Theorem sseqfv2 30456

Description: Value of the strong sequence builder function. (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 )
sseqfv2.4	|- ( ph -> N e. ( ZZ>= ` ( # ` M ) ) )

Assertion

Ref	Expression
sseqfv2	|- ( ph -> ( ( Mseqstr F ) ` N ) = ( lastS ` ( seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ` N ) ) )

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

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

Proof of Theorem sseqfv2

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseqval.1	. . . 4 |- ( ph -> S e. _V )
2		sseqval.2	. . . 4 |- ( ph -> M e. Word S )
3		sseqval.3	. . . 4 |- W = (Word S i^i ( `' # " ( ZZ>= ` ( # ` M ) ) ) )
4		sseqval.4	. . . 4 |- ( ph -> F : W --> S )
5	1, 2, 3, 4	sseqval 30450	. . 3 |- ( ph -> ( Mseqstr F ) = ( M u. ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ) )
6	5	fveq1d 6193	. 2 |- ( ph -> ( ( Mseqstr F ) ` N ) = ( ( M u. ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ) ` N ) )
7		wrdfn 13319	. . . 4 |- ( M e. Word S -> M Fn ( 0..^ ( # ` M ) ) )
8	2, 7	syl 17	. . 3 |- ( ph -> M Fn ( 0..^ ( # ` M ) ) )
9		fvex 6201	. . . . . 6 |- ( x ` ( ( # ` x ) - 1 ) ) e. _V
10		df-lsw 13300	. . . . . 6 |- lastS = ( x e. _V |-> ( x ` ( ( # ` x ) - 1 ) ) )
11	9, 10	fnmpti 6022	. . . . 5 |- lastS Fn _V
12	11	a1i 11	. . . 4 |- ( ph -> lastS Fn _V )
13		lencl 13324	. . . . . . 7 |- ( M e. Word S -> ( # ` M ) e. NN0 )
14	2, 13	syl 17	. . . . . 6 |- ( ph -> ( # ` M ) e. NN0 )
15	14	nn0zd 11480	. . . . 5 |- ( ph -> ( # ` M ) e. ZZ )
16		seqfn 12813	. . . . 5 |- ( ( # ` M ) e. ZZ -> seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) Fn ( ZZ>= ` ( # ` M ) ) )
17	15, 16	syl 17	. . . 4 |- ( ph -> seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) Fn ( ZZ>= ` ( # ` M ) ) )
18		ssv 3625	. . . . 5 |- ran seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) C_ _V
19	18	a1i 11	. . . 4 |- ( ph -> ran seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) C_ _V )
20		fnco 5999	. . . 4 |- ( ( lastS Fn _V /\ seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) Fn ( ZZ>= ` ( # ` M ) ) /\ ran seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) C_ _V ) -> ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) Fn ( ZZ>= ` ( # ` M ) ) )
21	12, 17, 19, 20	syl3anc 1326	. . 3 |- ( ph -> ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) Fn ( ZZ>= ` ( # ` M ) ) )
22		fzouzdisj 12504	. . . 4 |- ( ( 0..^ ( # ` M ) ) i^i ( ZZ>= ` ( # ` M ) ) ) = (/)
23	22	a1i 11	. . 3 |- ( ph -> ( ( 0..^ ( # ` M ) ) i^i ( ZZ>= ` ( # ` M ) ) ) = (/) )
24		sseqfv2.4	. . 3 |- ( ph -> N e. ( ZZ>= ` ( # ` M ) ) )
25		fvun2 6270	. . 3 |- ( ( M Fn ( 0..^ ( # ` M ) ) /\ ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) Fn ( ZZ>= ` ( # ` M ) ) /\ ( ( ( 0..^ ( # ` M ) ) i^i ( ZZ>= ` ( # ` M ) ) ) = (/) /\ N e. ( ZZ>= ` ( # ` M ) ) ) ) -> ( ( M u. ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ) ` N ) = ( ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ` N ) )
26	8, 21, 23, 24, 25	syl112anc 1330	. 2 |- ( ph -> ( ( M u. ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ) ` N ) = ( ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ` N ) )
27		fnfun 5988	. . . 4 |- ( seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) Fn ( ZZ>= ` ( # ` M ) ) -> Fun seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) )
28	17, 27	syl 17	. . 3 |- ( ph -> Fun seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) )
29		fvexd 6203	. . . . . 6 |- ( ph -> ( ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ` ( # ` M ) ) e. _V )
30		ovexd 6680	. . . . . 6 |- ( ( ph /\ ( a e. _V /\ b e. _V ) ) -> ( a ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) b ) e. _V )
31		eqid 2622	. . . . . 6 |- ( ZZ>= ` ( # ` M ) ) = ( ZZ>= ` ( # ` M ) )
32		fvexd 6203	. . . . . 6 |- ( ( ph /\ a e. ( ZZ>= ` ( ( # ` M ) + 1 ) ) ) -> ( ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ` a ) e. _V )
33	29, 30, 31, 15, 32	seqf2 12820	. . . . 5 |- ( ph -> seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) : ( ZZ>= ` ( # ` M ) ) --> _V )
34		fdm 6051	. . . . 5 |- ( seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) : ( ZZ>= ` ( # ` M ) ) --> _V -> dom seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) = ( ZZ>= ` ( # ` M ) ) )
35	33, 34	syl 17	. . . 4 |- ( ph -> dom seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) = ( ZZ>= ` ( # ` M ) ) )
36	24, 35	eleqtrrd 2704	. . 3 |- ( ph -> N e. dom seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) )
37		fvco 6274	. . 3 |- ( ( Fun seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) /\ N e. dom 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 ) "> ) } ) ) ) ` N ) = ( lastS ` ( seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ` N ) ) )
38	28, 36, 37	syl2anc 693	. 2 |- ( ph -> ( ( lastS o. seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ) ` N ) = ( lastS ` ( seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ` N ) ) )
39	6, 26, 38	3eqtrd 2660	1 |- ( ph -> ( ( Mseqstr F ) ` N ) = ( lastS ` ( seq ( # ` M ) ( ( x e. _V , y e. _V |-> ( x ++ <" ( F ` x ) "> ) ) , ( NN0 X. { ( M ++ <" ( F ` M ) "> ) } ) ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687  ..^cfzo 12465   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  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  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-int 4476  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-riota 6611  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-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-s1 13302  df-sseq 30446

This theorem is referenced by:  sseqp1  30457