Metamath Proof Explorer


Theorem xaddmnf2

Description: Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xaddmnf2
|- ( ( A e. RR* /\ A =/= +oo ) -> ( -oo +e A ) = -oo )

Proof

Step Hyp Ref Expression
1 mnfxr
 |-  -oo e. RR*
2 xaddval
 |-  ( ( -oo e. RR* /\ A e. RR* ) -> ( -oo +e A ) = if ( -oo = +oo , if ( A = -oo , 0 , +oo ) , if ( -oo = -oo , if ( A = +oo , 0 , -oo ) , if ( A = +oo , +oo , if ( A = -oo , -oo , ( -oo + A ) ) ) ) ) )
3 1 2 mpan
 |-  ( A e. RR* -> ( -oo +e A ) = if ( -oo = +oo , if ( A = -oo , 0 , +oo ) , if ( -oo = -oo , if ( A = +oo , 0 , -oo ) , if ( A = +oo , +oo , if ( A = -oo , -oo , ( -oo + A ) ) ) ) ) )
4 mnfnepnf
 |-  -oo =/= +oo
5 ifnefalse
 |-  ( -oo =/= +oo -> if ( -oo = +oo , if ( A = -oo , 0 , +oo ) , if ( -oo = -oo , if ( A = +oo , 0 , -oo ) , if ( A = +oo , +oo , if ( A = -oo , -oo , ( -oo + A ) ) ) ) ) = if ( -oo = -oo , if ( A = +oo , 0 , -oo ) , if ( A = +oo , +oo , if ( A = -oo , -oo , ( -oo + A ) ) ) ) )
6 4 5 ax-mp
 |-  if ( -oo = +oo , if ( A = -oo , 0 , +oo ) , if ( -oo = -oo , if ( A = +oo , 0 , -oo ) , if ( A = +oo , +oo , if ( A = -oo , -oo , ( -oo + A ) ) ) ) ) = if ( -oo = -oo , if ( A = +oo , 0 , -oo ) , if ( A = +oo , +oo , if ( A = -oo , -oo , ( -oo + A ) ) ) )
7 eqid
 |-  -oo = -oo
8 7 iftruei
 |-  if ( -oo = -oo , if ( A = +oo , 0 , -oo ) , if ( A = +oo , +oo , if ( A = -oo , -oo , ( -oo + A ) ) ) ) = if ( A = +oo , 0 , -oo )
9 6 8 eqtri
 |-  if ( -oo = +oo , if ( A = -oo , 0 , +oo ) , if ( -oo = -oo , if ( A = +oo , 0 , -oo ) , if ( A = +oo , +oo , if ( A = -oo , -oo , ( -oo + A ) ) ) ) ) = if ( A = +oo , 0 , -oo )
10 ifnefalse
 |-  ( A =/= +oo -> if ( A = +oo , 0 , -oo ) = -oo )
11 9 10 eqtrid
 |-  ( A =/= +oo -> if ( -oo = +oo , if ( A = -oo , 0 , +oo ) , if ( -oo = -oo , if ( A = +oo , 0 , -oo ) , if ( A = +oo , +oo , if ( A = -oo , -oo , ( -oo + A ) ) ) ) ) = -oo )
12 3 11 sylan9eq
 |-  ( ( A e. RR* /\ A =/= +oo ) -> ( -oo +e A ) = -oo )