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Theorem ofcs2 26944
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 12473 . . . 4  |-  <" A B ">  =  (
<" A "> concat  <" B "> )
21oveq1i 6099 . . 3  |-  ( <" A B ">𝑓/𝑐 R C )  =  ( ( <" A "> concat  <" B "> )𝑓/𝑐 R C )
3 simp1 988 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  A  e.  S )
43s1cld 12292 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  <" A ">  e. Word  S )
5 simp2 989 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  B  e.  S )
65s1cld 12292 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  <" B ">  e. Word  S )
7 simp3 990 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  C  e.  T )
84, 6, 7ofcccat 26940 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( ( <" A "> concat  <" B "> )𝑓/𝑐 R C )  =  ( ( <" A ">𝑓/𝑐 R C ) concat  ( <" B ">𝑓/𝑐 R C ) ) )
92, 8syl5eq 2485 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" A B ">𝑓/𝑐 R C )  =  ( ( <" A ">𝑓/𝑐 R C ) concat  ( <" B ">𝑓/𝑐 R C ) ) )
10 ofcs1 26942 . . . . 5  |-  ( ( A  e.  S  /\  C  e.  T )  ->  ( <" A ">𝑓/𝑐 R C )  =  <" ( A R C ) "> )
113, 7, 10syl2anc 661 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" A ">𝑓/𝑐 R C )  =  <" ( A R C ) "> )
12 ofcs1 26942 . . . . 5  |-  ( ( B  e.  S  /\  C  e.  T )  ->  ( <" B ">𝑓/𝑐 R C )  =  <" ( B R C ) "> )
135, 7, 12syl2anc 661 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( <" B ">𝑓/𝑐 R C )  =  <" ( B R C ) "> )
1411, 13oveq12d 6107 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( ( <" A ">𝑓/𝑐 R C ) concat  ( <" B ">𝑓/𝑐 R C ) )  =  ( <" ( A R C ) "> concat  <" ( B R C ) "> ) )
15 df-s2 12473 . . 3  |-  <" ( A R C ) ( B R C ) ">  =  (
<" ( A R C ) "> concat  <" ( B R C ) "> )
1614, 15syl6eqr 2491 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  T )  ->  ( ( <" A ">𝑓/𝑐 R C ) concat  ( <" B ">𝑓/𝑐 R C ) )  = 
<" ( A R C ) ( B R C ) "> )
179, 16eqtrd 2473 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 965    = wceq 1369    e. wcel 1756  (class class class)co 6089   concat cconcat 12221   <"cs1 12222   <"cs2 12466  ∘𝑓/𝑐cofc 26535
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-concat 12229  df-s1 12230  df-s2 12473  df-ofc 26536
This theorem is referenced by: (None)
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