ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xaddass Unicode version

Theorem xaddass 9539
Description: Associativity of extended real addition. The correct condition here is "it is not the case that both +oo and -oo appear as one of  A ,  B ,  C, i.e.  -.  { +oo , -oo }  C_  { A ,  B ,  C }", but this condition is difficult to work with, so we break the theorem into two parts: this one, where -oo is not present in  A ,  B ,  C, and xaddass2 9540, where +oo is not present. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddass  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )

Proof of Theorem xaddass
StepHypRef Expression
1 recn 7671 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
2 recn 7671 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  CC )
3 recn 7671 . . . . . . . . . 10  |-  ( C  e.  RR  ->  C  e.  CC )
4 addass 7668 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
51, 2, 3, 4syl3an 1239 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
653expa 1162 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
7 readdcl 7664 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
8 rexadd 9522 . . . . . . . . 9  |-  ( ( ( A  +  B
)  e.  RR  /\  C  e.  RR )  ->  ( ( A  +  B ) +e
C )  =  ( ( A  +  B
)  +  C ) )
97, 8sylan 279 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  +  B ) +e C )  =  ( ( A  +  B )  +  C
) )
10 readdcl 7664 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
11 rexadd 9522 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  +  C
)  e.  RR )  ->  ( A +e ( B  +  C ) )  =  ( A  +  ( B  +  C ) ) )
1210, 11sylan2 282 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( A +e ( B  +  C ) )  =  ( A  +  ( B  +  C
) ) )
1312anassrs 395 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A +e ( B  +  C ) )  =  ( A  +  ( B  +  C ) ) )
146, 9, 133eqtr4d 2155 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  +  B ) +e C )  =  ( A +e
( B  +  C
) ) )
15 rexadd 9522 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
1615adantr 272 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A +e B )  =  ( A  +  B
) )
1716oveq1d 5741 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A +e B ) +e C )  =  ( ( A  +  B ) +e C ) )
18 rexadd 9522 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  =  ( B  +  C ) )
1918adantll 465 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( B +e C )  =  ( B  +  C
) )
2019oveq2d 5742 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A +e ( B +e C ) )  =  ( A +e ( B  +  C ) ) )
2114, 17, 203eqtr4d 2155 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )
2221adantll 465 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  ( A  e.  RR  /\  B  e.  RR ) )  /\  C  e.  RR )  ->  (
( A +e
B ) +e
C )  =  ( A +e ( B +e C ) ) )
23 oveq2 5734 . . . . . . . . 9  |-  ( C  = +oo  ->  (
( A +e
B ) +e
C )  =  ( ( A +e
B ) +e +oo ) )
24 simp1l 986 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  A  e.  RR* )
25 simp2l 988 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  B  e.  RR* )
26 xaddcl 9530 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
2724, 25, 26syl2anc 406 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A +e B )  e.  RR* )
28 xaddnemnf 9527 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )
)  ->  ( A +e B )  =/= -oo )
29283adant3 982 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A +e B )  =/= -oo )
30 xaddpnf1 9516 . . . . . . . . . 10  |-  ( ( ( A +e
B )  e.  RR*  /\  ( A +e
B )  =/= -oo )  ->  ( ( A +e B ) +e +oo )  = +oo )
3127, 29, 30syl2anc 406 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( ( A +e B ) +e +oo )  = +oo )
3223, 31sylan9eqr 2167 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  C  = +oo )  ->  ( ( A +e B ) +e C )  = +oo )
33 xaddpnf1 9516 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
34333ad2ant1 983 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A +e +oo )  = +oo )
3534adantr 272 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  C  = +oo )  ->  ( A +e +oo )  = +oo )
3632, 35eqtr4d 2148 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  C  = +oo )  ->  ( ( A +e B ) +e C )  =  ( A +e +oo ) )
37 oveq2 5734 . . . . . . . . 9  |-  ( C  = +oo  ->  ( B +e C )  =  ( B +e +oo ) )
38 xaddpnf1 9516 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
39383ad2ant2 984 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( B +e +oo )  = +oo )
4037, 39sylan9eqr 2167 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  C  = +oo )  ->  ( B +e
C )  = +oo )
4140oveq2d 5742 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  C  = +oo )  ->  ( A +e
( B +e
C ) )  =  ( A +e +oo ) )
4236, 41eqtr4d 2148 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  C  = +oo )  ->  ( ( A +e B ) +e C )  =  ( A +e
( B +e
C ) ) )
4342adantlr 466 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  ( A  e.  RR  /\  B  e.  RR ) )  /\  C  = +oo )  ->  (
( A +e
B ) +e
C )  =  ( A +e ( B +e C ) ) )
44 simp3 964 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( C  e.  RR*  /\  C  =/= -oo ) )
