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Theorem ofcs2 29222
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 12929 . . . 4  |-  <" A B ">  =  (
<" A "> ++  <" B "> )
21oveq1i 6315 . . 3  |-  ( <" A B ">𝑓/𝑐 R C )  =  ( ( <" A "> ++  <" B "> )𝑓/𝑐 R C )
3 simp1 1005 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  A  e.  S )
43s1cld 12729 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  <" A ">  e. Word  S )
5 simp2 1006 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  B  e.  S )
65s1cld 12729 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  <" B ">  e. Word  S )
7 simp3 1007 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  C  e.  T )
84, 6, 7ofcccat 29218 . . 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 2482 . 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 29220 . . . . 5  |-  ( ( A  e.  S  /\  C  e.  T )  ->  ( <" A ">𝑓/𝑐 R C )  =  <" ( A R C ) "> )
113, 7, 10syl2anc 665 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" A ">𝑓/𝑐 R C )  =  <" ( A R C ) "> )
12 ofcs1 29220 . . . . 5  |-  ( ( B  e.  S  /\  C  e.  T )  ->  ( <" B ">𝑓/𝑐 R C )  =  <" ( B R C ) "> )
135, 7, 12syl2anc 665 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" B ">𝑓/𝑐 R C )  =  <" ( B R C ) "> )
1411, 13oveq12d 6323 . . 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 12929 . . 3  |-  <" ( A R C ) ( B R C ) ">  =  (
<" ( A R C ) "> ++  <" ( B R C ) "> )
1614, 15syl6eqr 2488 . 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 2470 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 982    = wceq 1437    e. wcel 1870  (class class class)co 6305   ++ cconcat 12645   <"cs1 12646   <"cs2 12922  ∘𝑓/𝑐cofc 28755
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-concat 12653  df-s1 12654  df-s2 12929  df-ofc 28756
This theorem is referenced by: (None)
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