Theorem splval2 13508

Description: Value of a splice, assuming the input word S has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.)

Hypotheses

Ref	Expression
splval2.a	|- ( ph -> A e. Word X )
splval2.b	|- ( ph -> B e. Word X )
splval2.c	|- ( ph -> C e. Word X )
splval2.r	|- ( ph -> R e. Word X )
splval2.s	|- ( ph -> S = ( ( A ++ B ) ++ C ) )
splval2.f	|- ( ph -> F = ( # ` A ) )
splval2.t	|- ( ph -> T = ( F + ( # ` B ) ) )

Assertion

Ref	Expression
splval2	|- ( ph -> ( S splice <. F , T , R >. ) = ( ( A ++ R ) ++ C ) )

Proof of Theorem splval2

Step	Hyp	Ref	Expression
1		splval2.s	. . . 4 |- ( ph -> S = ( ( A ++ B ) ++ C ) )
2		splval2.a	. . . . . 6 |- ( ph -> A e. Word X )
3		splval2.b	. . . . . 6 |- ( ph -> B e. Word X )
4		ccatcl 13359	. . . . . 6 |- ( ( A e. Word X /\ B e. Word X ) -> ( A ++ B ) e. Word X )
5	2, 3, 4	syl2anc 693	. . . . 5 |- ( ph -> ( A ++ B ) e. Word X )
6		splval2.c	. . . . 5 |- ( ph -> C e. Word X )
7		ccatcl 13359	. . . . 5 |- ( ( ( A ++ B ) e. Word X /\ C e. Word X ) -> ( ( A ++ B ) ++ C ) e. Word X )
8	5, 6, 7	syl2anc 693	. . . 4 |- ( ph -> ( ( A ++ B ) ++ C ) e. Word X )
9	1, 8	eqeltrd 2701	. . 3 |- ( ph -> S e. Word X )
10		splval2.f	. . . 4 |- ( ph -> F = ( # ` A ) )
11		lencl 13324	. . . . 5 |- ( A e. Word X -> ( # ` A ) e. NN0 )
12	2, 11	syl 17	. . . 4 |- ( ph -> ( # ` A ) e. NN0 )
13	10, 12	eqeltrd 2701	. . 3 |- ( ph -> F e. NN0 )
14		splval2.t	. . . 4 |- ( ph -> T = ( F + ( # ` B ) ) )
15		lencl 13324	. . . . . 6 |- ( B e. Word X -> ( # ` B ) e. NN0 )
16	3, 15	syl 17	. . . . 5 |- ( ph -> ( # ` B ) e. NN0 )
17	13, 16	nn0addcld 11355	. . . 4 |- ( ph -> ( F + ( # ` B ) ) e. NN0 )
18	14, 17	eqeltrd 2701	. . 3 |- ( ph -> T e. NN0 )
19		splval2.r	. . 3 |- ( ph -> R e. Word X )
20		splval 13502	. . 3 |- ( ( S e. Word X /\ ( F e. NN0 /\ T e. NN0 /\ R e. Word X ) ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) )
21	9, 13, 18, 19, 20	syl13anc 1328	. 2 |- ( ph -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) )
22		nn0uz 11722	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
23	13, 22	syl6eleq 2711	. . . . . . . . 9 |- ( ph -> F e. ( ZZ>= ` 0 ) )
24		eluzfz1 12348	. . . . . . . . 9 |- ( F e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... F ) )
25	23, 24	syl 17	. . . . . . . 8 |- ( ph -> 0 e. ( 0 ... F ) )
26	13	nn0zd 11480	. . . . . . . . . . . 12 |- ( ph -> F e. ZZ )
27		uzid 11702	. . . . . . . . . . . 12 |- ( F e. ZZ -> F e. ( ZZ>= ` F ) )
28	26, 27	syl 17	. . . . . . . . . . 11 |- ( ph -> F e. ( ZZ>= ` F ) )
29		uzaddcl 11744	. . . . . . . . . . 11 |- ( ( F e. ( ZZ>= ` F ) /\ ( # ` B ) e. NN0 ) -> ( F + ( # ` B ) ) e. ( ZZ>= ` F ) )
30	28, 16, 29	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( F + ( # ` B ) ) e. ( ZZ>= ` F ) )