45 xrnemnf 9451 . . . . . . 7  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  <->  ( C  e.  RR  \/  C  = +oo ) )
4644, 45sylib 121 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( C  e.  RR  \/  C  = +oo ) )
4746adantr 272 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  ( C  e.  RR  \/  C  = +oo ) )
4822, 43, 47mpjaodan 770 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )
4948anassrs 395 . . 3  |-  ( ( ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  e.  RR )  /\  B  e.  RR )  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )
50 xaddpnf2 9517 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
51503ad2ant3 985 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( +oo +e C )  = +oo )
5251, 34eqtr4d 2148 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( +oo +e C )  =  ( A +e +oo ) )
5352adantr 272 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( +oo +e
C )  =  ( A +e +oo ) )
54 oveq2 5734 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
5554, 34sylan9eqr 2167 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
5655oveq1d 5741 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( ( A +e B ) +e C )  =  ( +oo +e
C ) )
57 oveq1 5733 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e C )  =  ( +oo +e C ) )
5857, 51sylan9eqr 2167 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( B +e
C )  = +oo )
5958oveq2d 5742 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( A +e
( B +e
C ) )  =  ( A +e +oo ) )
6053, 56, 593eqtr4d 2155 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( ( A +e B ) +e C )  =  ( A +e
( B +e
C ) ) )
6160adantlr 466 . . 3  |-  ( ( ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )
62 simpl2 966 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  e.  RR )  ->  ( B  e.  RR*  /\  B  =/= -oo )
)
63 xrnemnf 9451 . . . 4  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  <->  ( B  e.  RR  \/  B  = +oo ) )
6462, 63sylib 121 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  e.  RR )  ->  ( B  e.  RR  \/  B  = +oo ) )
6549, 61, 64mpjaodan 770 . 2  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  e.  RR )  ->  ( ( A +e B ) +e C )  =  ( A +e
( B +e
C ) ) )
66 simpl3 967 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( C  e.  RR*  /\  C  =/= -oo )
)
6766, 50syl 14 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( +oo +e
C )  = +oo )
68 simpl2l 1015 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  B  e.  RR* )
69 simpl3l 1017 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  C  e.  RR* )
70 xaddcl 9530 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B +e C )  e.  RR* )
7168, 69, 70syl2anc 406 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( B +e
C )  e.  RR* )
72 simpl2 966 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( B  e.  RR*  /\  B  =/= -oo )
)
73 xaddnemnf 9527 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( B +e C )  =/= -oo )
7472, 66, 73syl2anc 406 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( B +e
C )  =/= -oo )
75 xaddpnf2 9517 . . . . 5  |-  ( ( ( B +e
C )  e.  RR*  /\  ( B +e
C )  =/= -oo )  ->  ( +oo +e ( B +e C ) )  = +oo )
7671, 74, 75syl2anc 406 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( +oo +e
( B +e
C ) )  = +oo )
7767, 76eqtr4d 2148 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( +oo +e
C )  =  ( +oo +e ( B +e C ) ) )
78 simpr 109 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  A  = +oo )
7978oveq1d 5741 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( A +e
B )  =  ( +oo +e B ) )
80 xaddpnf2 9517 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
8172, 80syl 14 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( +oo +e
B )  = +oo )
8279, 81eqtrd 2145 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( A +e
B )  = +oo )
8382oveq1d 5741 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( ( A +e B ) +e C )  =  ( +oo +e
C ) )
8478oveq1d 5741 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( A +e
( B +e
C ) )  =  ( +oo +e
( B +e
C ) ) )
8577, 83, 843eqtr4d 2155 . 2  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( ( A +e B ) +e C )  =  ( A +e
( B +e
C ) ) )
86 simp1 962 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A  e.  RR*  /\  A  =/= -oo ) )
87 xrnemnf 9451 . . 3  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )
8886, 87sylib 121 . 2  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A  e.  RR  \/  A  = +oo ) )
8965, 85, 88mpjaodan 770 1  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    /\ w3a 943    = wceq 1312    e. wcel 1461    =/= wne 2280  (class class class)co 5726   CCcc 7539   RRcr 7540    + caddc 7544   +oocpnf 7715   -oocmnf 7716   RR*cxr 7717   +ecxad 9444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1re 7633  ax-addrcl 7636  ax-addass 7641  ax-rnegex 7648
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-pnf 7720  df-mnf 7721  df-xr 7722  df-xadd 9447
This theorem is referenced by:  xaddass2  9540  xpncan  9541  xadd4d  9555
  Copyright terms: Public domain W3C validator