Metamath Proof Explorer


Theorem cncfshiftioo

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)

Ref Expression
Hypotheses cncfshiftioo.a φ A
cncfshiftioo.b φ B
cncfshiftioo.c C = A B
cncfshiftioo.t φ T
cncfshiftioo.d D = A + T B + T
cncfshiftioo.f φ F : C cn
cncfshiftioo.g G = x D F x T
Assertion cncfshiftioo φ G : D cn

Proof

Step Hyp Ref Expression
1 cncfshiftioo.a φ A
2 cncfshiftioo.b φ B
3 cncfshiftioo.c C = A B
4 cncfshiftioo.t φ T
5 cncfshiftioo.d D = A + T B + T
6 cncfshiftioo.f φ F : C cn
7 cncfshiftioo.g G = x D F x T
8 ioosscn A B
9 8 a1i φ A B
10 4 recnd φ T
11 eqeq1 w = x w = z + T x = z + T
12 11 rexbidv w = x z A B w = z + T z A B x = z + T
13 oveq1 z = y z + T = y + T
14 13 eqeq2d z = y x = z + T x = y + T
15 14 cbvrexvw z A B x = z + T y A B x = y + T
16 12 15 bitrdi w = x z A B w = z + T y A B x = y + T
17 16 cbvrabv w | z A B w = z + T = x | y A B x = y + T
18 3 oveq1i C cn = A B cn
19 6 18 eleqtrdi φ F : A B cn
20 eqid x w | z A B w = z + T F x T = x w | z A B w = z + T F x T
21 9 10 17 19 20 cncfshift φ x w | z A B w = z + T F x T : w | z A B w = z + T cn
22 1 2 4 iooshift φ A + T B + T = w | z A B w = z + T
23 5 22 syl5eq φ D = w | z A B w = z + T
24 23 mpteq1d φ x D F x T = x w | z A B w = z + T F x T
25 7 24 syl5eq φ G = x w | z A B w = z + T F x T
26 23 oveq1d φ D cn = w | z A B w = z + T cn
27 21 25 26 3eltr4d φ G : D cn