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Theorem ofcs1 26949
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 4538 . . . 4  |-  { 0 }  e.  _V
21a1i 11 . . 3  |-  ( ( A  e.  S  /\  B  e.  T )  ->  { 0 }  e.  _V )
3 simpr 461 . . 3  |-  ( ( A  e.  S  /\  B  e.  T )  ->  B  e.  T )
4 simpll 753 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  T
)  /\  i  e.  { 0 } )  ->  A  e.  S )
5 s1val 12295 . . . . 5  |-  ( A  e.  S  ->  <" A ">  =  { <. 0 ,  A >. } )
6 0nn0 10599 . . . . . 6  |-  0  e.  NN0
7 fmptsn 5904 . . . . . 6  |-  ( ( 0  e.  NN0  /\  A  e.  S )  ->  { <. 0 ,  A >. }  =  ( i  e.  { 0 } 
|->  A ) )
86, 7mpan 670 . . . . 5  |-  ( A  e.  S  ->  { <. 0 ,  A >. }  =  ( i  e. 
{ 0 }  |->  A ) )
95, 8eqtrd 2475 . . . 4  |-  ( A  e.  S  ->  <" A ">  =  ( i  e.  { 0 } 
|->  A ) )
109adantr 465 . . 3  |-  ( ( A  e.  S  /\  B  e.  T )  ->  <" A ">  =  ( i  e. 
{ 0 }  |->  A ) )
112, 3, 4, 10ofcfval2 26551 . 2  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( <" A ">𝑓/𝑐 R B )  =  ( i  e.  { 0 }  |->  ( A R B ) ) )
12 ovex 6121 . . . 4  |-  ( A R B )  e. 
_V
13 s1val 12295 . . . 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 5904 . . . 4  |-  ( ( 0  e.  NN0  /\  ( A R B )  e.  _V )  ->  { <. 0 ,  ( A R B )
>. }  =  ( i  e.  { 0 } 
|->  ( A R B ) ) )
166, 12, 15mp2an 672 . . 3  |-  { <. 0 ,  ( A R B ) >. }  =  ( i  e.  {
0 }  |->  ( A R B ) )
1714, 16eqtri 2463 . 2  |-  <" ( A R B ) ">  =  ( i  e.  { 0 } 
|->  ( A R B ) )
1811, 17syl6eqr 2493 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 369    = wceq 1369    e. wcel 1756   _Vcvv 2977   {csn 3882   <.cop 3888    e. cmpt 4355  (class class class)co 6096   0cc0 9287   NN0cn0 10584   <"cs1 12229  ∘𝑓/𝑐cofc 26542
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-mulcl 9349  ax-i2m1 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  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-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-n0 10585  df-s1 12237  df-ofc 26543
This theorem is referenced by:  ofcs2  26951
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