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Theorem 2swrdeqwrdeq 12658
Description: Two words are equal if and only if they have the same prefix and the same suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
Assertion
Ref Expression
2swrdeqwrdeq  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( W  =  S  <->  ( ( # `  W )  =  (
# `  S )  /\  ( ( W substr  <. 0 ,  I >. )  =  ( S substr  <. 0 ,  I >. )  /\  ( W substr  <. I ,  ( # `  W ) >. )  =  ( S substr  <. I ,  ( # `  W
) >. ) ) ) ) )

Proof of Theorem 2swrdeqwrdeq
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 eqwrd 12562 . . 3  |-  ( ( W  e. Word  V  /\  S  e. Word  V )  ->  ( W  =  S  <-> 
( ( # `  W
)  =  ( # `  S )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
) ) ) )
213adant3 1016 . 2  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( W  =  S  <->  ( ( # `  W )  =  (
# `  S )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
) ) ) )
3 elfzofz 11823 . . . . . . . . 9  |-  ( I  e.  ( 0..^ (
# `  W )
)  ->  I  e.  ( 0 ... ( # `
 W ) ) )
4 fzosplit 11838 . . . . . . . . 9  |-  ( I  e.  ( 0 ... ( # `  W
) )  ->  (
0..^ ( # `  W
) )  =  ( ( 0..^ I )  u.  ( I..^ (
# `  W )
) ) )
53, 4syl 16 . . . . . . . 8  |-  ( I  e.  ( 0..^ (
# `  W )
)  ->  ( 0..^ ( # `  W
) )  =  ( ( 0..^ I )  u.  ( I..^ (
# `  W )
) ) )
653ad2ant3 1019 . . . . . . 7  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( 0..^ (
# `  W )
)  =  ( ( 0..^ I )  u.  ( I..^ ( # `  W ) ) ) )
76adantr 465 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  (
0..^ ( # `  W
) )  =  ( ( 0..^ I )  u.  ( I..^ (
# `  W )
) ) )
87raleqdv 3069 . . . . 5  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
)  <->  A. i  e.  ( ( 0..^ I )  u.  ( I..^ (
# `  W )
) ) ( W `
 i )  =  ( S `  i
) ) )
9 ralunb 3690 . . . . 5  |-  ( A. i  e.  ( (
0..^ I )  u.  ( I..^ ( # `  W ) ) ) ( W `  i
)  =  ( S `
 i )  <->  ( A. i  e.  ( 0..^ I ) ( W `
 i )  =  ( S `  i
)  /\  A. i  e.  ( I..^ ( # `  W ) ) ( W `  i )  =  ( S `  i ) ) )
108, 9syl6bb 261 . . . 4  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
)  <->  ( A. i  e.  ( 0..^ I ) ( W `  i
)  =  ( S `
 i )  /\  A. i  e.  ( I..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
) ) ) )
11 3simpa 993 . . . . . . 7  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( W  e. Word  V  /\  S  e. Word  V
) )
1211adantr 465 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  ( W  e. Word  V  /\  S  e. Word  V ) )
13 elfzonn0 11847 . . . . . . . . 9  |-  ( I  e.  ( 0..^ (
# `  W )
)  ->  I  e.  NN0 )
14133ad2ant3 1019 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  I  e.  NN0 )
1514adantr 465 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  I  e.  NN0 )
16 0nn0 10822 . . . . . . 7  |-  0  e.  NN0
1715, 16jctil 537 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  (
0  e.  NN0  /\  I  e.  NN0 ) )
18 elfzo0le 11846 . . . . . . . 8  |-  ( I  e.  ( 0..^ (
# `  W )
)  ->  I  <_  (
# `  W )
)
19183ad2ant3 1019 . . . . . . 7  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  I  <_  ( # `
 W ) )
2019adantr 465 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  I  <_  ( # `  W
) )
21 breq2 4457 . . . . . . . 8  |-  ( (
# `  W )  =  ( # `  S
)  ->  ( I  <_  ( # `  W
)  <->  I  <_  ( # `  S ) ) )
2221adantl 466 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  (
I  <_  ( # `  W
)  <->  I  <_  ( # `  S ) ) )
2320, 22mpbid 210 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  I  <_  ( # `  S
) )
24 swrdspsleq 12653 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V )  /\  ( 0  e. 
NN0  /\  I  e.  NN0 )  /\  ( I  <_  ( # `  W
)  /\  I  <_  (
# `  S )
) )  ->  (
( W substr  <. 0 ,  I >. )  =  ( S substr  <. 0 ,  I >. )  <->  A. i  e.  ( 0..^ I ) ( W `  i )  =  ( S `  i ) ) )
2524bicomd 201 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V )  /\  ( 0  e. 
NN0  /\  I  e.  NN0 )  /\  ( I  <_  ( # `  W
)  /\  I  <_  (
# `  S )
) )  ->  ( A. i  e.  (
0..^ I ) ( W `  i )  =  ( S `  i )  <->  ( W substr  <.
