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Theorem swrds2 11880
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 957 . . . . 5  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  W  e. Word  A )
2 simp2 958 . . . . . 6  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  I  e.  NN0 )
3 elfzo0 11171 . . . . . . . 8  |-  ( ( I  +  1 )  e.  ( 0..^ (
# `  W )
)  <->  ( ( I  +  1 )  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  ( I  +  1
)  <  ( # `  W
) ) )
43simp2bi 973 . . . . . . 7  |-  ( ( I  +  1 )  e.  ( 0..^ (
# `  W )
)  ->  ( # `  W
)  e.  NN )
543ad2ant3 980 . . . . . 6  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  ( # `
 W )  e.  NN )
62nn0red 10275 . . . . . . 7  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  I  e.  RR )
7 peano2nn0 10260 . . . . . . . . 9  |-  ( I  e.  NN0  ->  ( I  +  1 )  e. 
NN0 )
82, 7syl 16 . . . . . . . 8  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
I  +  1 )  e.  NN0 )
98nn0red 10275 . . . . . . 7  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
I  +  1 )  e.  RR )
105nnred 10015 . . . . . . 7  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  ( # `
 W )  e.  RR )
116lep1d 9942 . . . . . . 7  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  I  <_  ( I  +  1 ) )
12 elfzolt2 11148 . . . . . . . 8  |-  ( ( I  +  1 )  e.  ( 0..^ (
# `  W )
)  ->  ( I  +  1 )  < 
( # `  W ) )
13123ad2ant3 980 . . . . . . 7  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
I  +  1 )  <  ( # `  W
) )
146, 9, 10, 11, 13lelttrd 9228 . . . . . 6  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  I  <  ( # `  W
) )
15 elfzo0 11171 . . . . . 6  |-  ( I  e.  ( 0..^ (
# `  W )
)  <->  ( I  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  I  <  ( # `  W
) ) )
162, 5, 14, 15syl3anbrc 1138 . . . . 5  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  I  e.  ( 0..^ ( # `  W ) ) )
17 swrds1 11787 . . . . 5  |-  ( ( W  e. Word  A  /\  I  e.  ( 0..^ ( # `  W
) ) )  -> 
( W substr  <. I ,  ( I  +  1 ) >. )  =  <" ( W `  I
) "> )
181, 16, 17syl2anc 643 . . . 4  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  ( W substr  <. I ,  ( I  +  1 )
>. )  =  <" ( W `  I
) "> )
19 nn0cn 10231 . . . . . . . . 9  |-  ( I  e.  NN0  ->  I  e.  CC )
20193ad2ant2 979 . . . . . . . 8  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  I  e.  CC )
21 ax-1cn 9048 . . . . . . . . . 10  |-  1  e.  CC
22 addass 9077 . . . . . . . . . 10  |-  ( ( I  e.  CC  /\  1  e.  CC  /\  1  e.  CC )  ->  (
( I  +  1 )  +  1 )  =  ( I  +  ( 1  +  1 ) ) )
2321, 21, 22mp3an23 1271 . . . . . . . . 9  |-  ( I  e.  CC  ->  (
( I  +  1 )  +  1 )  =  ( I  +  ( 1  +  1 ) ) )
24 df-2 10058 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
2524oveq2i 6092 . . . . . . . . 9  |-  ( I  +  2 )  =  ( I  +  ( 1  +  1 ) )
2623, 25syl6reqr 2487 . . . . . . . 8  |-  ( I  e.  CC  ->  (
I  +  2 )  =  ( ( I  +  1 )  +  1 ) )
2720, 26syl 16 . . . . . . 7  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
I  +  2 )  =  ( ( I  +  1 )  +  1 ) )
2827opeq2d 3991 . . . . . 6  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  <. (
I  +  1 ) ,  ( I  + 
2 ) >.  =  <. ( I  +  1 ) ,  ( ( I  +  1 )  +  1 ) >. )
2928oveq2d 6097 . . . . 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 11787 . . . . . 6  |-  ( ( W  e. Word  A  /\  ( I  +  1
)  e.  ( 0..^ ( # `  W
) ) )  -> 
( W substr  <. ( I  +  1 ) ,  ( ( I  + 
1 )  +  1 ) >. )  =  <" ( W `  (
I  +  1 ) ) "> )
31303adant2 976 . . . . 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 2468 . . . 4  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  ( W substr  <. ( I  + 
1 ) ,  ( I  +  2 )
>. )  =  <" ( W `  (
I  +  1 ) ) "> )
3318, 32oveq12d 6099 . . 3  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
( W substr  <. I ,  ( I  +  1 ) >. ) concat  ( W substr  <.
