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Theorem pfxccat1 11156
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 11052 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2 lencl 11000 . . . . . 6 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3 lencl 11000 . . . . . 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 10243 . . . . . . 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 10244 . . . . 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 11054 . . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
1110oveq2d 5962 . . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0 ... (♯ ` ( S ++ T ) ) ) = ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
129, 11eleqtrrd 2285 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( 0 ... (♯ ` ( S ++ T ) ) ) )
13 pfxres 11135 . . 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 11060 . . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
162nn0zd 9495 . . . . . . 7 |- ( S e. Word B -> (♯ ` S ) e. ZZ )
1716uzidd 9665 . . . . . 6 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
18 uzaddcl 9709 . . . . . 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 10298 . . . . 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 5391 . . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) Fn ( 0..^ (♯ ` S ) ) )
23 wrdfn 11011 . . . 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 5602 . . . . 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 11056 . . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` k ) = ( S ` k ) )
28273expa 1206 . . . 4 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` k ) = ( S ` k ) )
2926, 28eqtrd 2238 . . 3 |- ( ( ( S e. Word B /\ T e. Word B ) /\ k e. ( 0..^ (♯ ` S ) ) ) -> ( ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) ` k ) = ( S ` k ) )
3022, 24, 29eqfnfvd 5682 . 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) |` ( 0..^ (♯ ` S ) ) ) = S )
3114, 30eqtrd 2238 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 1373   e. wcel 2176   C_ wss 3166   |` cres 4678   Fn wfn 5267   ` cfv 5272  (class class class)co 5946   0cc0 7927   + caddc 7930   NN0cn0 9297   ZZ>=cuz 9650   ...cfz 10132  ..^cfzo 10266  ♯chash 10922  Word cword 10996   ++ cconcat 11049   prefix cpfx 11128
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 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043
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 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-fz 10133  df-fzo 10267  df-ihash 10923  df-word 10997  df-concat 11050  df-substr 11102  df-pfx 11129
This theorem is referenced by: (None)
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