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Theorem ofs1 26955
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 4545 . . . 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 12302 . . . . 5  |-  ( A  e.  S  ->  <" A ">  =  { <. 0 ,  A >. } )
6 0nn0 10606 . . . . . 6  |-  0  e.  NN0
7 fmptsn 5911 . . . . . 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 ) )
11 s1val 12302 . . . . 5  |-  ( B  e.  T  ->  <" B ">  =  { <. 0 ,  B >. } )
12 fmptsn 5911 . . . . . 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 2475 . . . 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 6348 . 2  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( <" A ">  oF R
<" B "> )  =  ( i  e.  { 0 }  |->  ( A R B ) ) )
17 ovex 6128 . . . 4  |-  ( A R B )  e. 
_V
18 s1val 12302 . . . 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 5911 . . . 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 2463 . 2  |-  <" ( A R B ) ">  =  ( i  e.  { 0 } 
|->  ( A R B ) )
2316, 22syl6eqr 2493 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 1369    e. wcel 1756   _Vcvv 2984   {csn 3889   <.cop 3895    e. cmpt 4362  (class class class)co 6103    oFcof 6330   0cc0 9294   NN0cn0 10591   <"cs1 12236
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-mulcl 9356  ax-i2m1 9362
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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-n0 10592  df-s1 12244
This theorem is referenced by:  ofs2  26957
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