( I  +  1 ) ,  ( I  +  2 ) >.
) )  =  (
<" ( W `  I ) "> concat  <" ( W `  ( I  +  1
) ) "> ) )
34 df-s2 11812 . . 3  |-  <" ( W `  I )
( W `  (
I  +  1 ) ) ">  =  ( <" ( W `
 I ) "> concat  <" ( W `
 ( I  + 
1 ) ) "> )
3533, 34syl6reqr 2487 . 2  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  <" ( W `  I )
( W `  (
I  +  1 ) ) ">  =  ( ( W substr  <. I ,  ( I  +  1 ) >. ) concat  ( W substr  <.
( I  +  1 ) ,  ( I  +  2 ) >.
) ) )
36 elfz2nn0 11082 . . . 4  |-  ( I  e.  ( 0 ... ( I  +  1 ) )  <->  ( I  e.  NN0  /\  ( I  +  1 )  e. 
NN0  /\  I  <_  ( I  +  1 ) ) )
372, 8, 11, 36syl3anbrc 1138 . . 3  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  I  e.  ( 0 ... (
I  +  1 ) ) )
38 peano2nn0 10260 . . . . . 6  |-  ( ( I  +  1 )  e.  NN0  ->  ( ( I  +  1 )  +  1 )  e. 
NN0 )
398, 38syl 16 . . . . 5  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
( I  +  1 )  +  1 )  e.  NN0 )
4027, 39eqeltrd 2510 . . . 4  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
I  +  2 )  e.  NN0 )
419lep1d 9942 . . . . 5  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
I  +  1 )  <_  ( ( I  +  1 )  +  1 ) )
4241, 27breqtrrd 4238 . . . 4  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
I  +  1 )  <_  ( I  + 
2 ) )
43 elfz2nn0 11082 . . . 4  |-  ( ( I  +  1 )  e.  ( 0 ... ( I  +  2 ) )  <->  ( (
I  +  1 )  e.  NN0  /\  (
I  +  2 )  e.  NN0  /\  (
I  +  1 )  <_  ( I  + 
2 ) ) )
448, 40, 42, 43syl3anbrc 1138 . . 3  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
I  +  1 )  e.  ( 0 ... ( I  +  2 ) ) )
45 fzofzp1 11189 . . . . 5  |-  ( ( I  +  1 )  e.  ( 0..^ (
# `  W )
)  ->  ( (
I  +  1 )  +  1 )  e.  ( 0 ... ( # `
 W ) ) )
46453ad2ant3 980 . . . 4  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
( I  +  1 )  +  1 )  e.  ( 0 ... ( # `  W
) ) )
4727, 46eqeltrd 2510 . . 3  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
I  +  2 )  e.  ( 0 ... ( # `  W
) ) )
48 ccatswrd 11773 . . 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 ) >.
) concat  ( W substr  <. (
I  +  1 ) ,  ( I  + 
2 ) >. )
)  =  ( W substr  <. I ,  ( I  +  2 ) >.
) )
491, 37, 44, 47, 48syl13anc 1186 . 2  |-  ( ( W  e. Word  A  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  (
( W substr  <. I ,  ( I  +  1 ) >. ) concat  ( W substr  <.
( I  +  1 ) ,  ( I  +  2 ) >.
) )  =  ( W substr  <. I ,  ( I  +  2 )
>. ) )
5035, 49eqtr2d 2469 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 set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3817   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121   NNcn 10000   2c2 10049   NN0cn0 10221   ...cfz 11043  ..^cfzo 11135   #chash 11618  Word cword 11717   concat cconcat 11718   <"cs1 11719   substr csubstr 11720   <"cs2 11805
This theorem is referenced by:  psgnunilem2  27395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-concat 11724  df-s1 11725  df-substr 11726  df-s2 11812
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