31	14, 30	eqeltrd 2701	. . . . . . . . 9 |- ( ph -> T e. ( ZZ>= ` F ) )
32		elfzuzb 12336	. . . . . . . . 9 |- ( F e. ( 0 ... T ) <-> ( F e. ( ZZ>= ` 0 ) /\ T e. ( ZZ>= ` F ) ) )
33	23, 31, 32	sylanbrc 698	. . . . . . . 8 |- ( ph -> F e. ( 0 ... T ) )
34	18, 22	syl6eleq 2711	. . . . . . . . 9 |- ( ph -> T e. ( ZZ>= ` 0 ) )
35		ccatlen 13360	. . . . . . . . . . . 12 |- ( ( ( A ++ B ) e. Word X /\ C e. Word X ) -> ( # ` ( ( A ++ B ) ++ C ) ) = ( ( # ` ( A ++ B ) ) + ( # ` C ) ) )
36	5, 6, 35	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( # ` ( ( A ++ B ) ++ C ) ) = ( ( # ` ( A ++ B ) ) + ( # ` C ) ) )
37	1	fveq2d 6195	. . . . . . . . . . 11 |- ( ph -> ( # ` S ) = ( # ` ( ( A ++ B ) ++ C ) ) )
38	10	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ph -> ( F + ( # ` B ) ) = ( ( # ` A ) + ( # ` B ) ) )
39		ccatlen 13360	. . . . . . . . . . . . . 14 |- ( ( A e. Word X /\ B e. Word X ) -> ( # ` ( A ++ B ) ) = ( ( # ` A ) + ( # ` B ) ) )
40	2, 3, 39	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> ( # ` ( A ++ B ) ) = ( ( # ` A ) + ( # ` B ) ) )
41	38, 14, 40	3eqtr4d 2666	. . . . . . . . . . . 12 |- ( ph -> T = ( # ` ( A ++ B ) ) )
42	41	oveq1d 6665	. . . . . . . . . . 11 |- ( ph -> ( T + ( # ` C ) ) = ( ( # ` ( A ++ B ) ) + ( # ` C ) ) )
43	36, 37, 42	3eqtr4d 2666	. . . . . . . . . 10 |- ( ph -> ( # ` S ) = ( T + ( # ` C ) ) )
44	18	nn0zd 11480	. . . . . . . . . . . 12 |- ( ph -> T e. ZZ )
45		uzid 11702	. . . . . . . . . . . 12 |- ( T e. ZZ -> T e. ( ZZ>= ` T ) )
46	44, 45	syl 17	. . . . . . . . . . 11 |- ( ph -> T e. ( ZZ>= ` T ) )
47		lencl 13324	. . . . . . . . . . . 12 |- ( C e. Word X -> ( # ` C ) e. NN0 )
48	6, 47	syl 17	. . . . . . . . . . 11 |- ( ph -> ( # ` C ) e. NN0 )
49		uzaddcl 11744	. . . . . . . . . . 11 |- ( ( T e. ( ZZ>= ` T ) /\ ( # ` C ) e. NN0 ) -> ( T + ( # ` C ) ) e. ( ZZ>= ` T ) )
50	46, 48, 49	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( T + ( # ` C ) ) e. ( ZZ>= ` T ) )
51	43, 50	eqeltrd 2701	. . . . . . . . 9 |- ( ph -> ( # ` S ) e. ( ZZ>= ` T ) )
52		elfzuzb 12336	. . . . . . . . 9 |- ( T e. ( 0 ... ( # ` S ) ) <-> ( T e. ( ZZ>= ` 0 ) /\ ( # ` S ) e. ( ZZ>= ` T ) ) )
53	34, 51, 52	sylanbrc 698	. . . . . . . 8 |- ( ph -> T e. ( 0 ... ( # ` S ) ) )
54		ccatswrd 13456	. . . . . . . 8 |- ( ( S e. Word X /\ ( 0 e. ( 0 ... F ) /\ F e. ( 0 ... T ) /\ T e. ( 0 ... ( # ` S ) ) ) ) -> ( ( S substr <. 0 , F >. ) ++ ( S substr <. F , T >. ) ) = ( S substr <. 0 , T >. ) )
55	9, 25, 33, 53, 54	syl13anc 1328	. . . . . . 7 |- ( ph -> ( ( S substr <. 0 , F >. ) ++ ( S substr <. F , T >. ) ) = ( S substr <. 0 , T >. ) )
56		eluzfz1 12348	. . . . . . . . . . . 12 |- ( T e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... T ) )
