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Theorem swrds2 13092
Description: Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
swrds2 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. I , ( I + 2 ) >. ) = <" ( W ` I ) ( W ` ( I + 1 ) ) "> )
Proof of Theorem swrds2
StepHypRef Expression
1 simp1 1030 . . . . 5 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> W e. Word A )
2 simp2 1031 . . . . . 6 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> I e. NN0 )
3 elfzo0 11984 . . . . . . . 8 |- ( ( I + 1 ) e. ( 0..^ ( # ` W ) ) <-> ( ( I + 1 ) e. NN0 /\ ( # ` W ) e. NN /\ ( I + 1 ) < ( # ` W ) ) )
43simp2bi 1046 . . . . . . 7 |- ( ( I + 1 ) e. ( 0..^ ( # ` W ) ) -> ( # ` W ) e. NN )
543ad2ant3 1053 . . . . . 6 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( # ` W ) e. NN )
62nn0red 10950 . . . . . . 7 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> I e. RR )
7 peano2nn0 10934 . . . . . . . . 9 |- ( I e. NN0 -> ( I + 1 ) e. NN0 )
82, 7syl 17 . . . . . . . 8 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( I + 1 ) e. NN0 )
98nn0red 10950 . . . . . . 7 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( I + 1 ) e. RR )
105nnred 10646 . . . . . . 7 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( # ` W ) e. RR )
116lep1d 10560 . . . . . . 7 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> I <_ ( I + 1 ) )
12 elfzolt2 11956 . . . . . . . 8 |- ( ( I + 1 ) e. ( 0..^ ( # ` W ) ) -> ( I + 1 ) < ( # ` W ) )
13123ad2ant3 1053 . . . . . . 7 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( I + 1 ) < ( # ` W ) )
146, 9, 10, 11, 13lelttrd 9810 . . . . . 6 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> I < ( # ` W ) )
15 elfzo0 11984 . . . . . 6 |- ( I e. ( 0..^ ( # ` W ) ) <-> ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) )
162, 5, 14, 15syl3anbrc 1214 . . . . 5 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> I e. ( 0..^ ( # ` W ) ) )
17 swrds1 12861 . . . . 5 |- ( ( W e. Word A /\ I e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. I , ( I + 1 ) >. ) = <" ( W ` I ) "> )
181, 16, 17syl2anc 673 . . . 4 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. I , ( I + 1 ) >. ) = <" ( W ` I ) "> )
19 nn0cn 10903 . . . . . . . . 9 |- ( I e. NN0 -> I e. CC )
20193ad2ant2 1052 . . . . . . . 8 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> I e. CC )
21 ax-1cn 9615 . . . . . . . . . 10 |- 1 e. CC
22 addass 9644 . . . . . . . . . 10 |- ( ( I e. CC /\ 1 e. CC /\ 1 e. CC ) -> ( ( I + 1 ) + 1 ) = ( I + ( 1 + 1 ) ) )
2321, 21, 22mp3an23 1382 . . . . . . . . 9 |- ( I e. CC -> ( ( I + 1 ) + 1 ) = ( I + ( 1 + 1 ) ) )
24 df-2 10690 . . . . . . . . . 10 |- 2 = ( 1 + 1 )
2524oveq2i 6319 . . . . . . . . 9 |- ( I + 2 ) = ( I + ( 1 + 1 ) )
2623, 25syl6reqr 2524 . . . . . . . 8 |- ( I e. CC -> ( I + 2 ) = ( ( I + 1 ) + 1 ) )
2720, 26syl 17 . . . . . . 7 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( I + 2 ) = ( ( I + 1 ) + 1 ) )
2827opeq2d 4165 . . . . . 6 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> <. ( I + 1 ) , ( I + 2 ) >. = <. ( I + 1 ) , ( ( I + 1 ) + 1 ) >. )
2928oveq2d 6324 . . . . 5 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. ( I + 1 ) , ( I + 2 ) >. ) = ( W substr <. ( I + 1 ) , ( ( I + 1 ) + 1 ) >. ) )
30 swrds1 12861 . . . . . 6 |- ( ( W e. Word A /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. ( I + 1 ) , ( ( I + 1 ) + 1 ) >. ) = <" ( W ` ( I + 1 ) ) "> )
