Theorem swrdccat2 11459

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 11377	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2		lencl 11324	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3	2	nn0zd 9771	. . . . . 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 11324	. . . . . . . 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 9629	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. NN0 )
9	8	nn0zd 9771	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
10		swrdclg 11438	. . . . 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 1278	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) e. Word B )
12		wrdfn 11335	. . . 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 9967	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
15	2, 14	eleqtrdi 2331	. . . . . . . . 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 9947	. . . . . . . . 9 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
18		uzaddcl 9996	. . . . . . . . 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 10433	. . . . . . . 8 |- ( (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) <-> ( (♯ ` S ) e. ( ZZ>= ` 0 ) /\ ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) ) )
21	16, 19, 20	sylanbrc 421	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
22		nn0addcl 9603	. . . . . . . . . . 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 2331	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` 0 ) )
25	23	nn0zd 9771	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
26	25	uzidd 9947	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) )
27		elfzuzb 10433	. . . . . . . . 9 |- ( ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) <-> ( ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` 0 ) /\ ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) ) )
28	24, 26, 27	sylanbrc 421	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
29		ccatlen 11379	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
30	29	oveq2d 6101	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... (♯ ` ( S ++ T ) ) ) = ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
31	28, 30	eleqtrrd 2318	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) )
32		swrdlen 11440	. . . . . . 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 1278	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) )
34	2	nn0cnd 9627	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. CC )
35	6	nn0cnd 9627	. . . . . . 7 |- ( T e. Word B -> (♯ ` T ) e. CC )
36		pncan2 8535	. . . . . . 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 2271	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) = (♯ ` T ) )
39	38	oveq2d 6101	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) ) = ( 0..^ (♯ ` T ) ) )
40	39	fneq2d 5472	. . 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 11335	. . 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 1208	. . . 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 6101	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) ) = ( 0..^ (♯ ` T ) ) )
46	45	eleq2d 2308	. . . . 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 11441	. . . 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 603	. . 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 11383	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ k e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( k + (♯ ` S ) ) ) = ( T ` k ) )
51	50	3expa 1234	. . 3 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( k + (♯ ` S ) ) ) = ( T ` k ) )
52	49, 51	eqtrd 2271	. 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 5809	1 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) = T )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   <.cop 3712   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   CCcc 8178   0cc0 8180   + caddc 8183   - cmin 8499   NN0cn0 9568   ZZcz 9649   ZZ>=cuz 9931   ...cfz 10422  ..^cfzo 10560  ♯chash 11230  Word cword 11320   ++ cconcat 11374   substr csubstr 11433

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-n0 9569  df-z 9650  df-uz 9932  df-fz 10423  df-fzo 10561  df-ihash 11231  df-word 11321  df-concat 11375  df-substr 11434

This theorem is used by:  ccatopth  11504