Step |
Hyp |
Ref |
Expression |
1 |
|
reelznn0nn |
|- ( A e. ZZ <-> ( A e. NN0 \/ ( A e. RR /\ ( 0 -R A ) e. NN ) ) ) |
2 |
|
reelznn0nn |
|- ( B e. ZZ <-> ( B e. NN0 \/ ( B e. RR /\ ( 0 -R B ) e. NN ) ) ) |
3 |
|
nn0mulcom |
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A x. B ) = ( B x. A ) ) |
4 |
|
zmulcomlem |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( A x. B ) = ( B x. A ) ) |
5 |
|
zmulcomlem |
|- ( ( ( B e. RR /\ ( 0 -R B ) e. NN ) /\ A e. NN0 ) -> ( B x. A ) = ( A x. B ) ) |
6 |
5
|
eqcomd |
|- ( ( ( B e. RR /\ ( 0 -R B ) e. NN ) /\ A e. NN0 ) -> ( A x. B ) = ( B x. A ) ) |
7 |
6
|
ancoms |
|- ( ( A e. NN0 /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( A x. B ) = ( B x. A ) ) |
8 |
|
nnmulcom |
|- ( ( ( 0 -R A ) e. NN /\ ( 0 -R B ) e. NN ) -> ( ( 0 -R A ) x. ( 0 -R B ) ) = ( ( 0 -R B ) x. ( 0 -R A ) ) ) |
9 |
8
|
ad2ant2l |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R A ) x. ( 0 -R B ) ) = ( ( 0 -R B ) x. ( 0 -R A ) ) ) |
10 |
9
|
oveq2d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( ( 0 -R A ) x. ( 0 -R B ) ) ) = ( 0 -R ( ( 0 -R B ) x. ( 0 -R A ) ) ) ) |
11 |
|
rernegcl |
|- ( A e. RR -> ( 0 -R A ) e. RR ) |
12 |
11
|
ad2antrr |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R A ) e. RR ) |
13 |
|
simprr |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R B ) e. NN ) |
14 |
12 13
|
renegmulnnass |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R B ) ) = ( 0 -R ( ( 0 -R A ) x. ( 0 -R B ) ) ) ) |
15 |
|
rernegcl |
|- ( B e. RR -> ( 0 -R B ) e. RR ) |
16 |
15
|
ad2antrl |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R B ) e. RR ) |
17 |
|
simplr |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R A ) e. NN ) |
18 |
16 17
|
renegmulnnass |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R A ) ) = ( 0 -R ( ( 0 -R B ) x. ( 0 -R A ) ) ) ) |
19 |
10 14 18
|
3eqtr4d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R B ) ) = ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R A ) ) ) |
20 |
19
|
oveq2d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R B ) ) ) = ( 0 -R ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R A ) ) ) ) |
21 |
|
rernegcl |
|- ( ( 0 -R A ) e. RR -> ( 0 -R ( 0 -R A ) ) e. RR ) |
22 |
11 21
|
syl |
|- ( A e. RR -> ( 0 -R ( 0 -R A ) ) e. RR ) |
23 |
22
|
ad2antrr |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( 0 -R A ) ) e. RR ) |
24 |
23 16
|
remulneg2d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R ( 0 -R B ) ) ) = ( 0 -R ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R B ) ) ) ) |
25 |
|
rernegcl |
|- ( ( 0 -R B ) e. RR -> ( 0 -R ( 0 -R B ) ) e. RR ) |
26 |
15 25
|
syl |
|- ( B e. RR -> ( 0 -R ( 0 -R B ) ) e. RR ) |
27 |
26
|
ad2antrl |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( 0 -R B ) ) e. RR ) |
28 |
27 12
|
remulneg2d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R ( 0 -R A ) ) ) = ( 0 -R ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R A ) ) ) ) |
29 |
20 24 28
|
3eqtr4d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R ( 0 -R B ) ) ) = ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R ( 0 -R A ) ) ) ) |
30 |
|
renegneg |
|- ( A e. RR -> ( 0 -R ( 0 -R A ) ) = A ) |
31 |
30
|
ad2antrr |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( 0 -R A ) ) = A ) |
32 |
|
renegneg |
|- ( B e. RR -> ( 0 -R ( 0 -R B ) ) = B ) |
33 |
32
|
ad2antrl |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( 0 -R B ) ) = B ) |
34 |
31 33
|
oveq12d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R ( 0 -R B ) ) ) = ( A x. B ) ) |
35 |
33 31
|
oveq12d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R ( 0 -R A ) ) ) = ( B x. A ) ) |
36 |
29 34 35
|
3eqtr3d |
|- ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( A x. B ) = ( B x. A ) ) |
37 |
3 4 7 36
|
ccase |
|- ( ( ( A e. NN0 \/ ( A e. RR /\ ( 0 -R A ) e. NN ) ) /\ ( B e. NN0 \/ ( B e. RR /\ ( 0 -R B ) e. NN ) ) ) -> ( A x. B ) = ( B x. A ) ) |
38 |
1 2 37
|
syl2anb |
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A x. B ) = ( B x. A ) ) |