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Theorem s2eq2s1eq 12654
Description: Two length 2 words are equal iff the corresponding singleton words consisting of their symbols are equal. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
Assertion
Ref Expression
s2eq2s1eq  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  -> 
( <" A B ">  =  <" C D ">  <->  ( <" A ">  =  <" C ">  /\  <" B ">  =  <" D "> ) ) )

Proof of Theorem s2eq2s1eq
StepHypRef Expression
1 df-s2 12586 . . . 4  |-  <" A B ">  =  (
<" A "> concat  <" B "> )
21a1i 11 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  ->  <" A B ">  =  ( <" A "> concat  <" B "> ) )
3 df-s2 12586 . . . 4  |-  <" C D ">  =  (
<" C "> concat  <" D "> )
43a1i 11 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  ->  <" C D ">  =  ( <" C "> concat  <" D "> ) )
52, 4eqeq12d 2473 . 2  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  -> 
( <" A B ">  =  <" C D ">  <->  ( <" A "> concat  <" B "> )  =  ( <" C "> concat  <" D "> ) ) )
6 s1cl 12404 . . . . 5  |-  ( A  e.  V  ->  <" A ">  e. Word  V )
7 s1cl 12404 . . . . 5  |-  ( B  e.  V  ->  <" B ">  e. Word  V )
86, 7anim12i 566 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( <" A ">  e. Word  V  /\  <" B ">  e. Word  V ) )
98adantr 465 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  -> 
( <" A ">  e. Word  V  /\  <" B ">  e. Word  V ) )
10 s1cl 12404 . . . . 5  |-  ( C  e.  V  ->  <" C ">  e. Word  V )
11 s1cl 12404 . . . . 5  |-  ( D  e.  V  ->  <" D ">  e. Word  V )
1210, 11anim12i 566 . . . 4  |-  ( ( C  e.  V  /\  D  e.  V )  ->  ( <" C ">  e. Word  V  /\  <" D ">  e. Word  V ) )
1312adantl 466 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  -> 
( <" C ">  e. Word  V  /\  <" D ">  e. Word  V ) )
14 s1len 12407 . . . . 5  |-  ( # `  <" A "> )  =  1
15 s1len 12407 . . . . 5  |-  ( # `  <" C "> )  =  1
1614, 15eqtr4i 2483 . . . 4  |-  ( # `  <" A "> )  =  ( # `
 <" C "> )
1716a1i 11 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  -> 
( # `  <" A "> )  =  (
# `  <" C "> ) )
18 ccatopth 12475 . . 3  |-  ( ( ( <" A ">  e. Word  V  /\  <" B ">  e. Word  V )  /\  ( <" C ">  e. Word  V  /\  <" D ">  e. Word  V )  /\  ( # `  <" A "> )  =  ( # `  <" C "> )
)  ->  ( ( <" A "> concat  <" B "> )  =  ( <" C "> concat  <" D "> )  <->  ( <" A ">  =  <" C ">  /\ 
<" B ">  =  <" D "> ) ) )
199, 13, 17, 18syl3anc 1219 . 2  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  -> 
( ( <" A "> concat  <" B "> )  =  ( <" C "> concat  <" D "> ) 
<->  ( <" A ">  =  <" C ">  /\  <" B ">  =  <" D "> ) ) )
205, 19bitrd 253 1  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( C  e.  V  /\  D  e.  V ) )  -> 
( <" A B ">  =  <" C D ">  <->  ( <" A ">  =  <" C ">  /\  <" B ">  =  <" D "> ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   1c1 9387   #chash 12213  Word cword 12332   concat cconcat 12334   <"cs1 12335   <"cs2 12579
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-substr 12344  df-s2 12586
This theorem is referenced by:  s2eq2seq  12655  2swrd2eqwrdeq  12664
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