Theorem sseqval 30450

Description: Value of the strong sequence builder function. The set W represents here the words of length greater than or equal to the lenght of the initial sequence M. (Contributed by Thierry Arnoux, 21-Apr-2019.)

Hypotheses

Ref	Expression
sseqval.1	|- ( ph -> S e. _V )
sseqval.2	|- ( ph -> M e. Word S )
sseqval.3	|- W = (Word S i^i ( `' # " ( ZZ>= ` ( # ` M ) ) ) )
sseqval.4	|- ( ph -> F : W --> S )

Assertion

Ref	Expression
sseqval	|- ( ph -> ( Mseqstr F ) = ( M u. ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ) )

Distinct variable groups:   x, y, F   x, M, y   ph, x, y

Allowed substitution hints:   S( x, y)   W( x, y)

Proof of Theorem sseqval

Dummy variables f m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sseq 30446	. . 3 |- seqstr = ( m e. _V , f e. _V |-> ( m u. ( lastS o. seq ( # ` m ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( f ` x ) "> ) ) , ( NN0 X. { ( m ++ <" ( f ` m ) "> ) } ) ) ) ) )
2	1	a1i 11	. 2 |- ( ph -> seqstr = ( m e. _V , f e. _V |-> ( m u. ( lastS o. seq ( # ` m ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( f ` x ) "> ) ) , ( NN0 X. { ( m ++ <" ( f ` m ) "> ) } ) ) ) ) ) )
3		simprl 794	. . 3 |- ( ( ph /\ ( m = M /\ f = F ) ) -> m = M )
4	3	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( m = M /\ f = F ) ) -> ( # ` m ) = ( # ` M ) )
5		simp1rr 1127	. . . . . . . . 9 |- ( ( ( ph /\ ( m = M /\ f = F ) ) /\ x e. _V /\ y e. _V ) -> f = F )
6	5	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ ( m = M /\ f = F ) ) /\ x e. _V /\ y e. _V ) -> ( f ` x ) = ( F ` x ) )
7	6	s1eqd 13381	. . . . . . 7 |- ( ( ( ph /\ ( m = M /\ f = F ) ) /\ x e. _V /\ y e. _V ) -> <" ( f ` x ) "> = <" ( F ` x ) "> )
8	7	oveq2d 6666	. . . . . 6 |- ( ( ( ph /\ ( m = M /\ f = F ) ) /\ x e. _V /\ y e. _V ) -> ( x ++ <" ( f ` x ) "> ) = ( x ++ <" ( F ` x ) "> ) )
9	8	mpt2eq3dva 6719	. . . . 5 |- ( ( ph /\ ( m = M /\ f = F ) ) -> ( x e. _V , y e. _V |-> ( x ++ <" ( f ` x ) "> ) ) = ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) )
10		simprr 796	. . . . . . . . . 10 |- ( ( ph /\ ( m = M /\ f = F ) ) -> f = F )
11	10, 3	fveq12d 6197	. . . . . . . . 9 |- ( ( ph /\ ( m = M /\ f = F ) ) -> ( f ` m ) = ( F ` M ) )
12	11	s1eqd 13381	. . . . . . . 8 |- ( ( ph /\ ( m = M /\ f = F ) ) -> <" ( f ` m ) "> = <" ( F ` M ) "> )
13	3, 12	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( m = M /\ f = F ) ) -> ( m ++ <" ( f ` m ) "> ) = ( M ++ <" ( F ` M ) "> ) )
14	13	sneqd 4189	. . . . . 6 |- ( ( ph /\ ( m = M /\ f = F ) ) -> { ( m ++ <" ( f ` m ) "> ) } = { ( M ++ <" ( F ` M ) "> ) } )
15	14	xpeq2d 5139	. . . . 5 |- ( ( ph /\ ( m = M /\ f = F ) ) -> ( NN0 X. { ( m ++ <" ( f ` m ) "> ) } ) = ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) )
16	4, 9, 15	seqeq123d 12810	. . . 4 |- ( ( ph /\ ( m = M /\ f = F ) ) -> seq ( # ` m ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( f ` x ) "> ) ) , ( NN0 X. { ( m ++ <" ( f ` m ) "> ) } ) ) = seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) )
17	16	coeq2d 5284	. . 3 |- ( ( ph /\ ( m = M /\ f = F ) ) -> ( lastS o. seq ( # ` m ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( f ` x ) "> ) ) , ( NN0 X. { ( m ++ <" ( f ` m ) "> ) } ) ) ) = ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) )
18	3, 17	uneq12d 3768	. 2 |- ( ( ph /\ ( m = M /\ f = F ) ) -> ( m u. ( lastS o. seq ( # ` m ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( f ` x ) "> ) ) , ( NN0 X. { ( m ++ <" ( f ` m ) "> ) } ) ) ) ) = ( M u. ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ) )
19		sseqval.2	. . 3 |- ( ph -> M e. Word S )
20		elex 3212	. . 3 |- ( M e. Word S -> M e. _V )
21	19, 20	syl 17	. 2 |- ( ph -> M e. _V )
22		sseqval.4	. . 3 |- ( ph -> F : W --> S )
23		sseqval.3	. . . 4 |- W = (Word S i^i ( `' # " ( ZZ>= ` ( # ` M ) ) ) )
24		sseqval.1	. . . . 5 |- ( ph -> S e. _V )
25		wrdexg 13315	. . . . 5 |- ( S e. _V -> Word S e. _V )
26		inex1g 4801	. . . . 5 |- (Word S e. _V -> (Word S i^i ( `' # " ( ZZ>= ` ( # ` M ) ) ) ) e. _V )
27	24, 25, 26	3syl 18	. . . 4 |- ( ph -> (Word S i^i ( `' # " ( ZZ>= ` ( # ` M ) ) ) ) e. _V )
28	23, 27	syl5eqel 2705	. . 3 |- ( ph -> W e. _V )
29		fex 6490	. . 3 |- ( ( F : W --> S /\ W e. _V ) -> F e. _V )
30	22, 28, 29	syl2anc 693	. 2 |- ( ph -> F e. _V )
31		df-lsw 13300	. . . . . 6 |- lastS = ( x e. _V |-> ( x ` ( ( # ` x ) - 1 ) ) )
32	31	funmpt2 5927	. . . . 5 |- Fun lastS
33	32	a1i 11	. . . 4 |- ( ph -> Fun lastS )
34		seqex 12803	. . . . 5 |- seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) e. _V
35	34	a1i 11	. . . 4 |- ( ph -> seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) e. _V )
36		cofunexg 7130	. . . 4 |- ( ( Fun lastS /\ seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) e. _V ) -> ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) e. _V )
37	33, 35, 36	syl2anc 693	. . 3 |- ( ph -> ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) e. _V )
38		unexg 6959	. . 3 |- ( ( M e. _V /\ ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) e. _V ) -> ( M u. ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ) e. _V )
39	21, 37, 38	syl2anc 693	. 2 |- ( ph -> ( M u. ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ) e. _V )
40	2, 18, 21, 30, 39	ovmpt2d 6788	1 |- ( ph -> ( Mseqstr F ) = ( M u. ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   {csn 4177   X. cxp 5112   `'ccnv 5113   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1c1 9937   - cmin 10266   NN0cn0 11292   ZZ>=cuz 11687   seqcseq 12801   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294  seq_strcsseq 30445

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993

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-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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  df-pm 7860  df-neg 10269  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-word 13299  df-lsw 13300  df-s1 13302  df-sseq 30446

This theorem is referenced by:  sseqfv1  30451  sseqfn  30452  sseqf  30454  sseqfv2  30456