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Theorem swrdccat3b 13496
Description: A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 30-May-2018.)
Hypothesis
Ref Expression
swrdccatin12.l |- L = ( # ` A )
Assertion
Ref Expression
swrdccat3b |- ( ( A e. Word V /\ B e. Word V ) -> ( M e. ( 0 ... ( L + ( # ` B ) ) ) -> ( ( A ++ B ) substr <. M , ( L + ( # ` B ) ) >. ) = if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) ) )
Proof of Theorem swrdccat3b
StepHypRef Expression
1 simpl 473 . . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( A e. Word V /\ B e. Word V ) )
2 simpr 477 . . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) -> M e. ( 0 ... ( L + ( # ` B ) ) ) )
3 elfzubelfz 12353 . . . . 5 |- ( M e. ( 0 ... ( L + ( # ` B ) ) ) -> ( L + ( # ` B ) ) e. ( 0 ... ( L + ( # ` B ) ) ) )
43adantl 482 . . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( L + ( # ` B ) ) e. ( 0 ... ( L + ( # ` B ) ) ) )
5 swrdccatin12.l . . . . . 6 |- L = ( # ` A )
65swrdccat3 13492 . . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( M e. ( 0 ... ( L + ( # ` B ) ) ) /\ ( L + ( # ` B ) ) e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , ( L + ( # ` B ) ) >. ) = if ( ( L + ( # ` B ) ) <_ L , ( A substr <. M , ( L + ( # ` B ) ) >. ) , if ( L <_ M , ( B substr <. ( M - L ) , ( ( L + ( # ` B ) ) - L ) >. ) , ( ( A substr <. M , L >. ) ++ ( B substr <. 0 , ( ( L + ( # ` B ) ) - L ) >. ) ) ) ) ) )
76imp 445 . . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ ( M e. ( 0 ... ( L + ( # ` B ) ) ) /\ ( L + ( # ` B ) ) e. ( 0 ... ( L + ( # ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , ( L + ( # ` B ) ) >. ) = if ( ( L + ( # ` B ) ) <_ L , ( A substr <. M , ( L + ( # ` B ) ) >. ) , if ( L <_ M , ( B substr <. ( M - L ) , ( ( L + ( # ` B ) ) - L ) >. ) , ( ( A substr <. M , L >. ) ++ ( B substr <. 0 , ( ( L + ( # ` B ) ) - L ) >. ) ) ) ) )
81, 2, 4, 7syl12anc 1324 . . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , ( L + ( # ` B ) ) >. ) = if ( ( L + ( # ` B ) ) <_ L , ( A substr <. M , ( L + ( # ` B ) ) >. ) , if ( L <_ M , ( B substr <. ( M - L ) , ( ( L + ( # ` B ) ) - L ) >. ) , ( ( A substr <. M , L >. ) ++ ( B substr <. 0 , ( ( L + ( # ` B ) ) - L ) >. ) ) ) ) )
95swrdccat3blem 13495 . . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ ( L + ( # ` B ) ) <_ L ) -> if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + ( # ` B ) ) >. ) )
10 iftrue 4092 . . . . . 6 |- ( L <_ M -> if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( B substr <. ( M - L ) , ( # ` B ) >. ) )
11103ad2ant3 1084 . . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ L <_ M ) -> if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( B substr <. ( M - L ) , ( # ` B ) >. ) )
12 lencl 13324 . . . . . . . . . . . 12 |- ( A e. Word V -> ( # ` A ) e. NN0 )
1312nn0cnd 11353 . . . . . . . . . . 11 |- ( A e. Word V -> ( # ` A ) e. CC )
14 lencl 13324 . . . . . . . . . . . 12 |- ( B e. Word V -> ( # ` B ) e. NN0 )
1514nn0cnd 11353 . . . . . . . . . . 11 |- ( B e. Word V -> ( # ` B ) e. CC )
165eqcomi 2631 . . . . . . . . . . . . 13 |- ( # ` A ) = L
1716eleq1i 2692 . . . . . . . . . . . 12 |- ( ( # ` A ) e. CC <-> L e. CC )
18 pncan2 10288 . . . . . . . . . . . 12 |- ( ( L e. CC /\ ( # ` B ) e. CC ) -> ( ( L + ( # ` B ) ) - L ) = ( # ` B ) )
1917, 18sylanb 489 . . . . . . . . . . 11 |- ( ( ( # ` A ) e. CC /\ ( # ` B ) e. CC ) -> ( ( L + ( # ` B ) ) - L ) = ( # ` B ) )
2013, 15, 19syl2an 494 . . . . . . . . . 10 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + ( # ` B ) ) - L ) = ( # ` B ) )
2120eqcomd 2628 . . . . . . . . 9 |- ( ( A e. Word V /\ B e. Word V ) -> ( # ` B ) = ( ( L + ( # ` B ) ) - L ) )
2221adantr 481 . . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( # ` B ) = ( ( L + ( # ` B ) ) - L ) )
