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Theorem ofcs1 29220
Description: Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
Assertion
Ref Expression
ofcs1  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( <" A ">𝑓/𝑐 R B )  =  <" ( A R B ) "> )

Proof of Theorem ofcs1
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 snex 4663 . . . 4  |-  { 0 }  e.  _V
21a1i 11 . . 3  |-  ( ( A  e.  S  /\  B  e.  T )  ->  { 0 }  e.  _V )
3 simpr 462 . . 3  |-  ( ( A  e.  S  /\  B  e.  T )  ->  B  e.  T )
4 simpll 758 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  T
)  /\  i  e.  { 0 } )  ->  A  e.  S )
5 s1val 12724 . . . . 5  |-  ( A  e.  S  ->  <" A ">  =  { <. 0 ,  A >. } )
6 0nn0 10884 . . . . . 6  |-  0  e.  NN0
7 fmptsn 6099 . . . . . 6  |-  ( ( 0  e.  NN0  /\  A  e.  S )  ->  { <. 0 ,  A >. }  =  ( i  e.  { 0 } 
|->  A ) )
86, 7mpan 674 . . . . 5  |-  ( A  e.  S  ->  { <. 0 ,  A >. }  =  ( i  e. 
{ 0 }  |->  A ) )
95, 8eqtrd 2470 . . . 4  |-  ( A  e.  S  ->  <" A ">  =  ( i  e.  { 0 } 
|->  A ) )
109adantr 466 . . 3  |-  ( ( A  e.  S  /\  B  e.  T )  ->  <" A ">  =  ( i  e. 
{ 0 }  |->  A ) )
112, 3, 4, 10ofcfval2 28764 . 2  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( <" A ">𝑓/𝑐 R B )  =  ( i  e.  { 0 }  |->  ( A R B ) ) )
12 ovex 6333 . . . 4  |-  ( A R B )  e. 
_V
13 s1val 12724 . . . 4  |-  ( ( A R B )  e.  _V  ->  <" ( A R B ) ">  =  { <. 0 ,  ( A R B ) >. } )
1412, 13ax-mp 5 . . 3  |-  <" ( A R B ) ">  =  { <. 0 ,  ( A R B ) >. }
15 fmptsn 6099 . . . 4  |-  ( ( 0  e.  NN0  /\  ( A R B )  e.  _V )  ->  { <. 0 ,  ( A R B )
>. }  =  ( i  e.  { 0 } 
|->  ( A R B ) ) )
166, 12, 15mp2an 676 . . 3  |-  { <. 0 ,  ( A R B ) >. }  =  ( i  e.  {
0 }  |->  ( A R B ) )
1714, 16eqtri 2458 . 2  |-  <" ( A R B ) ">  =  ( i  e.  { 0 } 
|->  ( A R B ) )
1811, 17syl6eqr 2488 1  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( <" A ">𝑓/𝑐 R B )  =  <" ( A R B ) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   {csn 4002   <.cop 4008    |-> cmpt 4484  (class class class)co 6305   0cc0 9538   NN0cn0 10869   <"cs1 12646  ∘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-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-mulcl 9600  ax-i2m1 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  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-ral 2787  df-rex 2788  df-reu 2789  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-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  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-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-n0 10870  df-s1 12654  df-ofc 28756
This theorem is referenced by:  ofcs2  29222
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