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Theorem ofs2 29247
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 12918 . . . 4  |-  <" A B ">  =  (
<" A "> ++  <" B "> )
2 df-s2 12918 . . . 4  |-  <" C D ">  =  (
<" C "> ++  <" D "> )
31, 2oveq12i 6308 . . 3  |-  ( <" A B ">  oF R <" C D "> )  =  ( ( <" A "> ++  <" B "> )  oF R (
<" C "> ++  <" D "> ) )
4 simpll 758 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  A  e.  S )
54s1cld 12718 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  <" A ">  e. Word  S )
6 simplr 760 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  B  e.  S )
76s1cld 12718 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  <" B ">  e. Word  S )
8 simprl 762 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  C  e.  T )
98s1cld 12718 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  <" C ">  e. Word  T )
10 simprr 764 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  D  e.  T )
1110s1cld 12718 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  ->  <" D ">  e. Word  T )
12 s1len 12720 . . . . . 6  |-  ( # `  <" A "> )  =  1
13 s1len 12720 . . . . . 6  |-  ( # `  <" C "> )  =  1
1412, 13eqtr4i 2452 . . . . 5  |-  ( # `  <" A "> )  =  ( # `
 <" C "> )
1514a1i 11 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( # `  <" A "> )  =  (
# `  <" C "> ) )
16 s1len 12720 . . . . . 6  |-  ( # `  <" B "> )  =  1
17 s1len 12720 . . . . . 6  |-  ( # `  <" D "> )  =  1
1816, 17eqtr4i 2452 . . . . 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 29243 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( ( <" A "> ++  <" B "> )  oF
R ( <" C "> ++  <" D "> ) )  =  ( ( <" A ">  oF R
<" C "> ) ++  ( <" B ">  oF R
<" D "> ) ) )
213, 20syl5eq 2473 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( <" A B ">  oF R <" C D "> )  =  ( ( <" A ">  oF R
<" C "> ) ++  ( <" B ">  oF R
<" D "> ) ) )
22 ofs1 29245 . . . . 5  |-  ( ( A  e.  S  /\  C  e.  T )  ->  ( <" A ">  oF R
<" C "> )  =  <" ( A R C ) "> )
234, 8, 22syl2anc 665 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( <" A ">  oF R <" C "> )  =  <" ( A R C ) "> )
24 ofs1 29245 . . . . 5  |-  ( ( B  e.  S  /\  D  e.  T )  ->  ( <" B ">  oF R
<" D "> )  =  <" ( B R D ) "> )
256, 10, 24syl2anc 665 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( <" B ">  oF R <" D "> )  =  <" ( B R D ) "> )
2623, 25oveq12d 6314 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( ( <" A ">  oF R
<" C "> ) ++  ( <" B ">  oF R
<" D "> ) )  =  (
<" ( A R C ) "> ++  <" ( B R D ) "> ) )
27 df-s2 12918 . . 3  |-  <" ( A R C ) ( B R D ) ">  =  (
<" ( A R C ) "> ++  <" ( B R D ) "> )
2826, 27syl6eqr 2479 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  T  /\  D  e.  T ) )  -> 
( ( <" A ">  oF R
<" C "> ) ++  ( <" B ">  oF R
<" D "> ) )  =  <" ( A R C ) ( B R D ) "> )
2921, 28eqtrd 2461 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 370    = wceq 1437    e. wcel 1867   ` cfv 5592  (class class class)co 6296    oFcof 6534   1c1 9529   #chash 12501   ++ cconcat 12634   <"cs1 12635   <"cs2 12911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-fzo 11903  df-hash 12502  df-word 12640  df-concat 12642  df-s1 12643  df-s2 12918
This theorem is referenced by: (None)
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