Theorem swrdccat2 11124

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 11049	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2		lencl 10998	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3	2	nn0zd 9493	. . . . . 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 10998	. . . . . . . 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 9352	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. NN0 )
9	8	nn0zd 9493	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
10		swrdclg 11103	. . . . 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 11009	. . . 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 9683	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
15	2, 14	eleqtrdi 2298	. . . . . . . . 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 9663	. . . . . . . . 9 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
18		uzaddcl 9707	. . . . . . . . 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 10141	. . . . . . . 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 9330	. . . . . . . . . . 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 2298	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` 0 ) )
25	23	nn0zd 9493	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
26	25	uzidd 9663	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) )
27		elfzuzb 10141	. . . . . . . . 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 11051	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
30	29	oveq2d 5960	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... (♯ ` ( S ++ T ) ) ) = ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
31	28, 30	eleqtrrd 2285	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) )
32		swrdlen 11105	. . . . . . 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 9350	. . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. CC )
35	6	nn0cnd 9350	. . . . . . 7 |- ( T e. Word B -> (♯ ` T ) e. CC )
36		pncan2 8279	. . . . . . 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 2238	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) = (♯ ` T ) )
39	38	oveq2d 5960	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ (♯ ` ( ( S ++ T ) substr <. (♯ ` S ) , ( (♯ ` S ) + (♯ ` T ) ) >. ) ) ) = ( 0..^ (♯ ` T ) ) )
40	39	fneq2d 5365	. . 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 11009	. . 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 5960	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) - (♯ ` S ) ) ) = ( 0..^ (♯ ` T ) ) )
46	45	eleq2d 2275	. . . . 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 11106	. . . 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 11055	. . . 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 2238	. 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 5680	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 2176   <.cop 3636   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   CCcc 7923   0cc0 7925   + caddc 7928   - cmin 8243   NN0cn0 9295   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130  ..^cfzo 10264  ♯chash 10920  Word cword 10994   ++ cconcat 11046   substr csubstr 11098

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131  df-fzo 10265  df-ihash 10921  df-word 10995  df-concat 11047  df-substr 11099

This theorem is referenced by: (None)