Theorem spllen 13505

Description: The length of a splice. (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 )

Assertion

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

Proof of Theorem spllen

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	6	fveq2d 6195	. 2 |- ( ph -> ( # ` ( S splice <. F , T , R >. ) ) = ( # ` ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) ) )
8		swrdcl 13419	. . . . 5 |- ( S e. Word A -> ( S substr <. 0 , F >. ) e. Word A )
9	1, 8	syl 17	. . . 4 |- ( ph -> ( S substr <. 0 , F >. ) e. Word A )
10		ccatcl 13359	. . . 4 |- ( ( ( S substr <. 0 , F >. ) e. Word A /\ R e. Word A ) -> ( ( S substr <. 0 , F >. ) ++ R ) e. Word A )
11	9, 4, 10	syl2anc 693	. . 3 |- ( ph -> ( ( S substr <. 0 , F >. ) ++ R ) e. Word A )
12		swrdcl 13419	. . . 4 |- ( S e. Word A -> ( S substr <. T , ( # ` S ) >. ) e. Word A )
13	1, 12	syl 17	. . 3 |- ( ph -> ( S substr <. T , ( # ` S ) >. ) e. Word A )
14		ccatlen 13360	. . 3 |- ( ( ( ( S substr <. 0 , F >. ) ++ R ) e. Word A /\ ( S substr <. T , ( # ` S ) >. ) e. Word A ) -> ( # ` ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) ) = ( ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) + ( # ` ( S substr <. T , ( # ` S ) >. ) ) ) )
15	11, 13, 14	syl2anc 693	. 2 |- ( ph -> ( # ` ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) ) = ( ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) + ( # ` ( S substr <. T , ( # ` S ) >. ) ) ) )
16		lencl 13324	. . . . . . 7 |- ( R e. Word A -> ( # ` R ) e. NN0 )
17	4, 16	syl 17	. . . . . 6 |- ( ph -> ( # ` R ) e. NN0 )
18	17	nn0cnd 11353	. . . . 5 |- ( ph -> ( # ` R ) e. CC )
19		elfzelz 12342	. . . . . . 7 |- ( F e. ( 0 ... T ) -> F e. ZZ )
20	2, 19	syl 17	. . . . . 6 |- ( ph -> F e. ZZ )
21	20	zcnd 11483	. . . . 5 |- ( ph -> F e. CC )
22	18, 21	addcld 10059	. . . 4 |- ( ph -> ( ( # ` R ) + F ) e. CC )
23		elfzel2 12340	. . . . . 6 |- ( T e. ( 0 ... ( # ` S ) ) -> ( # ` S ) e. ZZ )
24	3, 23	syl 17	. . . . 5 |- ( ph -> ( # ` S ) e. ZZ )
25	24	zcnd 11483	. . . 4 |- ( ph -> ( # ` S ) e. CC )
26		elfzelz 12342	. . . . . 6 |- ( T e. ( 0 ... ( # ` S ) ) -> T e. ZZ )
27	3, 26	syl 17	. . . . 5 |- ( ph -> T e. ZZ )
28	27	zcnd 11483	. . . 4 |- ( ph -> T e. CC )
29	22, 25, 28	addsub12d 10415	. . 3 |- ( ph -> ( ( ( # ` R ) + F ) + ( ( # ` S ) - T ) ) = ( ( # ` S ) + ( ( ( # ` R ) + F ) - T ) ) )
30		ccatlen 13360	. . . . . 6 |- ( ( ( S substr <. 0 , F >. ) e. Word A /\ R e. Word A ) -> ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) = ( ( # ` ( S substr <. 0 , F >. ) ) + ( # ` R ) ) )
31	9, 4, 30	syl2anc 693	. . . . 5 |- ( ph -> ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) = ( ( # ` ( S substr <. 0 , F >. ) ) + ( # ` R ) ) )
32		elfzuz 12338	. . . . . . . . . 10 |- ( F e. ( 0 ... T ) -> F e. ( ZZ>= ` 0 ) )
33	2, 32	syl 17	. . . . . . . . 9 |- ( ph -> F e. ( ZZ>= ` 0 ) )
34		eluzfz1 12348	. . . . . . . . 9 |- ( F e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... F ) )
35	33, 34	syl 17	. . . . . . . 8 |- ( ph -> 0 e. ( 0 ... F ) )
