Theorem ccatfval 13358

Description: Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Assertion

Ref	Expression
ccatfval	|- ( ( S e. V /\ T e. W ) -> ( S ++ T ) = ( x e. ( 0..^ ( ( # ` S ) + ( # ` T ) ) ) |-> if ( x e. ( 0..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) ) )

Distinct variable groups:   x, S   x, T

Allowed substitution hints:   V( x)   W( x)

Proof of Theorem ccatfval

Dummy variables t s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( S e. V -> S e. _V )
2		elex 3212	. 2 |- ( T e. W -> T e. _V )
3		fveq2 6191	. . . . . 6 |- ( s = S -> ( # ` s ) = ( # ` S ) )
4		fveq2 6191	. . . . . 6 |- ( t = T -> ( # ` t ) = ( # ` T ) )
5	3, 4	oveqan12d 6669	. . . . 5 |- ( ( s = S /\ t = T ) -> ( ( # ` s ) + ( # ` t ) ) = ( ( # ` S ) + ( # ` T ) ) )
6	5	oveq2d 6666	. . . 4 |- ( ( s = S /\ t = T ) -> ( 0..^ ( ( # ` s ) + ( # ` t ) ) ) = ( 0..^ ( ( # ` S ) + ( # ` T ) ) ) )
7	3	oveq2d 6666	. . . . . . 7 |- ( s = S -> ( 0..^ ( # ` s ) ) = ( 0..^ ( # ` S ) ) )
8	7	eleq2d 2687	. . . . . 6 |- ( s = S -> ( x e. ( 0..^ ( # ` s ) ) <-> x e. ( 0..^ ( # ` S ) ) ) )
9	8	adantr 481	. . . . 5 |- ( ( s = S /\ t = T ) -> ( x e. ( 0..^ ( # ` s ) ) <-> x e. ( 0..^ ( # ` S ) ) ) )
10		fveq1 6190	. . . . . 6 |- ( s = S -> ( s ` x ) = ( S ` x ) )
11	10	adantr 481	. . . . 5 |- ( ( s = S /\ t = T ) -> ( s ` x ) = ( S ` x ) )
12		simpr 477	. . . . . 6 |- ( ( s = S /\ t = T ) -> t = T )
13	3	oveq2d 6666	. . . . . . 7 |- ( s = S -> ( x - ( # ` s ) ) = ( x - ( # ` S ) ) )
14	13	adantr 481	. . . . . 6 |- ( ( s = S /\ t = T ) -> ( x - ( # ` s ) ) = ( x - ( # ` S ) ) )
15	12, 14	fveq12d 6197	. . . . 5 |- ( ( s = S /\ t = T ) -> ( t ` ( x - ( # ` s ) ) ) = ( T ` ( x - ( # ` S ) ) ) )
16	9, 11, 15	ifbieq12d 4113	. . . 4 |- ( ( s = S /\ t = T ) -> if ( x e. ( 0..^ ( # ` s ) ) , ( s ` x ) , ( t ` ( x - ( # ` s ) ) ) ) = if ( x e. ( 0..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) )
17	6, 16	mpteq12dv 4733	. . 3 |- ( ( s = S /\ t = T ) -> ( x e. ( 0..^ ( ( # ` s ) + ( # ` t ) ) ) |-> if ( x e. ( 0..^ ( # ` s ) ) , ( s ` x ) , ( t ` ( x - ( # ` s ) ) ) ) ) = ( x e. ( 0..^ ( ( # ` S ) + ( # ` T ) ) ) |-> if ( x e. ( 0..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) ) )
18		df-concat 13301	. . 3 |- ++ = ( s e. _V , t e. _V |-> ( x e. ( 0..^ ( ( # ` s ) + ( # ` t ) ) ) |-> if ( x e. ( 0..^ ( # ` s ) ) , ( s ` x ) , ( t ` ( x - ( # ` s ) ) ) ) ) )
19		ovex 6678	. . . 4 |- ( 0..^ ( ( # ` S ) + ( # ` T ) ) ) e. _V
20	19	mptex 6486	. . 3 |- ( x e. ( 0..^ ( ( # ` S ) + ( # ` T ) ) ) |-> if ( x e. ( 0..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) ) e. _V
21	17, 18, 20	ovmpt2a 6791	. 2 |- ( ( S e. _V /\ T e. _V ) -> ( S ++ T ) = ( x e. ( 0..^ ( ( # ` S ) + ( # ` T ) ) ) |-> if ( x e. ( 0..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) ) )
22	1, 2, 21	syl2an 494	1 |- ( ( S e. V /\ T e. W ) -> ( S ++ T ) = ( x e. ( 0..^ ( ( # ` S ) + ( # ` T ) ) ) |-> if ( x e. ( 0..^ ( # ` S ) ) , ( S ` x ) , ( T ` ( x - ( # ` S ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   - cmin 10266  ..^cfzo 12465   #chash 13117   ++ cconcat 13293

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-rep 4771  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-ne 2795  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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-concat 13301

This theorem is referenced by:  ccatcl  13359  ccatlen  13360  ccatval1  13361  ccatval2  13362  ccatvalfn  13365  ccatalpha  13375  repswccat  13532  ccatco  13581  ofccat  13708  ccatmulgnn0dir  30619