23223ad2ant1 1082 . . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ L <_ M ) -> ( # ` B ) = ( ( L + ( # ` B ) ) - L ) )
2423opeq2d 4409 . . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ L <_ M ) -> <. ( M - L ) , ( # ` B ) >. = <. ( M - L ) , ( ( L + ( # ` B ) ) - L ) >. )
2524oveq2d 6666 . . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ L <_ M ) -> ( B substr <. ( M - L ) , ( # ` B ) >. ) = ( B substr <. ( M - L ) , ( ( L + ( # ` B ) ) - L ) >. ) )
2611, 25eqtrd 2656 . . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ L <_ M ) -> if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( B substr <. ( M - L ) , ( ( L + ( # ` B ) ) - L ) >. ) )
27 iffalse 4095 . . . . . 6 |- ( -. L <_ M -> if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( ( A substr <. M , L >. ) ++ B ) )
28273ad2ant3 1084 . . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ -. L <_ M ) -> if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( ( A substr <. M , L >. ) ++ B ) )
2920adantr 481 . . . . . . . . . 10 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( ( L + ( # ` B ) ) - L ) = ( # ` B ) )
30293ad2ant1 1082 . . . . . . . . 9 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ -. L <_ M ) -> ( ( L + ( # ` B ) ) - L ) = ( # ` B ) )
3130opeq2d 4409 . . . . . . . 8 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ -. L <_ M ) -> <. 0 , ( ( L + ( # ` B ) ) - L ) >. = <. 0 , ( # ` B ) >. )
3231oveq2d 6666 . . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ -. L <_ M ) -> ( B substr <. 0 , ( ( L + ( # ` B ) ) - L ) >. ) = ( B substr <. 0 , ( # ` B ) >. ) )
33 swrdid 13428 . . . . . . . . . 10 |- ( B e. Word V -> ( B substr <. 0 , ( # ` B ) >. ) = B )
3433adantl 482 . . . . . . . . 9 |- ( ( A e. Word V /\ B e. Word V ) -> ( B substr <. 0 , ( # ` B ) >. ) = B )
3534adantr 481 . . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( B substr <. 0 , ( # ` B ) >. ) = B )
36353ad2ant1 1082 . . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ -. L <_ M ) -> ( B substr <. 0 , ( # ` B ) >. ) = B )
3732, 36eqtr2d 2657 . . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ -. L <_ M ) -> B = ( B substr <. 0 , ( ( L + ( # ` B ) ) - L ) >. ) )
3837oveq2d 6666 . . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ -. L <_ M ) -> ( ( A substr <. M , L >. ) ++ B ) = ( ( A substr <. M , L >. ) ++ ( B substr <. 0 , ( ( L + ( # ` B ) ) - L ) >. ) ) )
3928, 38eqtrd 2656 . . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) /\ -. ( L + ( # ` B ) ) <_ L /\ -. L <_ M ) -> if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( ( A substr <. M , L >. ) ++ ( B substr <. 0 , ( ( L + ( # ` B ) ) - L ) >. ) ) )
409, 26, 392if2 4136 . . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) -> if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = if ( ( L + ( # ` B ) ) <_ L , ( A substr <. M , ( L + ( # ` B ) ) >. ) , if ( L <_ M , ( B substr <. ( M - L ) , ( ( L + ( # ` B ) ) - L ) >. ) , ( ( A substr <. M , L >. ) ++ ( B substr <. 0 , ( ( L + ( # ` B ) ) - L ) >. ) ) ) ) )
418, 40eqtr4d 2659 . 2 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + ( # ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , ( L + ( # ` B ) ) >. ) = if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) )
4241ex 450 1 |- ( ( A e. Word V /\ B e. Word V ) -> ( M e. ( 0 ... ( L + ( # ` B ) ) ) -> ( ( A ++ B ) substr <. M , ( L + ( # ` B ) ) >. ) = if ( L <_ M , ( B substr <. ( M - L ) , ( # ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ifcif 4086   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   <_ cle 10075   - cmin 10266   ...cfz 12326   #chash 13117  Word cword 13291   ++ cconcat 13293   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-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303
This theorem is referenced by: (None)
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