36		elfzuz3 12339	. . . . . . . . . . 11 |- ( T e. ( 0 ... ( # ` S ) ) -> ( # ` S ) e. ( ZZ>= ` T ) )
37	3, 36	syl 17	. . . . . . . . . 10 |- ( ph -> ( # ` S ) e. ( ZZ>= ` T ) )
38		elfzuz3 12339	. . . . . . . . . . 11 |- ( F e. ( 0 ... T ) -> T e. ( ZZ>= ` F ) )
39	2, 38	syl 17	. . . . . . . . . 10 |- ( ph -> T e. ( ZZ>= ` F ) )
40		uztrn 11704	. . . . . . . . . 10 |- ( ( ( # ` S ) e. ( ZZ>= ` T ) /\ T e. ( ZZ>= ` F ) ) -> ( # ` S ) e. ( ZZ>= ` F ) )
41	37, 39, 40	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( # ` S ) e. ( ZZ>= ` F ) )
42		elfzuzb 12336	. . . . . . . . 9 |- ( F e. ( 0 ... ( # ` S ) ) <-> ( F e. ( ZZ>= ` 0 ) /\ ( # ` S ) e. ( ZZ>= ` F ) ) )
43	33, 41, 42	sylanbrc 698	. . . . . . . 8 |- ( ph -> F e. ( 0 ... ( # ` S ) ) )
44		swrdlen 13423	. . . . . . . 8 |- ( ( S e. Word A /\ 0 e. ( 0 ... F ) /\ F e. ( 0 ... ( # ` S ) ) ) -> ( # ` ( S substr <. 0 , F >. ) ) = ( F - 0 ) )
45	1, 35, 43, 44	syl3anc 1326	. . . . . . 7 |- ( ph -> ( # ` ( S substr <. 0 , F >. ) ) = ( F - 0 ) )
46	21	subid1d 10381	. . . . . . 7 |- ( ph -> ( F - 0 ) = F )
47	45, 46	eqtrd 2656	. . . . . 6 |- ( ph -> ( # ` ( S substr <. 0 , F >. ) ) = F )
48	47	oveq1d 6665	. . . . 5 |- ( ph -> ( ( # ` ( S substr <. 0 , F >. ) ) + ( # ` R ) ) = ( F + ( # ` R ) ) )
49	21, 18	addcomd 10238	. . . . 5 |- ( ph -> ( F + ( # ` R ) ) = ( ( # ` R ) + F ) )
50	31, 48, 49	3eqtrd 2660	. . . 4 |- ( ph -> ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) = ( ( # ` R ) + F ) )
51		elfzuz2 12346	. . . . . . 7 |- ( T e. ( 0 ... ( # ` S ) ) -> ( # ` S ) e. ( ZZ>= ` 0 ) )
52	3, 51	syl 17	. . . . . 6 |- ( ph -> ( # ` S ) e. ( ZZ>= ` 0 ) )
53		eluzfz2 12349	. . . . . 6 |- ( ( # ` S ) e. ( ZZ>= ` 0 ) -> ( # ` S ) e. ( 0 ... ( # ` S ) ) )
54	52, 53	syl 17	. . . . 5 |- ( ph -> ( # ` S ) e. ( 0 ... ( # ` S ) ) )
55		swrdlen 13423	. . . . 5 |- ( ( S e. Word A /\ T e. ( 0 ... ( # ` S ) ) /\ ( # ` S ) e. ( 0 ... ( # ` S ) ) ) -> ( # ` ( S substr <. T , ( # ` S ) >. ) ) = ( ( # ` S ) - T ) )
56	1, 3, 54, 55	syl3anc 1326	. . . 4 |- ( ph -> ( # ` ( S substr <. T , ( # ` S ) >. ) ) = ( ( # ` S ) - T ) )
57	50, 56	oveq12d 6668	. . 3 |- ( ph -> ( ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) + ( # ` ( S substr <. T , ( # ` S ) >. ) ) ) = ( ( ( # ` R ) + F ) + ( ( # ` S ) - T ) ) )
58	18, 28, 21	subsub3d 10422	. . . 4 |- ( ph -> ( ( # ` R ) - ( T - F ) ) = ( ( ( # ` R ) + F ) - T ) )
59	58	oveq2d 6666	. . 3 |- ( ph -> ( ( # ` S ) + ( ( # ` R ) - ( T - F ) ) ) = ( ( # ` S ) + ( ( ( # ` R ) + F ) - T ) ) )
60	29, 57, 59	3eqtr4d 2666	. 2 |- ( ph -> ( ( # ` ( ( S substr <. 0 , F >. ) ++ R ) ) + ( # ` ( S substr <. T , ( # ` S ) >. ) ) ) = ( ( # ` S ) + ( ( # ` R ) - ( T - F ) ) ) )
61	7, 15, 60	3eqtrd 2660	1 |- ( ph -> ( # ` ( S splice <. F , T , R >. ) ) = ( ( # ` S ) + ( ( # ` R ) - ( T - F ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   <.cotp 4185   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   - cmin 10266   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   #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  efgtlen  18139