Theorem splfv2a 13507

Description: Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (Contributed by Stefan O'Rear, 23-Aug-2015.)

Hypotheses

Ref	Expression
spllen.s	|- ( ph -> S e. Word A )
spllen.f	|- ( ph -> F e. ( 0 ... T ) )
spllen.t	|- ( ph -> T e. ( 0 ... ( # ` S ) ) )
spllen.r	|- ( ph -> R e. Word A )
splfv2a.x	|- ( ph -> X e. ( 0..^ ( # ` R ) ) )

Assertion

Ref	Expression
splfv2a	|- ( ph -> ( ( S splice <. F , T , R >. ) ` ( F + X ) ) = ( R ` X ) )

Proof of Theorem splfv2a

Step	Hyp	Ref	Expression
1		spllen.s	. . . 4 |- ( ph -> S e. Word A )
2		spllen.f	. . . 4 |- ( ph -> F e. ( 0 ... T ) )
3		spllen.t	. . . 4 |- ( ph -> T e. ( 0 ... ( # ` S ) ) )
4		spllen.r	. . . 4 |- ( ph -> R e. Word A )
5		splval 13502	. . . 4 |- ( ( S e. Word A /\ ( F e. ( 0 ... T ) /\ T e. ( 0 ... ( # ` S ) ) /\ R e. Word A ) ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) )
6	1, 2, 3, 4, 5	syl13anc 1328	. . 3 |- ( ph -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) )
7		elfznn0 12433	. . . . . . 7 |- ( F e. ( 0 ... T ) -> F e. NN0 )
8	2, 7	syl 17	. . . . . 6 |- ( ph -> F e. NN0 )
9	8	nn0cnd 11353	. . . . 5 |- ( ph -> F e. CC )
10		splfv2a.x	. . . . . . 7 |- ( ph -> X e. ( 0..^ ( # ` R ) ) )
11		elfzoelz 12470	. . . . . . 7 |- ( X e. ( 0..^ ( # ` R ) ) -> X e. ZZ )
12	10, 11	syl 17	. . . . . 6 |- ( ph -> X e. ZZ )
13	12	zcnd 11483	. . . . 5 |- ( ph -> X e. CC )
14	9, 13	addcomd 10238	. . . 4 |- ( ph -> ( F + X ) = ( X + F ) )
15		nn0uz 11722	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
16	8, 15	syl6eleq 2711	. . . . . . . 8 |- ( ph -> F e. ( ZZ>= ` 0 ) )
17		eluzfz1 12348	. . . . . . . 8 |- ( F e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... F ) )
18	16, 17	syl 17	. . . . . . 7 |- ( ph -> 0 e. ( 0 ... F ) )
19		elfzuz3 12339	. . . . . . . . . 10 |- ( T e. ( 0 ... ( # ` S ) ) -> ( # ` S ) e. ( ZZ>= ` T ) )
20	3, 19	syl 17	. . . . . . . . 9 |- ( ph -> ( # ` S ) e. ( ZZ>= ` T ) )
21		elfzuz3 12339	. . . . . . . . . 10 |- ( F e. ( 0 ... T ) -> T e. ( ZZ>= ` F ) )
22	2, 21	syl 17	. . . . . . . . 9 |- ( ph -> T e. ( ZZ>= ` F ) )
23		uztrn 11704	. . . . . . . . 9 |- ( ( ( # ` S ) e. ( ZZ>= ` T ) /\ T e. ( ZZ>= ` F ) ) -> ( # ` S ) e. ( ZZ>= ` F ) )
24	20, 22, 23	syl2anc 693	. . . . . . . 8 |- ( ph -> ( # ` S ) e. ( ZZ>= ` F ) )
25		elfzuzb 12336	. . . . . . . 8 |- ( F e. ( 0 ... ( # ` S ) ) <-> ( F e. ( ZZ>= ` 0 ) /\ ( # ` S ) e. ( ZZ>= ` F ) ) )
26	16, 24, 25	sylanbrc 698	. . . . . . 7 |- ( ph -> F e. ( 0 ... ( # ` S ) ) )
27		swrdlen 13423	. . . . . . 7 |- ( ( S e. Word A /\ 0 e. ( 0 ... F ) /\ F e. ( 0 ... ( # ` S ) ) ) -> ( # ` ( S substr <. 0 , F >. ) ) = ( F - 0 ) )
