Theorem swrdccat2 11157

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 11082	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2		lencl 11030	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3	2	nn0zd 9523	. . . . . 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 11030	. . . . . . . 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 9382	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. NN0 )
9	8	nn0zd 9523	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
10		swrdclg 11136	. . . . 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 1250	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) e. Word B )
12		wrdfn 11041	. . . 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 9713	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
15	2, 14	eleqtrdi 2299	. . . . . . . . 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 9693	. . . . . . . . 9 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
18		uzaddcl 9737	. . . . . . . . 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 10171	. . . . . . . 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 9360	. . . . . . . . . . 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 2299	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` 0 ) )
25	23	nn0zd 9523	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
26	25	uzidd 9693	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) )
27		elfzuzb 10171	. . . . . . . . 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 11084	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
30	29	oveq2d 5978	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... (♯ ` ( S ++ T ) ) ) = ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
31	28, 30	eleqtrrd 2286	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) )
32		swrdlen 11138	. . . . . . 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 1250	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) )
34	2	nn0cnd 9380	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. CC )
35	6	nn0cnd 9380	. . . . . . 7 |- ( T e. Word B -> (♯ ` T ) e. CC )
36		pncan2 8309	. . . . . . 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 2239	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) = (♯ ` T ) )
39	38	oveq2d 5978	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) ) = ( 0..^ (♯ ` T ) ) )
40	39	fneq2d 5379	. . 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 11041	. . 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 1180	. . . 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 5978	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) ) = ( 0..^ (♯ ` T ) ) )
46	45	eleq2d 2276	. . . . 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 11139	. . . 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 595	. . 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 11088	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ k e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( k + (♯ ` S ) ) ) = ( T ` k ) )
51	50	3expa 1206	. . 3 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( k + (♯ ` S ) ) ) = ( T ` k ) )
52	49, 51	eqtrd 2239	. 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 5698	1 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) = T )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2177   <.cop 3641   Fn wfn 5280   ` cfv 5285  (class class class)co 5962   CCcc 7953   0cc0 7955   + caddc 7958   - cmin 8273   NN0cn0 9325   ZZcz 9402   ZZ>=cuz 9678   ...cfz 10160  ..^cfzo 10294  ♯chash 10952  Word cword 11026   ++ cconcat 11079   substr csubstr 11131

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-1o 6520  df-er 6638  df-en 6846  df-dom 6847  df-fin 6848  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-inn 9067  df-n0 9326  df-z 9403  df-uz 9679  df-fz 10161  df-fzo 10295  df-ihash 10953  df-word 11027  df-concat 11080  df-substr 11132

This theorem is referenced by:  ccatopth  11202