MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xaddass Structured version   Unicode version

Theorem xaddass 11466
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 11467, 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 9599 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
2 recn 9599 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  CC )
3 recn 9599 . . . . . . . . . 10  |-  ( C  e.  RR  ->  C  e.  CC )
4 addass 9596 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
51, 2, 3, 4syl3an 1270 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
653expa 1196 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
7 readdcl 9592 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
8 rexadd 11456 . . . . . . . . 9  |-  ( ( ( A  +  B
)  e.  RR  /\  C  e.  RR )  ->  ( ( A  +  B ) +e
C )  =  ( ( A  +  B
)  +  C ) )
97, 8sylan 471 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  +  B ) +e C )  =  ( ( A  +  B )  +  C
) )
10 readdcl 9592 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
11 rexadd 11456 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  +  C
)  e.  RR )  ->  ( A +e ( B  +  C ) )  =  ( A  +  ( B  +  C ) ) )
1210, 11sylan2 474 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  C  e.  RR ) )  ->  ( A +e ( B  +  C ) )  =  ( A  +  ( B  +  C
) ) )
1312anassrs 648 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A +e ( B  +  C ) )  =  ( A  +  ( B  +  C ) ) )
146, 9, 133eqtr4d 2508 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  +  B ) +e C )  =  ( A +e
( B  +  C
) ) )
15 rexadd 11456 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
1615adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A +e B )  =  ( A  +  B
) )
1716oveq1d 6311 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A +e B ) +e C )  =  ( ( A  +  B ) +e C ) )
18 rexadd 11456 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  =  ( B  +  C ) )
1918adantll 713 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( B +e C )  =  ( B  +  C
) )
2019oveq2d 6312 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A +e ( B +e C ) )  =  ( A +e ( B  +  C ) ) )
2114, 17, 203eqtr4d 2508 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A +e B ) +e C )  =  ( A +e ( B +e C ) ) )
2221adantll 713 . . . . 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 6304 . . . . . . . . 9  |-  ( C  = +oo  ->  (
( A +e
B ) +e
C )  =  ( ( A +e
B ) +e +oo ) )
24 simp1l 1020 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  A  e.  RR* )
25 simp2l 1022 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  B  e.  RR* )
26 xaddcl 11461 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
2724, 25, 26syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A +e B )  e.  RR* )
28 xaddnemnf 11458 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )
)  ->  ( A +e B )  =/= -oo )
29283adant3 1016 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A +e B )  =/= -oo )
30 xaddpnf1 11450 . . . . . . . . . 10  |-  ( ( ( A +e
B )  e.  RR*  /\  ( A +e
B )  =/= -oo )  ->  ( ( A +e B ) +e +oo )  = +oo )
3127, 29, 30syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( ( A +e B ) +e +oo )  = +oo )
3223, 31sylan9eqr 2520 . . . . . . . 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 11450 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
34333ad2ant1 1017 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A +e +oo )  = +oo )
3534adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  C  = +oo )  ->  ( A +e +oo )  = +oo )
3632, 35eqtr4d 2501 . . . . . . 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 6304 . . . . . . . . 9  |-  ( C  = +oo  ->  ( B +e C )  =  ( B +e +oo ) )
38 xaddpnf1 11450 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
39383ad2ant2 1018 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( B +e +oo )  = +oo )
4037, 39sylan9eqr 2520 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  C  = +oo )  ->  ( B +e
C )  = +oo )
4140oveq2d 6312 . . . . . . 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 2501 . . . . . 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 714 . . . . 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 998 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( C  e.  RR*  /\  C  =/= -oo ) )
45 xrnemnf 11353 . . . . . . 7  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  <->  ( C  e.  RR  \/  C  = +oo ) )
4644, 45sylib 196 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( C  e.  RR  \/  C  = +oo ) )
4746adantr 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  \/  C  = +oo ) )
4822, 43, 47mpjaodan 786 . . . 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 648 . . 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 11451 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
51503ad2ant3 1019 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( +oo +e C )  = +oo )
5251, 34eqtr4d 2501 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( +oo +e C )  =  ( A +e +oo ) )
5352adantr 465 . . . . 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 6304 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
5554, 34sylan9eqr 2520 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
5655oveq1d 6311 . . . . 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 6303 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e C )  =  ( +oo +e C ) )
5857, 51sylan9eqr 2520 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( B +e
C )  = +oo )
5958oveq2d 6312 . . . . 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 2508 . . . 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 714 . . 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 1000 . . . 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 11353 . . . 4  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  <->  ( B  e.  RR  \/  B  = +oo ) )
6462, 63sylib 196 . . 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 786 . 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 1001 . . . . 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 16 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( +oo +e
C )  = +oo )
68 simpl2l 1049 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  B  e.  RR* )
69 simpl3l 1051 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  C  e.  RR* )
70 xaddcl 11461 . . . . . 6  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B +e C )  e.  RR* )
7168, 69, 70syl2anc 661 . . . . 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 1000 . . . . . 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 11458 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( B +e C )  =/= -oo )
7472, 66, 73syl2anc 661 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( B +e
C )  =/= -oo )
75 xaddpnf2 11451 . . . . 5  |-  ( ( ( B +e
C )  e.  RR*  /\  ( B +e
C )  =/= -oo )  ->  ( +oo +e ( B +e C ) )  = +oo )
7671, 74, 75syl2anc 661 . . . 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 2501 . . 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 461 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  A  = +oo )
7978oveq1d 6311 . . . . 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 11451 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
8172, 80syl 16 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( +oo +e
B )  = +oo )
8279, 81eqtrd 2498 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( A +e
B )  = +oo )
8382oveq1d 6311 . . 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 6311 . . 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 2508 . 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 996 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A  e.  RR*  /\  A  =/= -oo ) )
87 xrnemnf 11353 . . 3  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )
8886, 87sylib 196 . 2  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A  e.  RR  \/  A  = +oo ) )
8965, 85, 88mpjaodan 786 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652  (class class class)co 6296   CCcc 9507   RRcr 9508    + caddc 9512   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644   +ecxad 11341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-addass 9574  ax-i2m1 9577  ax-1ne0 9578  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-xadd 11344
This theorem is referenced by:  xaddass2  11467  xpncan  11468  xadd4d  11520  xrs1mnd  18583  xlt2addrd  27735  xrge0addass  27838  xrge0npcan  27844  carsgclctunlem2  28461
  Copyright terms: Public domain W3C validator