31303adant2 1049 . . . . 5 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. ( I + 1 ) , ( ( I + 1 ) + 1 ) >. ) = <" ( W ` ( I + 1 ) ) "> )
3229, 31eqtrd 2505 . . . 4 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. ( I + 1 ) , ( I + 2 ) >. ) = <" ( W ` ( I + 1 ) ) "> )
3318, 32oveq12d 6326 . . 3 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( ( W substr <. I , ( I + 1 ) >. ) ++ ( W substr <. ( I + 1 ) , ( I + 2 ) >. ) ) = ( <" ( W ` I ) "> ++ <" ( W ` ( I + 1 ) ) "> ) )
34 df-s2 13003 . . 3 |- <" ( W ` I ) ( W ` ( I + 1 ) ) "> = ( <" ( W ` I ) "> ++ <" ( W ` ( I + 1 ) ) "> )
3533, 34syl6reqr 2524 . 2 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> <" ( W ` I ) ( W ` ( I + 1 ) ) "> = ( ( W substr <. I , ( I + 1 ) >. ) ++ ( W substr <. ( I + 1 ) , ( I + 2 ) >. ) ) )
36 elfz2nn0 11911 . . . 4 |- ( I e. ( 0 ... ( I + 1 ) ) <-> ( I e. NN0 /\ ( I + 1 ) e. NN0 /\ I <_ ( I + 1 ) ) )
372, 8, 11, 36syl3anbrc 1214 . . 3 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> I e. ( 0 ... ( I + 1 ) ) )
38 peano2nn0 10934 . . . . . 6 |- ( ( I + 1 ) e. NN0 -> ( ( I + 1 ) + 1 ) e. NN0 )
398, 38syl 17 . . . . 5 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( ( I + 1 ) + 1 ) e. NN0 )
4027, 39eqeltrd 2549 . . . 4 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( I + 2 ) e. NN0 )
419lep1d 10560 . . . . 5 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( I + 1 ) <_ ( ( I + 1 ) + 1 ) )
4241, 27breqtrrd 4422 . . . 4 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( I + 1 ) <_ ( I + 2 ) )
43 elfz2nn0 11911 . . . 4 |- ( ( I + 1 ) e. ( 0 ... ( I + 2 ) ) <-> ( ( I + 1 ) e. NN0 /\ ( I + 2 ) e. NN0 /\ ( I + 1 ) <_ ( I + 2 ) ) )
448, 40, 42, 43syl3anbrc 1214 . . 3 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( I + 1 ) e. ( 0 ... ( I + 2 ) ) )
45 fzofzp1 12037 . . . . 5 |- ( ( I + 1 ) e. ( 0..^ ( # ` W ) ) -> ( ( I + 1 ) + 1 ) e. ( 0 ... ( # ` W ) ) )
46453ad2ant3 1053 . . . 4 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( ( I + 1 ) + 1 ) e. ( 0 ... ( # ` W ) ) )
4727, 46eqeltrd 2549 . . 3 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( I + 2 ) e. ( 0 ... ( # ` W ) ) )
48 ccatswrd 12866 . . 3 |- ( ( W e. Word A /\ ( I e. ( 0 ... ( I + 1 ) ) /\ ( I + 1 ) e. ( 0 ... ( I + 2 ) ) /\ ( I + 2 ) e. ( 0 ... ( # ` W ) ) ) ) -> ( ( W substr <. I , ( I + 1 ) >. ) ++ ( W substr <. ( I + 1 ) , ( I + 2 ) >. ) ) = ( W substr <. I , ( I + 2 ) >. ) )
491, 37, 44, 47, 48syl13anc 1294 . 2 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( ( W substr <. I , ( I + 1 ) >. ) ++ ( W substr <. ( I + 1 ) , ( I + 2 ) >. ) ) = ( W substr <. I , ( I + 2 ) >. ) )
5035, 49eqtr2d 2506 1 |- ( ( W e. Word A /\ I e. NN0 /\ ( I + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. I , ( I + 2 ) >. ) = <" ( W ` I ) ( W ` ( I + 1 ) ) "> )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ w3a 1007   = wceq 1452   e. wcel 1904   <.cop 3965   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558   + caddc 9560   < clt 9693   <_ cle 9694   NNcn 10631   2c2 10681   NN0cn0 10893   ...cfz 11810  ..^cfzo 11942   #chash 12553  Word cword 12703   ++ cconcat 12705   <"cs1 12706   substr csubstr 12707   <"cs2 12996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-substr 12715  df-s2 13003
This theorem is referenced by:  swrd2lsw  13102  psgnunilem2  17214
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