Theorem swrdccat2 13458

Description: Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)

Assertion

Ref	Expression
swrdccat2	|- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) = T )

Proof of Theorem swrdccat2

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatcl 13359	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2		swrdcl 13419	. . . 4 |- ( ( S ++ T ) e. Word B -> ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) e. Word B )
3		wrdf 13310	. . . 4 |- ( ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) e. Word B -> ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) : ( 0..^ ( # ` ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ) ) --> B )
4		ffn 6045	. . . 4 |- ( ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) : ( 0..^ ( # ` ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ) ) --> B -> ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) Fn ( 0..^ ( # ` ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ) ) )
5	1, 2, 3, 4	4syl 19	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) Fn ( 0..^ ( # ` ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ) ) )
6		lencl 13324	. . . . . . . . . 10 |- ( S e. Word B -> ( # ` S ) e. NN0 )
7	6	adantr 481	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` S ) e. NN0 )
8		nn0uz 11722	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
9	7, 8	syl6eleq 2711	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` S ) e. ( ZZ>= ` 0 ) )
10	7	nn0zd 11480	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` S ) e. ZZ )
11		uzid 11702	. . . . . . . . . 10 |- ( ( # ` S ) e. ZZ -> ( # ` S ) e. ( ZZ>= ` ( # ` S ) ) )
12	10, 11	syl 17	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` S ) e. ( ZZ>= ` ( # ` S ) ) )
13		lencl 13324	. . . . . . . . . 10 |- ( T e. Word B -> ( # ` T ) e. NN0 )
14	13	adantl 482	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` T ) e. NN0 )
15		uzaddcl 11744	. . . . . . . . 9 |- ( ( ( # ` S ) e. ( ZZ>= ` ( # ` S ) ) /\ ( # ` T ) e. NN0 ) -> ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` ( # ` S ) ) )
16	12, 14, 15	syl2anc 693	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` ( # ` S ) ) )
17		elfzuzb 12336	. . . . . . . 8 |- ( ( # ` S ) e. ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) <-> ( ( # ` S ) e. ( ZZ>= ` 0 ) /\ ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` ( # ` S ) ) ) )
18	9, 16, 17	sylanbrc 698	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` S ) e. ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) )
19	7, 14	nn0addcld 11355	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( # ` S ) + ( # ` T ) ) e. NN0 )
20	19, 8	syl6eleq 2711	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` 0 ) )
21	19	nn0zd 11480	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( # ` S ) + ( # ` T ) ) e. ZZ )
22		uzid 11702	. . . . . . . . . 10 |- ( ( ( # ` S ) + ( # ` T ) ) e. ZZ -> ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` ( ( # ` S ) + ( # ` T ) ) ) )
23	21, 22	syl 17	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` ( ( # ` S ) + ( # ` T ) ) ) )
24		elfzuzb 12336	. . . . . . . . 9 |- ( ( ( # ` S ) + ( # ` T ) ) e. ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) <-> ( ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` 0 ) /\ ( ( # ` S ) + ( # ` T ) ) e. ( ZZ>= ` ( ( # ` S ) + ( # ` T ) ) ) ) )
25	20, 23, 24	sylanbrc 698	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( # ` S ) + ( # ` T ) ) e. ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) )
26		ccatlen 13360	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` ( S ++ T ) ) = ( ( # ` S ) + ( # ` T ) ) )
27	26	oveq2d 6666	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... ( # ` ( S ++ T ) ) ) = ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) )
28	25, 27	eleqtrrd 2704	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( # ` S ) + ( # ` T ) ) e. ( 0 ... ( # ` ( S ++ T ) ) ) )