28	1, 18, 26, 27	syl3anc 1326	. . . . . 6 |- ( ph -> ( # ` ( S substr <. 0 , F >. ) ) = ( F - 0 ) )
29	9	subid1d 10381	. . . . . 6 |- ( ph -> ( F - 0 ) = F )
30	28, 29	eqtrd 2656	. . . . 5 |- ( ph -> ( # ` ( S substr <. 0 , F >. ) ) = F )
31	30	oveq2d 6666	. . . 4 |- ( ph -> ( X + ( # ` ( S substr <. 0 , F >. ) ) ) = ( X + F ) )
32	14, 31	eqtr4d 2659	. . 3 |- ( ph -> ( F + X ) = ( X + ( # ` ( S substr <. 0 , F >. ) ) ) )
33	6, 32	fveq12d 6197	. 2 |- ( ph -> ( ( S splice <. F , T , R >. ) ` ( F + X ) ) = ( ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) ` ( X + ( # ` ( S substr <. 0 , F >. ) ) ) ) )
34		swrdcl 13419	. . . . 5 |- ( S e. Word A -> ( S substr <. 0 , F >. ) e. Word A )
35	1, 34	syl 17	. . . 4 |- ( ph -> ( S substr <. 0 , F >. ) e. Word A )
36		ccatcl 13359	. . . 4 |- ( ( ( S substr <. 0 , F >. ) e. Word A /\ R e. Word A ) -> ( ( S substr <. 0 , F >. ) ++ R ) e. Word A )
37	35, 4, 36	syl2anc 693	. . 3 |- ( ph -> ( ( S substr <. 0 , F >. ) ++ R ) e. Word A )
38		swrdcl 13419	. . . 4 |- ( S e. Word A -> ( S substr <. T , ( # ` S ) >. ) e. Word A )
39	1, 38	syl 17	. . 3 |- ( ph -> ( S substr <. T , ( # ` S ) >. ) e. Word A )
40		0nn0 11307	. . . . . . . 8 |- 0 e. NN0
41		nn0addcl 11328	. . . . . . . 8 |- ( ( 0 e. NN0 /\ F e. NN0 ) -> ( 0 + F ) e. NN0 )
42	40, 8, 41	sylancr 695	. . . . . . 7 |- ( ph -> ( 0 + F ) e. NN0 )
43		fzoss1 12495	. . . . . . . 8 |- ( ( 0 + F ) e. ( ZZ>= ` 0 ) -> ( ( 0 + F )..^ ( ( # ` R ) + F ) ) C_ ( 0..^ ( ( # ` R ) + F ) ) )
44	43, 15	eleq2s 2719	. . . . . . 7 |- ( ( 0 + F ) e. NN0 -> ( ( 0 + F )..^ ( ( # ` R ) + F ) ) C_ ( 0..^ ( ( # ` R ) + F ) ) )
45	42, 44	syl 17	. . . . . 6 |- ( ph -> ( ( 0 + F )..^ ( ( # ` R ) + F ) ) C_ ( 0..^ ( ( # ` R ) + F ) ) )
46		ccatlen 13360	. . . . . . . . 9 |- ( ( ( S substr <. 0 , F >. ) e. Word A /\ R e. Word A ) -> ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) = ( ( # ` ( S substr <. 0 , F >. ) ) + ( # ` R ) ) )
47	35, 4, 46	syl2anc 693	. . . . . . . 8 |- ( ph -> ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) = ( ( # ` ( S substr <. 0 , F >. ) ) + ( # ` R ) ) )
48	30	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( # ` ( S substr <. 0 , F >. ) ) + ( # ` R ) ) = ( F + ( # ` R ) ) )
49		wrdfin 13323	. . . . . . . . . . . 12 |- ( R e. Word A -> R e. Fin )
50	4, 49	syl 17	. . . . . . . . . . 11 |- ( ph -> R e. Fin )
51		hashcl 13147	. . . . . . . . . . 11 |- ( R e. Fin -> ( # ` R ) e. NN0 )
52	50, 51	syl 17	. . . . . . . . . 10 |- ( ph -> ( # ` R ) e. NN0 )
53	52	nn0cnd 11353	. . . . . . . . 9 |- ( ph -> ( # ` R ) e. CC )
54	9, 53	addcomd 10238	. . . . . . . 8 |- ( ph -> ( F + ( # ` R ) ) = ( ( # ` R ) + F ) )
