C_ CC
27	26	a1i 11	. . . . . 6 |- ( ph -> CC C_ CC )
28		dvmptresicc.fdv	. . . . . . . . 9 |- ( ph -> ( CC _D F ) = ( x e. CC |-> B ) )
29	28	dmeqd 5326	. . . . . . . 8 |- ( ph -> dom ( CC _D F ) = dom ( x e. CC |-> B ) )
30		dvmptresicc.b	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. CC )
31	30	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. x e. CC B e. CC )
32		dmmptg 5632	. . . . . . . . 9 |- ( A. x e. CC B e. CC -> dom ( x e. CC |-> B ) = CC )
33	31, 32	syl 17	. . . . . . . 8 |- ( ph -> dom ( x e. CC |-> B ) = CC )
34	29, 33	eqtr2d 2657	. . . . . . 7 |- ( ph -> CC = dom ( CC _D F ) )
35	7, 34	sseqtrd 3641	. . . . . 6 |- ( ph -> RR C_ dom ( CC _D F ) )
36		dvres3 23677	. . . . . 6 |- ( ( ( RR e. { RR , CC } /\ F : CC --> CC ) /\ ( CC C_ CC /\ RR C_ dom ( CC _D F ) ) ) -> ( RR _D ( F |` RR ) ) = ( ( CC _D F ) |` RR ) )
37	25, 16, 27, 35, 36	syl22anc 1327	. . . . 5 |- ( ph -> ( RR _D ( F |` RR ) ) = ( ( CC _D F ) |` RR ) )
38		iccntr 22624	. . . . . 6 |- ( ( C e. RR /\ D e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) = ( C (,) D ) )
39	3, 4, 38	syl2anc 693	. . . . 5 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) = ( C (,) D ) )
40	37, 39	reseq12d 5397	. . . 4 |- ( ph -> ( ( RR _D ( F |` RR ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) ) = ( ( ( CC _D F ) |` RR ) |` ( C (,) D ) ) )
41		ioossre 12235	. . . . 5 |- ( C (,) D ) C_ RR
42		resabs1 5427	. . . . 5 |- ( ( C (,) D ) C_ RR -> ( ( ( CC _D F ) |` RR ) |` ( C (,) D ) ) = ( ( CC _D F ) |` ( C (,) D ) ) )
43	41, 42	mp1i 13	. . . 4 |- ( ph -> ( ( ( CC _D F ) |` RR ) |` ( C (,) D ) ) = ( ( CC _D F ) |` ( C (,) D ) ) )
44	28	reseq1d 5395	. . . . 5 |- ( ph -> ( ( CC _D F ) |` ( C (,) D ) ) = ( ( x e. CC |-> B ) |` ( C (,) D ) ) )
45		ioosscn 39716	. . . . . 6 |- ( C (,) D ) C_ CC
46		resmpt 5449	. . . . . 6 |- ( ( C (,) D ) C_ CC -> ( ( x e. CC |-> B ) |` ( C (,) D ) ) = ( x e. ( C (,) D ) |-> B ) )
47	45, 46	mp1i 13	. . . . 5 |- ( ph -> ( ( x e. CC |-> B ) |` ( C (,) D ) ) = ( x e. ( C (,) D ) |-> B ) )
48	44, 47	eqtrd 2656	. . . 4 |- ( ph -> ( ( CC _D F ) |` ( C (,) D ) ) = ( x e. ( C (,) D ) |-> B ) )
49	40, 43, 48	3eqtrd 2660	. . 3 |- ( ph -> ( ( RR _D ( F |` RR ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( C [,] D ) ) ) = ( x e. ( C (,) D ) |-> B ) )
50	14, 23, 49	3eqtrd 2660	. 2 |- ( ph -> ( RR _D ( F |` ( C [,] D ) ) ) = ( x e. ( C (,) D ) |-> B ) )
51	11, 50	eqtr3d 2658	1 |- ( ph -> ( RR _D ( x e. ( C [,] D ) |-> A ) ) = ( x e. ( C (,) D ) |-> B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {cpr 4179   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   (,)cioo 12175   [,]cicc 12178   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   _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-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

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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  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-seq 12802  df-exp 12861  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-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  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-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-limc 23630  df-dv 23631

This theorem is referenced by:  itgsincmulx  40190