Theorem cshfn 13536

Description: Perform a cyclical shift for a function over a half-open range of nonnegative integers. (Contributed by AV, 20-May-2018.) (Revised by AV, 17-Nov-2018.)

Assertion

Ref	Expression
cshfn	|- ( ( W e. { f | E. l e. NN0 f Fn ( 0..^ l ) } /\ N e. ZZ ) -> ( W cyclShift N ) = if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )

Distinct variable group:   f, l

Allowed substitution hints:   N( f, l)   W( f, l)

Proof of Theorem cshfn

Dummy variables n w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . 4 |- ( w = W -> ( w = (/) <-> W = (/) ) )
2	1	adantr 481	. . 3 |- ( ( w = W /\ n = N ) -> ( w = (/) <-> W = (/) ) )
3		simpl 473	. . . . 5 |- ( ( w = W /\ n = N ) -> w = W )
4		simpr 477	. . . . . . 7 |- ( ( w = W /\ n = N ) -> n = N )
5		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( # ` w ) = ( # ` W ) )
6	5	adantr 481	. . . . . . 7 |- ( ( w = W /\ n = N ) -> ( # ` w ) = ( # ` W ) )
7	4, 6	oveq12d 6668	. . . . . 6 |- ( ( w = W /\ n = N ) -> ( n mod ( # ` w ) ) = ( N mod ( # ` W ) ) )
8	7, 6	opeq12d 4410	. . . . 5 |- ( ( w = W /\ n = N ) -> <. ( n mod ( # ` w ) ) , ( # ` w ) >. = <. ( N mod ( # ` W ) ) , ( # ` W ) >. )
9	3, 8	oveq12d 6668	. . . 4 |- ( ( w = W /\ n = N ) -> ( w substr <. ( n mod ( # ` w ) ) , ( # ` w ) >. ) = ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) )
10	7	opeq2d 4409	. . . . 5 |- ( ( w = W /\ n = N ) -> <. 0 , ( n mod ( # ` w ) ) >. = <. 0 , ( N mod ( # ` W ) ) >. )
11	3, 10	oveq12d 6668	. . . 4 |- ( ( w = W /\ n = N ) -> ( w substr <. 0 , ( n mod ( # ` w ) ) >. ) = ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) )
12	9, 11	oveq12d 6668	. . 3 |- ( ( w = W /\ n = N ) -> ( ( w substr <. ( n mod ( # ` w ) ) , ( # ` w ) >. ) ++ ( w substr <. 0 , ( n mod ( # ` w ) ) >. ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) )
13	2, 12	ifbieq2d 4111	. 2 |- ( ( w = W /\ n = N ) -> if ( w = (/) , (/) , ( ( w substr <. ( n mod ( # ` w ) ) , ( # ` w ) >. ) ++ ( w substr <. 0 , ( n mod ( # ` w ) ) >. ) ) ) = if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
14		df-csh 13535	. 2 |- cyclShift = ( w e. { f | E. l e. NN0 f Fn ( 0..^ l ) } , n e. ZZ |-> if ( w = (/) , (/) , ( ( w substr <. ( n mod ( # ` w ) ) , ( # ` w ) >. ) ++ ( w substr <. 0 , ( n mod ( # ` w ) ) >. ) ) ) )
15		0ex 4790	. . 3 |- (/) e. _V
16		ovex 6678	. . 3 |- ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) e. _V
17	15, 16	ifex 4156	. 2 |- if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) e. _V
18	13, 14, 17	ovmpt2a 6791	1 |- ( ( W e. { f | E. l e. NN0 f Fn ( 0..^ l ) } /\ N e. ZZ ) -> ( W cyclShift N ) = if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   (/)c0 3915   ifcif 4086   <.cop 4183   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   0cc0 9936   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   mod cmo 12668   #chash 13117   ++ cconcat 13293   substr csubstr 13295   cyclShift ccsh 13534

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-csh 13535

This theorem is referenced by:  cshword  13537  cshword2  41437