Database
REAL AND COMPLEX NUMBERS
Integer sets
Decimal arithmetic
decmul1
Metamath Proof Explorer
Description: The product of a numeral with a number (no carry). (Contributed by AV , 22-Jul-2021) (Revised by AV , 6-Sep-2021) Remove hypothesis
D e. NN0 . (Revised by Steven Nguyen , 7-Dec-2022)
Ref
Expression
Hypotheses
decmul1.p
|- P e. NN0
decmul1.a
|- A e. NN0
decmul1.b
|- B e. NN0
decmul1.n
|- N = ; A B
decmul1.c
|- ( A x. P ) = C
decmul1.d
|- ( B x. P ) = D
Assertion
decmul1
|- ( N x. P ) = ; C D
Proof
Step
Hyp
Ref
Expression
1
decmul1.p
|- P e. NN0
2
decmul1.a
|- A e. NN0
3
decmul1.b
|- B e. NN0
4
decmul1.n
|- N = ; A B
5
decmul1.c
|- ( A x. P ) = C
6
decmul1.d
|- ( B x. P ) = D
7
2 3
deccl
|- ; A B e. NN0
8
4 7
eqeltri
|- N e. NN0
9
8 1
num0u
|- ( N x. P ) = ( ( N x. P ) + 0 )
10
0nn0
|- 0 e. NN0
11
3 1
nn0mulcli
|- ( B x. P ) e. NN0
12
11
nn0cni
|- ( B x. P ) e. CC
13
12
addid1i
|- ( ( B x. P ) + 0 ) = ( B x. P )
14
13 6
eqtri
|- ( ( B x. P ) + 0 ) = D
15
2 3 10 4 1 5 14
decrmanc
|- ( ( N x. P ) + 0 ) = ; C D
16
9 15
eqtri
|- ( N x. P ) = ; C D