Metamath Proof Explorer


Theorem 1t10e1p1e11

Description: 11 is 1 times 10 to the power of 1, plus 1. (Contributed by AV, 4-Aug-2020) (Revised by AV, 9-Sep-2021)

Ref Expression
Assertion 1t10e1p1e11
|- ; 1 1 = ( ( 1 x. ( ; 1 0 ^ 1 ) ) + 1 )

Proof

Step Hyp Ref Expression
1 dfdec10
 |-  ; 1 1 = ( ( ; 1 0 x. 1 ) + 1 )
2 ax-1cn
 |-  1 e. CC
3 10nn
 |-  ; 1 0 e. NN
4 3 nncni
 |-  ; 1 0 e. CC
5 exp1
 |-  ( ; 1 0 e. CC -> ( ; 1 0 ^ 1 ) = ; 1 0 )
6 4 5 ax-mp
 |-  ( ; 1 0 ^ 1 ) = ; 1 0
7 6 eqcomi
 |-  ; 1 0 = ( ; 1 0 ^ 1 )
8 7 oveq2i
 |-  ( 1 x. ; 1 0 ) = ( 1 x. ( ; 1 0 ^ 1 ) )
9 2 4 8 mulcomli
 |-  ( ; 1 0 x. 1 ) = ( 1 x. ( ; 1 0 ^ 1 ) )
10 9 oveq1i
 |-  ( ( ; 1 0 x. 1 ) + 1 ) = ( ( 1 x. ( ; 1 0 ^ 1 ) ) + 1 )
11 1 10 eqtri
 |-  ; 1 1 = ( ( 1 x. ( ; 1 0 ^ 1 ) ) + 1 )