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Theorem ofs2 26964
Description: Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
Assertion
Ref Expression
ofs2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( <" A B ">  oF R <" C D "> )  = 
<" ( A R C ) ( B R D ) "> )

Proof of Theorem ofs2
StepHypRef Expression
1 df-s2 12494 . . . 4  |-  <" A B ">  =  (
<" A "> concat  <" B "> )
2 df-s2 12494 . . . 4  |-  <" C D ">  =  (
<" C "> concat  <" D "> )
31, 2oveq12i 6122 . . 3  |-  ( <" A B ">  oF R <" C D "> )  =  ( ( <" A "> concat  <" B "> )  oF R (
<" C "> concat  <" D "> ) )
4 simpll 753 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  A  e.  S )
54s1cld 12313 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  <" A ">  e. Word  S )
6 simplr 754 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  B  e.  S )
76s1cld 12313 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  <" B ">  e. Word  S )
8 simprl 755 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  C  e.  T )
98s1cld 12313 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  <" C ">  e. Word  T )
10 simprr 756 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  D  e.  T )
1110s1cld 12313 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  <" D ">  e. Word  T )
12 s1len 12315 . . . . . 6  |-  ( # `  <" A "> )  =  1
13 s1len 12315 . . . . . 6  |-  ( # `  <" C "> )  =  1
1412, 13eqtr4i 2466 . . . . 5  |-  ( # `  <" A "> )  =  ( # `
 <" C "> )
1514a1i 11 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( # `  <" A "> )  =  (
# `  <" C "> ) )
16 s1len 12315 . . . . . 6  |-  ( # `  <" B "> )  =  1
17 s1len 12315 . . . . . 6  |-  ( # `  <" D "> )  =  1
1816, 17eqtr4i 2466 . . . . 5  |-  ( # `  <" B "> )  =  ( # `
 <" D "> )
1918a1i 11 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( # `  <" B "> )  =  (
# `  <" D "> ) )
205, 7, 9, 11, 15, 19ofccat 26960 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( ( <" A "> concat  <" B "> )  oF
R ( <" C "> concat  <" D "> ) )  =  ( ( <" A ">  oF R
<" C "> ) concat  ( <" B ">  oF R
<" D "> ) ) )
213, 20syl5eq 2487 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( <" A B ">  oF R <" C D "> )  =  ( ( <" A ">  oF R
<" C "> ) concat  ( <" B ">  oF R
<" D "> ) ) )
22 ofs1 26962 . . . . 5  |-  ( ( A  e.  S  /\  C  e.  T )  ->  ( <" A ">  oF R
<" C "> )  =  <" ( A R C ) "> )
234, 8, 22syl2anc 661 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( <" A ">  oF R <" C "> )  =  <" ( A R C ) "> )
24 ofs1 26962 . . . . 5  |-  ( ( B  e.  S  /\  D  e.  T )  ->  ( <" B ">  oF R
<" D "> )  =  <" ( B R D ) "> )
256, 10, 24syl2anc 661 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( <" B ">  oF R <" D "> )  =  <" ( B R D ) "> )
2623, 25oveq12d 6128 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( ( <" A ">  oF R
<" C "> ) concat  ( <" B ">  oF R
<" D "> ) )  =  (
<" ( A R C ) "> concat  <" ( B R D ) "> ) )
27 df-s2 12494 . . 3  |-  <" ( A R C ) ( B R D ) ">  =  (
<" ( A R C ) "> concat  <" ( B R D ) "> )
2826, 27syl6eqr 2493 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( ( <" A ">  oF R
<" C "> ) concat  ( <" B ">  oF R
<" D "> ) )  =  <" ( A R C ) ( B R D ) "> )
2921, 28eqtrd 2475 1  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( <" A B ">  oF R <" C D "> )  = 
<" ( A R C ) ( B R D ) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5437  (class class class)co 6110    oFcof 6337   1c1 9302   #chash 12122   concat cconcat 12242   <"cs1 12243   <"cs2 12487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6339  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-n0 10599  df-z 10666  df-uz 10881  df-fz 11457  df-fzo 11568  df-hash 12123  df-word 12248  df-concat 12250  df-s1 12251  df-s2 12494
This theorem is referenced by: (None)
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