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Theorem pfxccat1 11234
Description: Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by AV, 6-May-2020.)
Assertion
Ref Expression
pfxccat1 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) prefix (♯ ` S ) ) = S )
Proof of Theorem pfxccat1
Dummy variable k is distinct from all other variables.
StepHypRef Expression
1 ccatcl 11128 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2 lencl 11075 . . . . . 6 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3 lencl 11075 . . . . . 6 |- ( T e. Word B -> (♯ ` T ) e. NN0 )
42, 3anim12i 338 . . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) e. NN0 /\ (♯ ` T ) e. NN0 ) )
5 nn0fz0 10315 . . . . . . 7 |- ( (♯ ` S ) e. NN0 <-> (♯ ` S ) e. ( 0 ... (♯ ` S ) ) )
62, 5sylib 122 . . . . . 6 |- ( S e. Word B -> (♯ ` S ) e. ( 0 ... (♯ ` S ) ) )
76adantr 276 . . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( 0 ... (♯ ` S ) ) )
8 elfz0add 10316 . . . . 5 |- ( ( (♯ ` S ) e. NN0 /\ (♯ ` T ) e. NN0 ) -> ( (♯ ` S ) e. ( 0 ... (♯ ` S ) ) -> (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) ) )
94, 7, 8sylc 62 . . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
10 ccatlen 11130 . . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
1110oveq2d 6017 . . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... (♯ ` ( S ++ T ) ) ) = ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
129, 11eleqtrrd 2309 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) )
13 pfxres 11213 . . 3 |- ( ( ( S ++ T ) e. Word B /\ (♯ ` S ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) ) -> ( ( S ++ T ) prefix (♯ ` S ) ) = ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) )
141, 12, 13syl2anc 411 . 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) prefix (♯ ` S ) ) = ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) )
15 ccatvalfn 11136 . . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
162nn0zd 9567 . . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. ZZ )
1716uzidd 9737 . . . . . 6 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
18 uzaddcl 9781 . . . . . 6 |- ( ( (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) /\ (♯ ` T ) e. NN0 ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) )
1917, 3, 18syl2an 289 . . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) )
20 fzoss2 10370 . . . . 5 |- ( ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) -> ( 0..^ (♯ ` S ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
2119, 20syl 14 . . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ (♯ ` S ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
2215, 21fnssresd 5437 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) Fn ( 0..^ (♯ ` S ) ) )
23 wrdfn 11086 . . . 4 |- ( S e. Word B -> S Fn ( 0..^ (♯ ` S ) ) )
2423adantr 276 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> S Fn ( 0..^ (♯ ` S ) ) )
25 fvres 5651 . . . . 5 |- ( k e. ( 0..^ (♯ ` S ) ) -> ( ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) ` k ) = ( ( S ++ T ) ` k ) )
2625adantl 277 . . . 4 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) ` k ) = ( ( S ++ T ) ` k ) )
27 ccatval1 11132 . . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` k ) = ( S ` k ) )
28273expa 1227 . . . 4 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` k ) = ( S ` k ) )
2926, 28eqtrd 2262 . . 3 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) ` k ) = ( S ` k ) )
3022, 24, 29eqfnfvd 5735 . 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) = S )
3114, 30eqtrd 2262 1 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) prefix (♯ ` S ) ) = S )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   C_ wss 3197   |` cres 4721   Fn wfn 5313   ` cfv 5318  (class class class)co 6001   0cc0 7999   + caddc 8002   NN0cn0 9369   ZZ>=cuz 9722   ...cfz 10204  ..^cfzo 10338  ♯chash 10997  Word cword 11071   ++ cconcat 11125   prefix cpfx 11204
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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115
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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205  df-fzo 10339  df-ihash 10998  df-word 11072  df-concat 11126  df-substr 11178  df-pfx 11205
This theorem is referenced by:  ccatopth  11248  reuccatpfxs1  11279
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