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Theorem 2swrdeqwrdeq 12469
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 12387 . . 3  |-  ( ( W  e. Word  V  /\  S  e. Word  V )  ->  ( W  =  S  <-> 
( ( # `  W
)  =  ( # `  S )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( S `  i
) ) ) )
213adant3 1008 . 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 11688 . . . . . . . . 9  |-  ( I  e.  ( 0..^ (
# `  W )
)  ->  I  e.  ( 0 ... ( # `
 W ) ) )
4 fzosplit 11703 . . . . . . . . 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 1011 . . . . . . 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 3029 . . . . 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 3648 . . . . 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 985 . . . . . . 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 11712 . . . . . . . . 9  |-  ( I  e.  ( 0..^ (
# `  W )
)  ->  I  e.  NN0 )
14133ad2ant3 1011 . . . . . . . 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 10709 . . . . . . 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 11711 . . . . . . . 8  |-  ( I  e.  ( 0..^ (
# `  W )
)  ->  I  <_  (
# `  W )
)
19183ad2ant3 1011 . . . . . . 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 4407 . . . . . . . 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 12464 . . . . . . 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 1223 . . . . 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 12371 . . . . . . . . 9  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
28273ad2ant1 1009 . . . . . . . 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 10703 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  RR )
3231leidd 10021 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  <_  ( # `  W
) )
3327, 32syl 16 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( # `
 W )  <_ 
( # `  W ) )
34333ad2ant1 1009 . . . . . . 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 4407 . . . . . . . 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 12464 . . . . . . 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 1223 . . . . 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 965    = wceq 1370    e. wcel 1758   A.wral 2799    u. cun 3437   <.cop 3994   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   0cc0 9397    <_ cle 9534   NN0cn0 10694   ...cfz 11558  ..^cfzo 11669   #chash 12224  Word cword 12343   substr csubstr 12347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-hash 12225  df-word 12351  df-substr 12355
This theorem is referenced by:  2swrd1eqwrdeq  12470  2swrd2eqwrdeq  12675
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