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Theorem ofs1 28250
Description: Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
Assertion
Ref Expression
ofs1  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( <" A ">  oF R
<" B "> )  =  <" ( A R B ) "> )

Proof of Theorem ofs1
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 snex 4688 . . . 4  |-  { 0 }  e.  _V
21a1i 11 . . 3  |-  ( ( A  e.  S  /\  B  e.  T )  ->  { 0 }  e.  _V )
3 simpll 753 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  T
)  /\  i  e.  { 0 } )  ->  A  e.  S )
4 simplr 754 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  T
)  /\  i  e.  { 0 } )  ->  B  e.  T )
5 s1val 12576 . . . . 5  |-  ( A  e.  S  ->  <" A ">  =  { <. 0 ,  A >. } )
6 0nn0 10811 . . . . . 6  |-  0  e.  NN0
7 fmptsn 6082 . . . . . 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 2508 . . . 4  |-  ( A  e.  S  ->  <" A ">  =  ( i  e.  { 0 } 
|->  A ) )
109adantr 465 . . 3  |-  ( ( A  e.  S  /\  B  e.  T )  ->  <" A ">  =  ( i  e. 
{ 0 }  |->  A ) )
11 s1val 12576 . . . . 5  |-  ( B  e.  T  ->  <" B ">  =  { <. 0 ,  B >. } )
12 fmptsn 6082 . . . . . 6  |-  ( ( 0  e.  NN0  /\  B  e.  T )  ->  { <. 0 ,  B >. }  =  ( i  e.  { 0 } 
|->  B ) )
136, 12mpan 670 . . . . 5  |-  ( B  e.  T  ->  { <. 0 ,  B >. }  =  ( i  e. 
{ 0 }  |->  B ) )
1411, 13eqtrd 2508 . . . 4  |-  ( B  e.  T  ->  <" B ">  =  ( i  e.  { 0 } 
|->  B ) )
1514adantl 466 . . 3  |-  ( ( A  e.  S  /\  B  e.  T )  ->  <" B ">  =  ( i  e. 
{ 0 }  |->  B ) )
162, 3, 4, 10, 15offval2 6541 . 2  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( <" A ">  oF R
<" B "> )  =  ( i  e.  { 0 }  |->  ( A R B ) ) )
17 ovex 6310 . . . 4  |-  ( A R B )  e. 
_V
18 s1val 12576 . . . 4  |-  ( ( A R B )  e.  _V  ->  <" ( A R B ) ">  =  { <. 0 ,  ( A R B ) >. } )
1917, 18ax-mp 5 . . 3  |-  <" ( A R B ) ">  =  { <. 0 ,  ( A R B ) >. }
20 fmptsn 6082 . . . 4  |-  ( ( 0  e.  NN0  /\  ( A R B )  e.  _V )  ->  { <. 0 ,  ( A R B )
>. }  =  ( i  e.  { 0 } 
|->  ( A R B ) ) )
216, 17, 20mp2an 672 . . 3  |-  { <. 0 ,  ( A R B ) >. }  =  ( i  e.  {
0 }  |->  ( A R B ) )
2219, 21eqtri 2496 . 2  |-  <" ( A R B ) ">  =  ( i  e.  { 0 } 
|->  ( A R B ) )
2316, 22syl6eqr 2526 1  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( <" A ">  oF R
<" B "> )  =  <" ( A R B ) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027   <.cop 4033    |-> cmpt 4505  (class class class)co 6285    oFcof 6523   0cc0 9493   NN0cn0 10796   <"cs1 12504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-mulcl 9555  ax-i2m1 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-n0 10797  df-s1 12512
This theorem is referenced by:  ofs2  28252
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