Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> B e. NN0 ) |
2 |
1
|
nn0cnd |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> B e. CC ) |
3 |
|
rernegcl |
|- ( A e. RR -> ( 0 -R A ) e. RR ) |
4 |
3
|
ad2antrr |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( 0 -R A ) e. RR ) |
5 |
4
|
recnd |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( 0 -R A ) e. CC ) |
6 |
|
simpll |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> A e. RR ) |
7 |
6
|
recnd |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> A e. CC ) |
8 |
2 5 7
|
addassd |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( B + ( 0 -R A ) ) + A ) = ( B + ( ( 0 -R A ) + A ) ) ) |
9 |
|
renegid2 |
|- ( A e. RR -> ( ( 0 -R A ) + A ) = 0 ) |
10 |
9
|
ad2antrr |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( 0 -R A ) + A ) = 0 ) |
11 |
10
|
oveq2d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( B + ( ( 0 -R A ) + A ) ) = ( B + 0 ) ) |
12 |
|
nn0re |
|- ( B e. NN0 -> B e. RR ) |
13 |
|
readdrid |
|- ( B e. RR -> ( B + 0 ) = B ) |
14 |
12 13
|
syl |
|- ( B e. NN0 -> ( B + 0 ) = B ) |
15 |
14
|
adantl |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( B + 0 ) = B ) |
16 |
8 11 15
|
3eqtrrd |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> B = ( ( B + ( 0 -R A ) ) + A ) ) |
17 |
9
|
oveq1d |
|- ( A e. RR -> ( ( ( 0 -R A ) + A ) + B ) = ( 0 + B ) ) |
18 |
17
|
adantr |
|- ( ( A e. RR /\ ( 0 -R A ) e. NN ) -> ( ( ( 0 -R A ) + A ) + B ) = ( 0 + B ) ) |
19 |
|
readdlid |
|- ( B e. RR -> ( 0 + B ) = B ) |
20 |
12 19
|
syl |
|- ( B e. NN0 -> ( 0 + B ) = B ) |
21 |
18 20
|
sylan9eq |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( ( 0 -R A ) + A ) + B ) = B ) |
22 |
|
nnnn0 |
|- ( ( 0 -R A ) e. NN -> ( 0 -R A ) e. NN0 ) |
23 |
|
nn0addcom |
|- ( ( ( 0 -R A ) e. NN0 /\ B e. NN0 ) -> ( ( 0 -R A ) + B ) = ( B + ( 0 -R A ) ) ) |
24 |
22 23
|
sylan |
|- ( ( ( 0 -R A ) e. NN /\ B e. NN0 ) -> ( ( 0 -R A ) + B ) = ( B + ( 0 -R A ) ) ) |
25 |
24
|
adantll |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( 0 -R A ) + B ) = ( B + ( 0 -R A ) ) ) |
26 |
25
|
oveq1d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( ( 0 -R A ) + B ) + A ) = ( ( B + ( 0 -R A ) ) + A ) ) |
27 |
16 21 26
|
3eqtr4d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( ( 0 -R A ) + A ) + B ) = ( ( ( 0 -R A ) + B ) + A ) ) |
28 |
5 7 2
|
addassd |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( ( 0 -R A ) + A ) + B ) = ( ( 0 -R A ) + ( A + B ) ) ) |
29 |
5 2 7
|
addassd |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( ( 0 -R A ) + B ) + A ) = ( ( 0 -R A ) + ( B + A ) ) ) |
30 |
27 28 29
|
3eqtr3d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( 0 -R A ) + ( A + B ) ) = ( ( 0 -R A ) + ( B + A ) ) ) |
31 |
7 2
|
addcld |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( A + B ) e. CC ) |
32 |
2 7
|
addcld |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( B + A ) e. CC ) |
33 |
5 31 32
|
sn-addcand |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( ( ( 0 -R A ) + ( A + B ) ) = ( ( 0 -R A ) + ( B + A ) ) <-> ( A + B ) = ( B + A ) ) ) |
34 |
30 33
|
mpbid |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( A + B ) = ( B + A ) ) |