Theorem signstfval 30641

Description: Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)

Hypotheses

Ref	Expression
signsv.p	|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
signsv.w	|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
signsv.t	|- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
signsv.v	|- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )

Assertion

Ref	Expression
signstfval	|- ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` F ) ` N ) = ( W gsumg ( i e. ( 0 ... N ) |-> (sgn ` ( F ` i ) ) ) ) )

Distinct variable groups:   f, i, n, F   i, N, n   f, W, n

Allowed substitution hints:   .+^ ( f, i, j, n, a, b)   T( f, i, j, n, a, b)   F( j, a, b)   N( f, j, a, b)   V( f, i, j, n, a, b)   W( i, j, a, b)

Proof of Theorem signstfval

Step	Hyp	Ref	Expression
1		signsv.p	. . . 4 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
2		signsv.w	. . . 4 |- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
3		signsv.t	. . . 4 |- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
4		signsv.v	. . . 4 |- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
5	1, 2, 3, 4	signstfv 30640	. . 3 |- ( F e. Word RR -> ( T ` F ) = ( n e. ( 0..^ ( # ` F ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( F ` i ) ) ) ) ) )
6	5	adantr 481	. 2 |- ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( T ` F ) = ( n e. ( 0..^ ( # ` F ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( F ` i ) ) ) ) ) )
7		simpr 477	. . . . 5 |- ( ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) /\ n = N ) -> n = N )
8	7	oveq2d 6666	. . . 4 |- ( ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) /\ n = N ) -> ( 0 ... n ) = ( 0 ... N ) )
9	8	mpteq1d 4738	. . 3 |- ( ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) /\ n = N ) -> ( i e. ( 0 ... n ) |-> (sgn ` ( F ` i ) ) ) = ( i e. ( 0 ... N ) |-> (sgn ` ( F ` i ) ) ) )
10	9	oveq2d 6666	. 2 |- ( ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) /\ n = N ) -> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( F ` i ) ) ) ) = ( W gsumg ( i e. ( 0 ... N ) |-> (sgn ` ( F ` i ) ) ) ) )
11		simpr 477	. 2 |- ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> N e. ( 0..^ ( # ` F ) ) )
12		ovexd 6680	. 2 |- ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( W gsumg ( i e. ( 0 ... N ) |-> (sgn ` ( F ` i ) ) ) ) e. _V )
13	6, 10, 11, 12	fvmptd 6288	1 |- ( ( F e. Word RR /\ N e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` F ) ` N ) = ( W gsumg ( i e. ( 0 ... N ) |-> (sgn ` ( F ` i ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   ifcif 4086   {cpr 4179   {ctp 4181   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   - cmin 10266   -ucneg 10267   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  sgncsgn 13826   sum_csu 14416   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101

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

This theorem is referenced by:  signstcl  30642  signstfvn  30646  signstfvp  30648