57	34, 56	syl 17	. . . . . . . . . . 11 |- ( ph -> 0 e. ( 0 ... T ) )
58		lencl 13324	. . . . . . . . . . . . . 14 |- ( S e. Word X -> ( # ` S ) e. NN0 )
59	9, 58	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( # ` S ) e. NN0 )
60	59, 22	syl6eleq 2711	. . . . . . . . . . . 12 |- ( ph -> ( # ` S ) e. ( ZZ>= ` 0 ) )
61		eluzfz2 12349	. . . . . . . . . . . 12 |- ( ( # ` S ) e. ( ZZ>= ` 0 ) -> ( # ` S ) e. ( 0 ... ( # ` S ) ) )
62	60, 61	syl 17	. . . . . . . . . . 11 |- ( ph -> ( # ` S ) e. ( 0 ... ( # ` S ) ) )
63		ccatswrd 13456	. . . . . . . . . . 11 |- ( ( S e. Word X /\ ( 0 e. ( 0 ... T ) /\ T e. ( 0 ... ( # ` S ) ) /\ ( # ` S ) e. ( 0 ... ( # ` S ) ) ) ) -> ( ( S substr <. 0 , T >. ) ++ ( S substr <. T , ( # ` S ) >. ) ) = ( S substr <. 0 , ( # ` S ) >. ) )
64	9, 57, 53, 62, 63	syl13anc 1328	. . . . . . . . . 10 |- ( ph -> ( ( S substr <. 0 , T >. ) ++ ( S substr <. T , ( # ` S ) >. ) ) = ( S substr <. 0 , ( # ` S ) >. ) )
65		swrdid 13428	. . . . . . . . . . 11 |- ( S e. Word X -> ( S substr <. 0 , ( # ` S ) >. ) = S )
66	9, 65	syl 17	. . . . . . . . . 10 |- ( ph -> ( S substr <. 0 , ( # ` S ) >. ) = S )
67	64, 66, 1	3eqtrd 2660	. . . . . . . . 9 |- ( ph -> ( ( S substr <. 0 , T >. ) ++ ( S substr <. T , ( # ` S ) >. ) ) = ( ( A ++ B ) ++ C ) )
68		swrdcl 13419	. . . . . . . . . . 11 |- ( S e. Word X -> ( S substr <. 0 , T >. ) e. Word X )
69	9, 68	syl 17	. . . . . . . . . 10 |- ( ph -> ( S substr <. 0 , T >. ) e. Word X )
70		swrdcl 13419	. . . . . . . . . . 11 |- ( S e. Word X -> ( S substr <. T , ( # ` S ) >. ) e. Word X )
71	9, 70	syl 17	. . . . . . . . . 10 |- ( ph -> ( S substr <. T , ( # ` S ) >. ) e. Word X )
72		swrd0len 13422	. . . . . . . . . . . 12 |- ( ( S e. Word X /\ T e. ( 0 ... ( # ` S ) ) ) -> ( # ` ( S substr <. 0 , T >. ) ) = T )
73	9, 53, 72	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( # ` ( S substr <. 0 , T >. ) ) = T )
74	73, 41	eqtrd 2656	. . . . . . . . . 10 |- ( ph -> ( # ` ( S substr <. 0 , T >. ) ) = ( # ` ( A ++ B ) ) )
75		ccatopth 13470	. . . . . . . . . 10 |- ( ( ( ( S substr <. 0 , T >. ) e. Word X /\ ( S substr <. T , ( # ` S ) >. ) e. Word X ) /\ ( ( A ++ B ) e. Word X /\ C e. Word X ) /\ ( # ` ( S substr <. 0 , T >. ) ) = ( # ` ( A ++ B ) ) ) -> ( ( ( S substr <. 0 , T >. ) ++ ( S substr <. T , ( # ` S ) >. ) ) = ( ( A ++ B ) ++ C ) <-> ( ( S substr <. 0 , T >. ) = ( A ++ B ) /\ ( S substr <. T , ( # ` S ) >. ) = C ) ) )
76	69, 71, 5, 6, 74, 75	syl221anc 1337	. . . . . . . . 9 |- ( ph -> ( ( ( S substr <. 0 , T >. ) ++ ( S substr <. T , ( # ` S ) >. ) ) = ( ( A ++ B ) ++ C ) <-> ( ( S substr <. 0 , T >. ) = ( A ++ B ) /\ ( S substr <. T , ( # ` S ) >. ) = C ) ) )