0 ,  I >. )  =  ( S substr  <. 0 ,  I >. ) ) )
2612, 17, 20, 23, 25syl112anc 1232 . . . . 5  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  ( A. i  e.  (
0..^ I ) ( W `  i )  =  ( S `  i )  <->  ( W substr  <.
0 ,  I >. )  =  ( S substr  <. 0 ,  I >. ) ) )
27 lencl 12543 . . . . . . . . 9  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
28273ad2ant1 1017 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( # `  W
)  e.  NN0 )
2914, 28jca 532 . . . . . . 7  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( I  e. 
NN0  /\  ( # `  W
)  e.  NN0 )
)
3029adantr 465 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  (
I  e.  NN0  /\  ( # `  W )  e.  NN0 ) )
31 nn0re 10816 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  RR )
3231leidd 10131 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  <_  ( # `  W
) )
3327, 32syl 16 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( # `
 W )  <_ 
( # `  W ) )
34333ad2ant1 1017 . . . . . . 7  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( # `  W
)  <_  ( # `  W
) )
3534adantr 465 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  ( # `
 W )  <_ 
( # `  W ) )
36 breq2 4457 . . . . . . . 8  |-  ( (
# `  W )  =  ( # `  S
)  ->  ( ( # `
 W )  <_ 
( # `  W )  <-> 
( # `  W )  <_  ( # `  S
) ) )
3736adantl 466 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  (
( # `  W )  <_  ( # `  W
)  <->  ( # `  W
)  <_  ( # `  S
) ) )
3835, 37mpbid 210 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  ( # `
 W )  <_ 
( # `  S ) )
39 swrdspsleq 12653 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V )  /\  ( I  e. 
NN0  /\  ( # `  W
)  e.  NN0 )  /\  ( ( # `  W
)  <_  ( # `  W
)  /\  ( # `  W
)  <_  ( # `  S
) ) )  -> 
( ( W substr  <. I ,  ( # `  W
) >. )  =  ( S substr  <. I ,  (
# `  W ) >. )  <->  A. i  e.  ( I..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
) ) )
4039bicomd 201 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V )  /\  ( I  e. 
NN0  /\  ( # `  W
)  e.  NN0 )  /\  ( ( # `  W
)  <_  ( # `  W
)  /\  ( # `  W
)  <_  ( # `  S
) ) )  -> 
( A. i  e.  ( I..^ ( # `  W ) ) ( W `  i )  =  ( S `  i )  <->  ( W substr  <.
I ,  ( # `  W ) >. )  =  ( S substr  <. I ,  ( # `  W
) >. ) ) )
4112, 30, 35, 38, 40syl112anc 1232 . . . . 5  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  ( A. i  e.  (
I..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
)  <->  ( W substr  <. I ,  ( # `  W
) >. )  =  ( S substr  <. I ,  (
# `  W ) >. ) ) )
4226, 41anbi12d 710 . . . 4  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  (
( A. i  e.  ( 0..^ I ) ( W `  i
)  =  ( S `
 i )  /\  A. i  e.  ( I..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
) )  <->  ( ( W substr  <. 0 ,  I >. )  =  ( S substr  <. 0 ,  I >. )  /\  ( W substr  <. I ,  ( # `  W
) >. )  =  ( S substr  <. I ,  (
# `  W ) >. ) ) ) )
4310, 42bitrd 253 . . 3  |-  ( ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  /\  ( # `  W )  =  ( # `  S
) )  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
)  <->  ( ( W substr  <. 0 ,  I >. )  =  ( S substr  <. 0 ,  I >. )  /\  ( W substr  <. I ,  (
# `  W ) >. )  =  ( S substr  <. I ,  ( # `  W ) >. )
) ) )
4443pm5.32da 641 . 2  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( (
# `  W )  =  ( # `  S
)  /\  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( S `  i ) )  <->  ( ( # `
 W )  =  ( # `  S
)  /\  ( ( W substr  <. 0 ,  I >. )  =  ( S substr  <. 0 ,  I >. )  /\  ( W substr  <. I ,  ( # `  W
) >. )  =  ( S substr  <. I ,  (
# `  W ) >. ) ) ) ) )
452, 44bitrd 253 1  |-  ( ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( W  =  S  <->  ( ( # `  W )  =  (
# `  S )  /\  ( ( W substr  <. 0 ,  I >. )  =  ( S substr  <. 0 ,  I >. )  /\  ( W substr  <. I ,  ( # `  W ) >. )  =  ( S substr  <. I ,  ( # `  W
) >. ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817    u. cun 3479   <.cop 4039   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   0cc0 9504    <_ cle 9641   NN0cn0 10807   ...cfz 11684  ..^cfzo 11804   #chash 12385  Word cword 12515   substr csubstr 12519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-substr 12527
This theorem is referenced by:  2swrd1eqwrdeq  12659  2swrd2eqwrdeq  12871
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