Metamath Proof Explorer


Theorem decmul1

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