77	67, 76	mpbid 222	. . . . . . . 8 |- ( ph -> ( ( S substr <. 0 , T >. ) = ( A ++ B ) /\ ( S substr <. T , ( # ` S ) >. ) = C ) )
78	77	simpld 475	. . . . . . 7 |- ( ph -> ( S substr <. 0 , T >. ) = ( A ++ B ) )
79	55, 78	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( S substr <. 0 , F >. ) ++ ( S substr <. F , T >. ) ) = ( A ++ B ) )
80		swrdcl 13419	. . . . . . . 8 |- ( S e. Word X -> ( S substr <. 0 , F >. ) e. Word X )
81	9, 80	syl 17	. . . . . . 7 |- ( ph -> ( S substr <. 0 , F >. ) e. Word X )
82		swrdcl 13419	. . . . . . . 8 |- ( S e. Word X -> ( S substr <. F , T >. ) e. Word X )
83	9, 82	syl 17	. . . . . . 7 |- ( ph -> ( S substr <. F , T >. ) e. Word X )
84		uztrn 11704	. . . . . . . . . . 11 |- ( ( ( # ` S ) e. ( ZZ>= ` T ) /\ T e. ( ZZ>= ` F ) ) -> ( # ` S ) e. ( ZZ>= ` F ) )
85	51, 31, 84	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( # ` S ) e. ( ZZ>= ` F ) )
86		elfzuzb 12336	. . . . . . . . . 10 |- ( F e. ( 0 ... ( # ` S ) ) <-> ( F e. ( ZZ>= ` 0 ) /\ ( # ` S ) e. ( ZZ>= ` F ) ) )
87	23, 85, 86	sylanbrc 698	. . . . . . . . 9 |- ( ph -> F e. ( 0 ... ( # ` S ) ) )
88		swrd0len 13422	. . . . . . . . 9 |- ( ( S e. Word X /\ F e. ( 0 ... ( # ` S ) ) ) -> ( # ` ( S substr <. 0 , F >. ) ) = F )
89	9, 87, 88	syl2anc 693	. . . . . . . 8 |- ( ph -> ( # ` ( S substr <. 0 , F >. ) ) = F )
90	89, 10	eqtrd 2656	. . . . . . 7 |- ( ph -> ( # ` ( S substr <. 0 , F >. ) ) = ( # ` A ) )
91		ccatopth 13470	. . . . . . 7 |- ( ( ( ( S substr <. 0 , F >. ) e. Word X /\ ( S substr <. F , T >. ) e. Word X ) /\ ( A e. Word X /\ B e. Word X ) /\ ( # ` ( S substr <. 0 , F >. ) ) = ( # ` A ) ) -> ( ( ( S substr <. 0 , F >. ) ++ ( S substr <. F , T >. ) ) = ( A ++ B ) <-> ( ( S substr <. 0 , F >. ) = A /\ ( S substr <. F , T >. ) = B ) ) )
92	81, 83, 2, 3, 90, 91	syl221anc 1337	. . . . . 6 |- ( ph -> ( ( ( S substr <. 0 , F >. ) ++ ( S substr <. F , T >. ) ) = ( A ++ B ) <-> ( ( S substr <. 0 , F >. ) = A /\ ( S substr <. F , T >. ) = B ) ) )
93	79, 92	mpbid 222	. . . . 5 |- ( ph -> ( ( S substr <. 0 , F >. ) = A /\ ( S substr <. F , T >. ) = B ) )
94	93	simpld 475	. . . 4 |- ( ph -> ( S substr <. 0 , F >. ) = A )
95	94	oveq1d 6665	. . 3 |- ( ph -> ( ( S substr <. 0 , F >. ) ++ R ) = ( A ++ R ) )
96	77	simprd 479	. . 3 |- ( ph -> ( S substr <. T , ( # ` S ) >. ) = C )
97	95, 96	oveq12d 6668	. 2 |- ( ph -> ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) = ( ( A ++ R ) ++ C ) )
98	21, 97	eqtrd 2656	1 |- ( ph -> ( S splice <. F , T , R >. ) = ( ( A ++ R ) ++ C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   <.cotp 4185   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   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:  efginvrel2  18140  efgredleme  18156  efgcpbllemb  18168  frgpnabllem1  18276