Step |
Hyp |
Ref |
Expression |
1 |
|
xrge0adddir |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> ( ( A +e B ) *e C ) = ( ( A *e C ) +e ( B *e C ) ) ) |
2 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
3 |
|
simp1 |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> A e. ( 0 [,] +oo ) ) |
4 |
2 3
|
sselid |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> A e. RR* ) |
5 |
|
simp2 |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> B e. ( 0 [,] +oo ) ) |
6 |
2 5
|
sselid |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> B e. RR* ) |
7 |
4 6
|
xaddcld |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> ( A +e B ) e. RR* ) |
8 |
|
simp3 |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> C e. ( 0 [,] +oo ) ) |
9 |
2 8
|
sselid |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> C e. RR* ) |
10 |
|
xmulcom |
|- ( ( ( A +e B ) e. RR* /\ C e. RR* ) -> ( ( A +e B ) *e C ) = ( C *e ( A +e B ) ) ) |
11 |
7 9 10
|
syl2anc |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> ( ( A +e B ) *e C ) = ( C *e ( A +e B ) ) ) |
12 |
|
xmulcom |
|- ( ( A e. RR* /\ C e. RR* ) -> ( A *e C ) = ( C *e A ) ) |
13 |
4 9 12
|
syl2anc |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> ( A *e C ) = ( C *e A ) ) |
14 |
|
xmulcom |
|- ( ( B e. RR* /\ C e. RR* ) -> ( B *e C ) = ( C *e B ) ) |
15 |
6 9 14
|
syl2anc |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> ( B *e C ) = ( C *e B ) ) |
16 |
13 15
|
oveq12d |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> ( ( A *e C ) +e ( B *e C ) ) = ( ( C *e A ) +e ( C *e B ) ) ) |
17 |
1 11 16
|
3eqtr3d |
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> ( C *e ( A +e B ) ) = ( ( C *e A ) +e ( C *e B ) ) ) |