Theorem cncfshiftioo 40105

Description: A periodic continuous function stays continuous if the domain is an open interval that is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
cncfshiftioo.a	|- ( ph -> A e. RR )
cncfshiftioo.b	|- ( ph -> B e. RR )
cncfshiftioo.c	|- C = ( A (,) B )
cncfshiftioo.t	|- ( ph -> T e. RR )
cncfshiftioo.d	|- D = ( ( A + T ) (,) ( B + T ) )
cncfshiftioo.f	|- ( ph -> F e. ( C -cn-> CC ) )
cncfshiftioo.g	|- G = ( x e. D |-> ( F ` ( x - T ) ) )

Assertion

Ref	Expression
cncfshiftioo	|- ( ph -> G e. ( D -cn-> CC ) )

Distinct variable groups:   x, A   x, B   x, D   x, F   x, T   ph, x

Allowed substitution hints:   C( x)   G( x)

Proof of Theorem cncfshiftioo

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioosscn 39716	. . . 4 |- ( A (,) B ) C_ CC
2	1	a1i 11	. . 3 |- ( ph -> ( A (,) B ) C_ CC )
3		cncfshiftioo.t	. . . 4 |- ( ph -> T e. RR )
4	3	recnd 10068	. . 3 |- ( ph -> T e. CC )
5		eqeq1 2626	. . . . . 6 |- ( w = x -> ( w = ( z + T ) <-> x = ( z + T ) ) )
6	5	rexbidv 3052	. . . . 5 |- ( w = x -> ( E. z e. ( A (,) B ) w = ( z + T ) <-> E. z e. ( A (,) B ) x = ( z + T ) ) )
7		oveq1 6657	. . . . . . 7 |- ( z = y -> ( z + T ) = ( y + T ) )
8	7	eqeq2d 2632	. . . . . 6 |- ( z = y -> ( x = ( z + T ) <-> x = ( y + T ) ) )
9	8	cbvrexv 3172	. . . . 5 |- ( E. z e. ( A (,) B ) x = ( z + T ) <-> E. y e. ( A (,) B ) x = ( y + T ) )
10	6, 9	syl6bb 276	. . . 4 |- ( w = x -> ( E. z e. ( A (,) B ) w = ( z + T ) <-> E. y e. ( A (,) B ) x = ( y + T ) ) )
11	10	cbvrabv 3199	. . 3 |- { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } = { x e. CC | E. y e. ( A (,) B ) x = ( y + T ) }
12		cncfshiftioo.f	. . . 4 |- ( ph -> F e. ( C -cn-> CC ) )
13		cncfshiftioo.c	. . . . 5 |- C = ( A (,) B )
14	13	oveq1i 6660	. . . 4 |- ( C -cn-> CC ) = ( ( A (,) B ) -cn-> CC )
15	12, 14	syl6eleq 2711	. . 3 |- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
16		eqid 2622	. . 3 |- ( x e. { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } |-> ( F ` ( x - T ) ) ) = ( x e. { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } |-> ( F ` ( x - T ) ) )
17	2, 4, 11, 15, 16	cncfshift 40087	. 2 |- ( ph -> ( x e. { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } |-> ( F ` ( x - T ) ) ) e. ( { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } -cn-> CC ) )
18		cncfshiftioo.g	. . 3 |- G = ( x e. D |-> ( F ` ( x - T ) ) )
19		cncfshiftioo.d	. . . . 5 |- D = ( ( A + T ) (,) ( B + T ) )
20		cncfshiftioo.a	. . . . . 6 |- ( ph -> A e. RR )
21		cncfshiftioo.b	. . . . . 6 |- ( ph -> B e. RR )
22	20, 21, 3	iooshift 39748	. . . . 5 |- ( ph -> ( ( A + T ) (,) ( B + T ) ) = { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } )
23	19, 22	syl5eq 2668	. . . 4 |- ( ph -> D = { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } )
24	23	mpteq1d 4738	. . 3 |- ( ph -> ( x e. D |-> ( F ` ( x - T ) ) ) = ( x e. { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } |-> ( F ` ( x - T ) ) ) )
25	18, 24	syl5eq 2668	. 2 |- ( ph -> G = ( x e. { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } |-> ( F ` ( x - T ) ) ) )
26	23	oveq1d 6665	. 2 |- ( ph -> ( D -cn-> CC ) = ( { w e. CC | E. z e. ( A (,) B ) w = ( z + T ) } -cn-> CC ) )
27	17, 25, 26	3eltr4d 2716	1 |- ( ph -> G e. ( D -cn-> CC ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   C_ wss 3574   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   - cmin 10266   (,)cioo 12175   -cn->ccncf 22679

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-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

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-ioo 12179  df-cncf 22681

This theorem is referenced by:  fourierdlem90  40413