Step |
Hyp |
Ref |
Expression |
1 |
|
simpll |
|- ( ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) /\ x e. RR+ ) -> ( A C_ RR /\ ( vol* ` A ) e. RR ) ) |
2 |
|
simplr |
|- ( ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) /\ x e. RR+ ) -> ( B C_ RR /\ ( vol* ` B ) e. RR ) ) |
3 |
|
simpr |
|- ( ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) /\ x e. RR+ ) -> x e. RR+ ) |
4 |
1 2 3
|
ovolunlem2 |
|- ( ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) /\ x e. RR+ ) -> ( vol* ` ( A u. B ) ) <_ ( ( ( vol* ` A ) + ( vol* ` B ) ) + x ) ) |
5 |
4
|
ralrimiva |
|- ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) -> A. x e. RR+ ( vol* ` ( A u. B ) ) <_ ( ( ( vol* ` A ) + ( vol* ` B ) ) + x ) ) |
6 |
|
unss |
|- ( ( A C_ RR /\ B C_ RR ) <-> ( A u. B ) C_ RR ) |
7 |
6
|
biimpi |
|- ( ( A C_ RR /\ B C_ RR ) -> ( A u. B ) C_ RR ) |
8 |
7
|
ad2ant2r |
|- ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) -> ( A u. B ) C_ RR ) |
9 |
|
ovolcl |
|- ( ( A u. B ) C_ RR -> ( vol* ` ( A u. B ) ) e. RR* ) |
10 |
8 9
|
syl |
|- ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) -> ( vol* ` ( A u. B ) ) e. RR* ) |
11 |
|
readdcl |
|- ( ( ( vol* ` A ) e. RR /\ ( vol* ` B ) e. RR ) -> ( ( vol* ` A ) + ( vol* ` B ) ) e. RR ) |
12 |
11
|
ad2ant2l |
|- ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) -> ( ( vol* ` A ) + ( vol* ` B ) ) e. RR ) |
13 |
|
xralrple |
|- ( ( ( vol* ` ( A u. B ) ) e. RR* /\ ( ( vol* ` A ) + ( vol* ` B ) ) e. RR ) -> ( ( vol* ` ( A u. B ) ) <_ ( ( vol* ` A ) + ( vol* ` B ) ) <-> A. x e. RR+ ( vol* ` ( A u. B ) ) <_ ( ( ( vol* ` A ) + ( vol* ` B ) ) + x ) ) ) |
14 |
10 12 13
|
syl2anc |
|- ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) -> ( ( vol* ` ( A u. B ) ) <_ ( ( vol* ` A ) + ( vol* ` B ) ) <-> A. x e. RR+ ( vol* ` ( A u. B ) ) <_ ( ( ( vol* ` A ) + ( vol* ` B ) ) + x ) ) ) |
15 |
5 14
|
mpbird |
|- ( ( ( A C_ RR /\ ( vol* ` A ) e. RR ) /\ ( B C_ RR /\ ( vol* ` B ) e. RR ) ) -> ( vol* ` ( A u. B ) ) <_ ( ( vol* ` A ) + ( vol* ` B ) ) ) |