Theorem swrdccat2 11388

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 11306	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2		lencl 11253	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3	2	nn0zd 9716	. . . . . 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 11253	. . . . . . . 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 9574	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. NN0 )
9	8	nn0zd 9716	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
10		swrdclg 11367	. . . . 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 1274	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) e. Word B )
12		wrdfn 11264	. . . 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 9907	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
15	2, 14	eleqtrdi 2327	. . . . . . . . 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 9887	. . . . . . . . 9 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
18		uzaddcl 9936	. . . . . . . . 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 10372	. . . . . . . 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 9548	. . . . . . . . . . 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 2327	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` 0 ) )
25	23	nn0zd 9716	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
26	25	uzidd 9887	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) )
27		elfzuzb 10372	. . . . . . . . 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 11308	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
30	29	oveq2d 6074	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... (♯ ` ( S ++ T ) ) ) = ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
31	28, 30	eleqtrrd 2314	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) )
32		swrdlen 11369	. . . . . . 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 1274	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) )
34	2	nn0cnd 9572	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. CC )
35	6	nn0cnd 9572	. . . . . . 7 |- ( T e. Word B -> (♯ ` T ) e. CC )
36		pncan2 8496	. . . . . . 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 2267	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) = (♯ ` T ) )
39	38	oveq2d 6074	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) ) = ( 0..^ (♯ ` T ) ) )
40	39	fneq2d 5452	. . 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 11264	. . 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 1204	. . . 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 6074	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) ) = ( 0..^ (♯ ` T ) ) )
46	45	eleq2d 2304	. . . . 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 11370	. . . 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 599	. . 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 11312	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ k e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( k + (♯ ` S ) ) ) = ( T ` k ) )
51	50	3expa 1230	. . 3 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( k + (♯ ` S ) ) ) = ( T ` k ) )
52	49, 51	eqtrd 2267	. 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 5783	1 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) = T )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   <.cop 3697   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   + caddc 8146   - cmin 8460   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361  ..^cfzo 10498  ♯chash 11163  Word cword 11249   ++ cconcat 11303   substr csubstr 11362

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-concat 11304  df-substr 11363

This theorem is referenced by:  ccatopth  11433