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Theorem pfxccat1 11349
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 11236 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2 lencl 11183 . . . . . 6 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3 lencl 11183 . . . . . 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 10416 . . . . . . 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 10417 . . . . 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 11238 . . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
1110oveq2d 6044 . . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... (♯ ` ( S ++ T ) ) ) = ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
129, 11eleqtrrd 2311 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) )
13 pfxres 11328 . . 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 11244 . . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
162nn0zd 9661 . . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. ZZ )
1716uzidd 9832 . . . . . 6 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
18 uzaddcl 9881 . . . . . 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 10471 . . . . 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 5453 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) Fn ( 0..^ (♯ ` S ) ) )
23 wrdfn 11194 . . . 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 5672 . . . . 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 11240 . . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` k ) = ( S ` k ) )
28273expa 1230 . . . 4 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` k ) = ( S ` k ) )
2926, 28eqtrd 2264 . . 3 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) ` k ) = ( S ` k ) )
3022, 24, 29eqfnfvd 5756 . 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) = S )
3114, 30eqtrd 2264 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 1398   e. wcel 2202   C_ wss 3201   |` cres 4733   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   0cc0 8092   + caddc 8095   NN0cn0 9461   ZZ>=cuz 9816   ...cfz 10305  ..^cfzo 10439  ♯chash 11100  Word cword 11179   ++ cconcat 11233   prefix cpfx 11319
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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208
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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306  df-fzo 10440  df-ihash 11101  df-word 11180  df-concat 11234  df-substr 11293  df-pfx 11320
This theorem is referenced by:  ccatopth  11363  reuccatpfxs1  11394
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