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Theorem ccats1swrdeq 13469
Description: The last symbol of a word concatenated with the subword of the word having length less by 1 than the word results in the word itself. (Contributed by Alexander van der Vekens, 24-Oct-2018.)
Assertion
Ref Expression
ccats1swrdeq |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( W = ( U substr <. 0 , ( # ` W ) >. ) -> U = ( W ++ <" ( lastS ` U ) "> ) ) )
Proof of Theorem ccats1swrdeq
StepHypRef Expression
1 oveq1 6657 . . . 4 |- ( W = ( U substr <. 0 , ( # ` W ) >. ) -> ( W ++ <" ( lastS ` U ) "> ) = ( ( U substr <. 0 , ( # ` W ) >. ) ++ <" ( lastS ` U ) "> ) )
21adantl 482 . . 3 |- ( ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) /\ W = ( U substr <. 0 , ( # ` W ) >. ) ) -> ( W ++ <" ( lastS ` U ) "> ) = ( ( U substr <. 0 , ( # ` W ) >. ) ++ <" ( lastS ` U ) "> ) )
3 oveq1 6657 . . . . . . . . . 10 |- ( ( # ` U ) = ( ( # ` W ) + 1 ) -> ( ( # ` U ) - 1 ) = ( ( ( # ` W ) + 1 ) - 1 ) )
433ad2ant3 1084 . . . . . . . . 9 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( ( # ` U ) - 1 ) = ( ( ( # ` W ) + 1 ) - 1 ) )
5 lencl 13324 . . . . . . . . . . 11 |- ( W e. Word V -> ( # ` W ) e. NN0 )
6 nn0cn 11302 . . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. CC )
7 pncan1 10454 . . . . . . . . . . 11 |- ( ( # ` W ) e. CC -> ( ( ( # ` W ) + 1 ) - 1 ) = ( # ` W ) )
85, 6, 73syl 18 . . . . . . . . . 10 |- ( W e. Word V -> ( ( ( # ` W ) + 1 ) - 1 ) = ( # ` W ) )
983ad2ant1 1082 . . . . . . . . 9 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( ( ( # ` W ) + 1 ) - 1 ) = ( # ` W ) )
104, 9eqtr2d 2657 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( # ` W ) = ( ( # ` U ) - 1 ) )
1110opeq2d 4409 . . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> <. 0 , ( # ` W ) >. = <. 0 , ( ( # ` U ) - 1 ) >. )
1211oveq2d 6666 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( U substr <. 0 , ( # ` W ) >. ) = ( U substr <. 0 , ( ( # ` U ) - 1 ) >. ) )
1312oveq1d 6665 . . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( ( U substr <. 0 , ( # ` W ) >. ) ++ <" ( lastS ` U ) "> ) = ( ( U substr <. 0 , ( ( # ` U ) - 1 ) >. ) ++ <" ( lastS ` U ) "> ) )
14 simp2 1062 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> U e. Word V )
15 nn0p1gt0 11322 . . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> 0 < ( ( # ` W ) + 1 ) )
165, 15syl 17 . . . . . . . . 9 |- ( W e. Word V -> 0 < ( ( # ` W ) + 1 ) )
17163ad2ant1 1082 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> 0 < ( ( # ` W ) + 1 ) )
18 breq2 4657 . . . . . . . . 9 |- ( ( # ` U ) = ( ( # ` W ) + 1 ) -> ( 0 < ( # ` U ) <-> 0 < ( ( # ` W ) + 1 ) ) )
19183ad2ant3 1084 . . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( 0 < ( # ` U ) <-> 0 < ( ( # ` W ) + 1 ) ) )
2017, 19mpbird 247 . . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> 0 < ( # ` U ) )
21 hashneq0 13155 . . . . . . . 8 |- ( U e. Word V -> ( 0 < ( # ` U ) <-> U =/= (/) ) )
22213ad2ant2 1083 . . . . . . 7 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( 0 < ( # ` U ) <-> U =/= (/) ) )
2320, 22mpbid 222 . . . . . 6 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> U =/= (/) )
24 swrdccatwrd 13468 . . . . . 6 |- ( ( U e. Word V /\ U =/= (/) ) -> ( ( U substr <. 0 , ( ( # ` U ) - 1 ) >. ) ++ <" ( lastS ` U ) "> ) = U )
2514, 23, 24syl2anc 693 . . . . 5 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( ( U substr <. 0 , ( ( # ` U ) - 1 ) >. ) ++ <" ( lastS ` U ) "> ) = U )
2613, 25eqtrd 2656 . . . 4 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( ( U substr <. 0 , ( # ` W ) >. ) ++ <" ( lastS ` U ) "> ) = U )
2726adantr 481 . . 3 |- ( ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) /\ W = ( U substr <. 0 , ( # ` W ) >. ) ) -> ( ( U substr <. 0 , ( # ` W ) >. ) ++ <" ( lastS ` U ) "> ) = U )
282, 27eqtr2d 2657 . 2 |- ( ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) /\ W = ( U substr <. 0 , ( # ` W ) >. ) ) -> U = ( W ++ <" ( lastS ` U ) "> ) )
2928ex 450 1 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( W = ( U substr <. 0 , ( # ` W ) >. ) -> U = ( W ++ <" ( lastS ` U ) "> ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   NN0cn0 11292   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303
This theorem is referenced by:  ccats1swrdeqrex  13478  ccats1swrdeqbi  13498
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