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Theorem ofcs2 29506
Description: Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 9-Oct-2018.)
Assertion
Ref Expression
ofcs2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" A B ">𝑓/𝑐 R C )  =  <" ( A R C ) ( B R C ) "> )

Proof of Theorem ofcs2
StepHypRef Expression
1 df-s2 13003 . . . 4  |-  <" A B ">  =  (
<" A "> ++  <" B "> )
21oveq1i 6318 . . 3  |-  ( <" A B ">𝑓/𝑐 R C )  =  ( ( <" A "> ++  <" B "> )𝑓/𝑐 R C )
3 simp1 1030 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  A  e.  S )
43s1cld 12795 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  <" A ">  e. Word  S )
5 simp2 1031 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  B  e.  S )
65s1cld 12795 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  <" B ">  e. Word  S )
7 simp3 1032 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  C  e.  T )
84, 6, 7ofcccat 29502 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( ( <" A "> ++  <" B "> )𝑓/𝑐 R C )  =  ( ( <" A ">𝑓/𝑐 R C ) ++  ( <" B ">𝑓/𝑐 R C ) ) )
92, 8syl5eq 2517 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" A B ">𝑓/𝑐 R C )  =  ( ( <" A ">𝑓/𝑐 R C ) ++  ( <" B ">𝑓/𝑐 R C ) ) )
10 ofcs1 29504 . . . . 5  |-  ( ( A  e.  S  /\  C  e.  T )  ->  ( <" A ">𝑓/𝑐 R C )  =  <" ( A R C ) "> )
113, 7, 10syl2anc 673 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" A ">𝑓/𝑐 R C )  =  <" ( A R C ) "> )
12 ofcs1 29504 . . . . 5  |-  ( ( B  e.  S  /\  C  e.  T )  ->  ( <" B ">𝑓/𝑐 R C )  =  <" ( B R C ) "> )
135, 7, 12syl2anc 673 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" B ">𝑓/𝑐 R C )  =  <" ( B R C ) "> )
1411, 13oveq12d 6326 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( ( <" A ">𝑓/𝑐 R C ) ++  ( <" B ">𝑓/𝑐 R C ) )  =  ( <" ( A R C ) "> ++  <" ( B R C ) "> ) )
15 df-s2 13003 . . 3  |-  <" ( A R C ) ( B R C ) ">  =  (
<" ( A R C ) "> ++  <" ( B R C ) "> )
1614, 15syl6eqr 2523 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( ( <" A ">𝑓/𝑐 R C ) ++  ( <" B ">𝑓/𝑐 R C ) )  = 
<" ( A R C ) ( B R C ) "> )
179, 16eqtrd 2505 1  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" A B ">𝑓/𝑐 R C )  =  <" ( A R C ) ( B R C ) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 1007    = wceq 1452    e. wcel 1904  (class class class)co 6308   ++ cconcat 12705   <"cs1 12706   <"cs2 12996  ∘𝑓/𝑐cofc 28990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-ofc 28991
This theorem is referenced by: (None)
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