Step |
Hyp |
Ref |
Expression |
1 |
|
hashnn0pnf |
|- ( A e. V -> ( ( # ` A ) e. NN0 \/ ( # ` A ) = +oo ) ) |
2 |
|
df-nel |
|- ( ( # ` A ) e/ NN0 <-> -. ( # ` A ) e. NN0 ) |
3 |
2
|
anbi2i |
|- ( ( ( ( # ` A ) = +oo \/ ( # ` A ) e. NN0 ) /\ ( # ` A ) e/ NN0 ) <-> ( ( ( # ` A ) = +oo \/ ( # ` A ) e. NN0 ) /\ -. ( # ` A ) e. NN0 ) ) |
4 |
|
pm5.61 |
|- ( ( ( ( # ` A ) = +oo \/ ( # ` A ) e. NN0 ) /\ -. ( # ` A ) e. NN0 ) <-> ( ( # ` A ) = +oo /\ -. ( # ` A ) e. NN0 ) ) |
5 |
3 4
|
sylbb |
|- ( ( ( ( # ` A ) = +oo \/ ( # ` A ) e. NN0 ) /\ ( # ` A ) e/ NN0 ) -> ( ( # ` A ) = +oo /\ -. ( # ` A ) e. NN0 ) ) |
6 |
5
|
ex |
|- ( ( ( # ` A ) = +oo \/ ( # ` A ) e. NN0 ) -> ( ( # ` A ) e/ NN0 -> ( ( # ` A ) = +oo /\ -. ( # ` A ) e. NN0 ) ) ) |
7 |
6
|
orcoms |
|- ( ( ( # ` A ) e. NN0 \/ ( # ` A ) = +oo ) -> ( ( # ` A ) e/ NN0 -> ( ( # ` A ) = +oo /\ -. ( # ` A ) e. NN0 ) ) ) |
8 |
1 7
|
syl |
|- ( A e. V -> ( ( # ` A ) e/ NN0 -> ( ( # ` A ) = +oo /\ -. ( # ` A ) e. NN0 ) ) ) |
9 |
8
|
imp |
|- ( ( A e. V /\ ( # ` A ) e/ NN0 ) -> ( ( # ` A ) = +oo /\ -. ( # ` A ) e. NN0 ) ) |
10 |
9
|
3adant2 |
|- ( ( A e. V /\ B e. W /\ ( # ` A ) e/ NN0 ) -> ( ( # ` A ) = +oo /\ -. ( # ` A ) e. NN0 ) ) |
11 |
|
oveq1 |
|- ( ( # ` A ) = +oo -> ( ( # ` A ) +e ( # ` B ) ) = ( +oo +e ( # ` B ) ) ) |
12 |
|
hashxrcl |
|- ( B e. W -> ( # ` B ) e. RR* ) |
13 |
|
hashnemnf |
|- ( B e. W -> ( # ` B ) =/= -oo ) |
14 |
12 13
|
jca |
|- ( B e. W -> ( ( # ` B ) e. RR* /\ ( # ` B ) =/= -oo ) ) |
15 |
14
|
3ad2ant2 |
|- ( ( A e. V /\ B e. W /\ ( # ` A ) e/ NN0 ) -> ( ( # ` B ) e. RR* /\ ( # ` B ) =/= -oo ) ) |
16 |
|
xaddpnf2 |
|- ( ( ( # ` B ) e. RR* /\ ( # ` B ) =/= -oo ) -> ( +oo +e ( # ` B ) ) = +oo ) |
17 |
15 16
|
syl |
|- ( ( A e. V /\ B e. W /\ ( # ` A ) e/ NN0 ) -> ( +oo +e ( # ` B ) ) = +oo ) |
18 |
11 17
|
sylan9eqr |
|- ( ( ( A e. V /\ B e. W /\ ( # ` A ) e/ NN0 ) /\ ( # ` A ) = +oo ) -> ( ( # ` A ) +e ( # ` B ) ) = +oo ) |
19 |
18
|
expcom |
|- ( ( # ` A ) = +oo -> ( ( A e. V /\ B e. W /\ ( # ` A ) e/ NN0 ) -> ( ( # ` A ) +e ( # ` B ) ) = +oo ) ) |
20 |
19
|
adantr |
|- ( ( ( # ` A ) = +oo /\ -. ( # ` A ) e. NN0 ) -> ( ( A e. V /\ B e. W /\ ( # ` A ) e/ NN0 ) -> ( ( # ` A ) +e ( # ` B ) ) = +oo ) ) |
21 |
10 20
|
mpcom |
|- ( ( A e. V /\ B e. W /\ ( # ` A ) e/ NN0 ) -> ( ( # ` A ) +e ( # ` B ) ) = +oo ) |