55	47, 48, 54	3eqtrd 2660	. . . . . . 7 |- ( ph -> ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) = ( ( # ` R ) + F ) )
56	55	oveq2d 6666	. . . . . 6 |- ( ph -> ( 0..^ ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) ) = ( 0..^ ( ( # ` R ) + F ) ) )
57	45, 56	sseqtr4d 3642	. . . . 5 |- ( ph -> ( ( 0 + F )..^ ( ( # ` R ) + F ) ) C_ ( 0..^ ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) ) )
58	8	nn0zd 11480	. . . . . 6 |- ( ph -> F e. ZZ )
59		fzoaddel 12520	. . . . . 6 |- ( ( X e. ( 0..^ ( # ` R ) ) /\ F e. ZZ ) -> ( X + F ) e. ( ( 0 + F )..^ ( ( # ` R ) + F ) ) )
60	10, 58, 59	syl2anc 693	. . . . 5 |- ( ph -> ( X + F ) e. ( ( 0 + F )..^ ( ( # ` R ) + F ) ) )
61	57, 60	sseldd 3604	. . . 4 |- ( ph -> ( X + F ) e. ( 0..^ ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) ) )
62	31, 61	eqeltrd 2701	. . 3 |- ( ph -> ( X + ( # ` ( S substr <. 0 , F >. ) ) ) e. ( 0..^ ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) ) )
63		ccatval1 13361	. . 3 |- ( ( ( ( S substr <. 0 , F >. ) ++ R ) e. Word A /\ ( S substr <. T , ( # ` S ) >. ) e. Word A /\ ( X + ( # ` ( S substr <. 0 , F >. ) ) ) e. ( 0..^ ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) ) ) -> ( ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) ` ( X + ( # ` ( S substr <. 0 , F >. ) ) ) ) = ( ( ( S substr <. 0 , F >. ) ++ R ) ` ( X + ( # ` ( S substr <. 0 , F >. ) ) ) ) )
64	37, 39, 62, 63	syl3anc 1326	. 2 |- ( ph -> ( ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) ` ( X + ( # ` ( S substr <. 0 , F >. ) ) ) ) = ( ( ( S substr <. 0 , F >. ) ++ R ) ` ( X + ( # ` ( S substr <. 0 , F >. ) ) ) ) )
65		ccatval3 13363	. . 3 |- ( ( ( S substr <. 0 , F >. ) e. Word A /\ R e. Word A /\ X e. ( 0..^ ( # ` R ) ) ) -> ( ( ( S substr <. 0 , F >. ) ++ R ) ` ( X + ( # ` ( S substr <. 0 , F >. ) ) ) ) = ( R ` X ) )
66	35, 4, 10, 65	syl3anc 1326	. 2 |- ( ph -> ( ( ( S substr <. 0 , F >. ) ++ R ) ` ( X + ( # ` ( S substr <. 0 , F >. ) ) ) ) = ( R ` X ) )
67	33, 64, 66	3eqtrd 2660	1 |- ( ph -> ( ( S splice <. F , T , R >. ) ` ( F + X ) ) = ( R ` X ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   <.cop 4183   <.cotp 4185   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   + caddc 9939   - cmin 10266   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295   splice csplice 13296

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-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-ot 4186  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-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-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-splice 13304

This theorem is referenced by:  psgnunilem2  17915