29		swrdlen 13423	. . . . . . 7 |- ( ( ( S ++ T ) e. Word B /\ ( # ` S ) e. ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) /\ ( ( # ` S ) + ( # ` T ) ) e. ( 0 ... ( # ` ( S ++ T ) ) ) ) -> ( # ` ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ) = ( ( ( # ` S ) + ( # ` T ) ) - ( # ` S ) ) )
30	1, 18, 28, 29	syl3anc 1326	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ) = ( ( ( # ` S ) + ( # ` T ) ) - ( # ` S ) ) )
31	7	nn0cnd 11353	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` S ) e. CC )
32	14	nn0cnd 11353	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` T ) e. CC )
33	31, 32	pncan2d 10394	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( ( # ` S ) + ( # ` T ) ) - ( # ` S ) ) = ( # ` T ) )
34	30, 33	eqtrd 2656	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( # ` ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ) = ( # ` T ) )
35	34	oveq2d 6666	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ ( # ` ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ) ) = ( 0..^ ( # ` T ) ) )
36	35	fneq2d 5982	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) Fn ( 0..^ ( # ` ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ) ) <-> ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) Fn ( 0..^ ( # ` T ) ) ) )
37	5, 36	mpbid 222	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) Fn ( 0..^ ( # ` T ) ) )
38		wrdf 13310	. . . 4 |- ( T e. Word B -> T : ( 0..^ ( # ` T ) ) --> B )
39	38	adantl 482	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> T : ( 0..^ ( # ` T ) ) --> B )
40		ffn 6045	. . 3 |- ( T : ( 0..^ ( # ` T ) ) --> B -> T Fn ( 0..^ ( # ` T ) ) )
41	39, 40	syl 17	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> T Fn ( 0..^ ( # ` T ) ) )
42	1	adantr 481	. . . 4 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ ( # ` T ) ) ) -> ( S ++ T ) e. Word B )
43	18	adantr 481	. . . 4 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ ( # ` T ) ) ) -> ( # ` S ) e. ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) )
44	28	adantr 481	. . . 4 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ ( # ` T ) ) ) -> ( ( # ` S ) + ( # ` T ) ) e. ( 0 ... ( # ` ( S ++ T ) ) ) )
45	33	oveq2d 6666	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ ( ( ( # ` S ) + ( # ` T ) ) - ( # ` S ) ) ) = ( 0..^ ( # ` T ) ) )
46	45	eleq2d 2687	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( k e. ( 0..^ ( ( ( # ` S ) + ( # ` T ) ) - ( # ` S ) ) ) <-> k e. ( 0..^ ( # ` T ) ) ) )
47	46	biimpar 502	. . . 4 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ ( # ` T ) ) ) -> k e. ( 0..^ ( ( ( # ` S ) + ( # ` T ) ) - ( # ` S ) ) ) )
48		swrdfv 13424	. . . 4 |- ( ( ( ( S ++ T ) e. Word B /\ ( # ` S ) e. ( 0 ... ( ( # ` S ) + ( # ` T ) ) ) /\ ( ( # ` S ) + ( # ` T ) ) e. ( 0 ... ( # ` ( S ++ T ) ) ) ) /\ k e. ( 0..^ ( ( ( # ` S ) + ( # ` T ) ) - ( # ` S ) ) ) ) -> ( ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ` k ) = ( ( S ++ T ) ` ( k + ( # ` S ) ) ) )
49	42, 43, 44, 47, 48	syl31anc 1329	. . 3 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ ( # ` T ) ) ) -> ( ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ` k ) = ( ( S ++ T ) ` ( k + ( # ` S ) ) ) )
50		ccatval3 13363	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ k e. ( 0..^ ( # ` T ) ) ) -> ( ( S ++ T ) ` ( k + ( # ` S ) ) ) = ( T ` k ) )
51	50	3expa 1265	. . 3 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ ( # ` T ) ) ) -> ( ( S ++ T ) ` ( k + ( # ` S ) ) ) = ( T ` k ) )
52	49, 51	eqtrd 2656	. 2 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ ( # ` T ) ) ) -> ( ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) ` k ) = ( T ` k ) )
53	37, 41, 52	eqfnfvd 6314	1 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) = T )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   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

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-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

This theorem is referenced by:  ccatopth  13470