Theorem splcl 13503

Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)

Assertion

Ref	Expression
splcl	|- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) e. Word A )

Proof of Theorem splcl

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

Step	Hyp	Ref	Expression
1		elex 3212	. . . 4 |- ( S e. Word A -> S e. _V )
2		otex 4933	. . . 4 |- <. F , T , R >. e. _V
3		id 22	. . . . . . . 8 |- ( s = S -> s = S )
4		fveq2 6191	. . . . . . . . . 10 |- ( b = <. F , T , R >. -> ( 1st ` b ) = ( 1st ` <. F , T , R >. ) )
5	4	fveq2d 6195	. . . . . . . . 9 |- ( b = <. F , T , R >. -> ( 1st ` ( 1st ` b ) ) = ( 1st ` ( 1st ` <. F , T , R >. ) ) )
6	5	opeq2d 4409	. . . . . . . 8 |- ( b = <. F , T , R >. -> <. 0 , ( 1st ` ( 1st ` b ) ) >. = <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. )
7	3, 6	oveqan12d 6669	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) = ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) )
8		simpr 477	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> b = <. F , T , R >. )
9	8	fveq2d 6195	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 2nd ` b ) = ( 2nd ` <. F , T , R >. ) )
10	7, 9	oveq12d 6668	. . . . . 6 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) ++ ( 2nd ` b ) ) = ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) )
11		simpl 473	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> s = S )
12	8	fveq2d 6195	. . . . . . . . 9 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 1st ` b ) = ( 1st ` <. F , T , R >. ) )
13	12	fveq2d 6195	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 2nd ` ( 1st ` b ) ) = ( 2nd ` ( 1st ` <. F , T , R >. ) ) )
14	11	fveq2d 6195	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( # ` s ) = ( # ` S ) )
15	13, 14	opeq12d 4410	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. = <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. )
16	11, 15	oveq12d 6668	. . . . . 6 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) = ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) )
17	10, 16	oveq12d 6668	. . . . 5 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) ++ ( 2nd ` b ) ) ++ ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) ++ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
18		df-splice 13304	. . . . 5 |- splice = ( s e. _V , b e. _V |-> ( ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) ++ ( 2nd ` b ) ) ++ ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) ) )
19		ovex 6678	. . . . 5 |- ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) ++ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. _V
20	17, 18, 19	ovmpt2a 6791	. . . 4 |- ( ( S e. _V /\ <. F , T , R >. e. _V ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) ++ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
21	1, 2, 20	sylancl 694	. . 3 |- ( S e. Word A -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) ++ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
22	21	adantr 481	. 2 |- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) ++ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
23		swrdcl 13419	. . . . 5 |- ( S e. Word A -> ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A )
24	23	adantr 481	. . . 4 |- ( ( S e. Word A /\ R e. Word A ) -> ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A )
25		ot3rdg 7184	. . . . . 6 |- ( R e. Word A -> ( 2nd ` <. F , T , R >. ) = R )
26	25	adantl 482	. . . . 5 |- ( ( S e. Word A /\ R e. Word A ) -> ( 2nd ` <. F , T , R >. ) = R )
27		simpr 477	. . . . 5 |- ( ( S e. Word A /\ R e. Word A ) -> R e. Word A )
28	26, 27	eqeltrd 2701	. . . 4 |- ( ( S e. Word A /\ R e. Word A ) -> ( 2nd ` <. F , T , R >. ) e. Word A )
29		ccatcl 13359	. . . 4 |- ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A /\ ( 2nd ` <. F , T , R >. ) e. Word A ) -> ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) e. Word A )
30	24, 28, 29	syl2anc 693	. . 3 |- ( ( S e. Word A /\ R e. Word A ) -> ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) e. Word A )
31		swrdcl 13419	. . . 4 |- ( S e. Word A -> ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A )
32	31	adantr 481	. . 3 |- ( ( S e. Word A /\ R e. Word A ) -> ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A )
33		ccatcl 13359	. . 3 |- ( ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) e. Word A /\ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A ) -> ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) ++ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. Word A )
34	30, 32, 33	syl2anc 693	. 2 |- ( ( S e. Word A /\ R e. Word A ) -> ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) ++ ( 2nd ` <. F , T , R >. ) ) ++ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. Word A )
35	22, 34	eqeltrd 2701	1 |- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) e. Word A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   <.cotp 4185   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   #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  efglem  18129  efgtf  18135  frgpuplem  18185