Metamath Proof Explorer

Theorem fmtno2

Description: The 2 nd Fermat number, see remark in ApostolNT p. 7. (Contributed by AV, 13-Jun-2021)

Ref Expression
Assertion fmtno2 
|- ( FermatNo ` 2 ) = ; 1 7

Proof

Step Hyp Ref Expression
1 2nn0 
 |-  2 e. NN0
2 fmtno 
 |-  ( 2 e. NN0 -> ( FermatNo ` 2 ) = ( ( 2 ^ ( 2 ^ 2 ) ) + 1 ) )
3 1 2 ax-mp 
 |-  ( FermatNo ` 2 ) = ( ( 2 ^ ( 2 ^ 2 ) ) + 1 )
4 sq2 
 |-  ( 2 ^ 2 ) = 4
5 4 oveq2i 
 |-  ( 2 ^ ( 2 ^ 2 ) ) = ( 2 ^ 4 )
6 5 oveq1i 
 |-  ( ( 2 ^ ( 2 ^ 2 ) ) + 1 ) = ( ( 2 ^ 4 ) + 1 )
7 2exp4 
 |-  ( 2 ^ 4 ) = ; 1 6
8 7 oveq1i 
 |-  ( ( 2 ^ 4 ) + 1 ) = ( ; 1 6 + 1 )
9 1nn0 
 |-  1 e. NN0
10 6nn0 
 |-  6 e. NN0
11 6p1e7 
 |-  ( 6 + 1 ) = 7
12 eqid 
 |-  ; 1 6 = ; 1 6
13 9 10 11 12 decsuc 
 |-  ( ; 1 6 + 1 ) = ; 1 7
14 6 8 13 3eqtri 
 |-  ( ( 2 ^ ( 2 ^ 2 ) ) + 1 ) = ; 1 7
15 3 14 eqtri 
 |-  ( FermatNo ` 2 ) = ; 1 7