Theorem swrdccat2 11242

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 11160	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2		lencl 11107	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3	2	nn0zd 9590	. . . . . 6 |- ( S e. Word B -> (♯ ` S ) e. ZZ )
4	3	adantr 276	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ZZ )
5	2	adantr 276	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. NN0 )
6		lencl 11107	. . . . . . . 8 |- ( T e. Word B -> (♯ ` T ) e. NN0 )
7	6	adantl 277	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` T ) e. NN0 )
8	5, 7	nn0addcld 9449	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. NN0 )
9	8	nn0zd 9590	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
10		swrdclg 11221	. . . . 5 |- ( ( ( S ++ T ) e. Word B /\ (♯ ` S ) e. ZZ /\ ( (♯ ` S ) + (♯ ` T ) ) e. ZZ ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) e. Word B )
11	1, 4, 9, 10	syl3anc 1271	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) e. Word B )
12		wrdfn 11118	. . . 4 |- ( ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) e. Word B -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) Fn ( 0..^ (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) ) )
13	11, 12	syl 14	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) Fn ( 0..^ (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) ) )
14		nn0uz 9781	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
15	2, 14	eleqtrdi 2322	. . . . . . . . 9 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` 0 ) )
16	15	adantr 276	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( ZZ>= ` 0 ) )
17	3	uzidd 9761	. . . . . . . . 9 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
18		uzaddcl 9810	. . . . . . . . 9 |- ( ( (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) /\ (♯ ` T ) e. NN0 ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) )
19	17, 6, 18	syl2an 289	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) )
20		elfzuzb 10244	. . . . . . . 8 |- ( (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) <-> ( (♯ ` S ) e. ( ZZ>= ` 0 ) /\ ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) ) )
21	16, 19, 20	sylanbrc 417	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
22		nn0addcl 9427	. . . . . . . . . . 11 |- ( ( (♯ ` S ) e. NN0 /\ (♯ ` T ) e. NN0 ) -> ( (♯ ` S ) + (♯ ` T ) ) e. NN0 )
23	2, 6, 22	syl2an 289	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. NN0 )
24	23, 14	eleqtrdi 2322	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` 0 ) )
25	23	nn0zd 9590	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
26	25	uzidd 9761	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) )
27		elfzuzb 10244	. . . . . . . . 9 |- ( ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) <-> ( ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` 0 ) /\ ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) ) )
28	24, 26, 27	sylanbrc 417	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
29		ccatlen 11162	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
30	29	oveq2d 6029	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... (♯ ` ( S ++ T ) ) ) = ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
31	28, 30	eleqtrrd 2309	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) )
32		swrdlen 11223	. . . . . . 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 ) ) )
33	1, 21, 31, 32	syl3anc 1271	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) )
34	2	nn0cnd 9447	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. CC )
35	6	nn0cnd 9447	. . . . . . 7 |- ( T e. Word B -> (♯ ` T ) e. CC )
36		pncan2 8376	. . . . . . 7 |- ( ( (♯ ` S ) e. CC /\ (♯ ` T ) e. CC ) -> ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) = (♯ ` T ) )
37	34, 35, 36	syl2an 289	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) = (♯ ` T ) )
38	33, 37	eqtrd 2262	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) = (♯ ` T ) )
39	38	oveq2d 6029	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) ) = ( 0..^ (♯ ` T ) ) )
40	39	fneq2d 5418	. . 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 ) ) ) )
41	13, 40	mpbid 147	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) Fn ( 0..^ (♯ ` T ) ) )
42		wrdfn 11118	. . 3 |- ( T e. Word B -> T Fn ( 0..^ (♯ ` T ) ) )
43	42	adantl 277	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> T Fn ( 0..^ (♯ ` T ) ) )
44	1, 21, 31	3jca 1201	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) e. Word B /\ (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) /\ ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) ) )
45	37	oveq2d 6029	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) ) = ( 0..^ (♯ ` T ) ) )
46	45	eleq2d 2299	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( k e. ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) ) <-> k e. ( 0..^ (♯ ` T ) ) ) )
47	46	biimpar 297	. . . 4 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` T ) ) ) -> k e. ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) ) )
48		swrdfv 11224	. . . 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	44, 47, 48	syl2an2r 597	. . 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 11166	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ k e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( k + (♯ ` S ) ) ) = ( T ` k ) )
51	50	3expa 1227	. . 3 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( k + (♯ ` S ) ) ) = ( T ` k ) )
52	49, 51	eqtrd 2262	. 2 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` T ) ) ) -> ( ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ` k ) = ( T ` k ) )
53	41, 43, 52	eqfnfvd 5743	1 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) = T )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   <.cop 3670   Fn wfn 5319   ` cfv 5324  (class class class)co 6013   CCcc 8020   0cc0 8022   + caddc 8025   - cmin 8340   NN0cn0 9392   ZZcz 9469   ZZ>=cuz 9745   ...cfz 10233  ..^cfzo 10367  ♯chash 11027  Word cword 11103   ++ cconcat 11157   substr csubstr 11216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-fzo 10368  df-ihash 11028  df-word 11104  df-concat 11158  df-substr 11217

This theorem is referenced by:  ccatopth  11287