Metamath Proof Explorer

Theorem hashinfxadd

Description: The extended real addition of the size of an infinite set with the size of an arbitrary set yields plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017)

Ref Expression
Assertion hashinfxadd 
|- ( ( A e. V /\ B e. W /\ ( # ` A ) e/ NN0 ) -> ( ( # ` A ) +e ( # ` B ) ) = +oo )

Proof

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 )