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Theorem xaddass 11324
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 11325, 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 9484 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
2 recn 9484 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  CC )
3 recn 9484 . . . . . . . . . 10  |-  ( C  e.  RR  ->  C  e.  CC )
4 addass 9481 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
51, 2, 3, 4syl3an 1261 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
653expa 1188 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
7 readdcl 9477 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
8 rexadd 11314 . . . . . . . . 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 9477 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
11 rexadd 11314 . . . . . . . . . 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 2505 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A  +  B ) +e C )  =  ( A +e
( B  +  C
) ) )
15 rexadd 11314 . . . . . . . . 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 6216 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( ( A +e B ) +e C )  =  ( ( A  +  B ) +e C ) )
18 rexadd 11314 . . . . . . . . 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 6217 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR )  ->  ( A +e ( B +e C ) )  =  ( A +e ( B  +  C ) ) )
2114, 17, 203eqtr4d 2505 . . . . . 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 6209 . . . . . . . . 9  |-  ( C  = +oo  ->  (
( A +e
B ) +e
C )  =  ( ( A +e
B ) +e +oo ) )
24 simp1l 1012 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  A  e.  RR* )
25 simp2l 1014 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  B  e.  RR* )
26 xaddcl 11319 . . . . . . . . . . 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 11316 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )
)  ->  ( A +e B )  =/= -oo )
29283adant3 1008 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A +e B )  =/= -oo )
30 xaddpnf1 11308 . . . . . . . . . 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 2517 . . . . . . . 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 11308 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
34333ad2ant1 1009 . . . . . . . . 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 2498 . . . . . . 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 6209 . . . . . . . . 9  |-  ( C  = +oo  ->  ( B +e C )  =  ( B +e +oo ) )
38 xaddpnf1 11308 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
39383ad2ant2 1010 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( B +e +oo )  = +oo )
4037, 39sylan9eqr 2517 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  C  = +oo )  ->  ( B +e
C )  = +oo )
4140oveq2d 6217 . . . . . . 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 2498 . . . . . 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 990 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( C  e.  RR*  /\  C  =/= -oo ) )
45 xrnemnf 11211 . . . . . . 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 784 . . . 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 11309 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
51503ad2ant3 1011 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( +oo +e C )  = +oo )
5251, 34eqtr4d 2498 . . . . . 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 6209 . . . . . . 7  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
5554, 34sylan9eqr 2517 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
5655oveq1d 6216 . . . . 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 6208 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e C )  =  ( +oo +e C ) )
5857, 51sylan9eqr 2517 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  B  = +oo )  ->  ( B +e
C )  = +oo )
5958oveq2d 6217 . . . . 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 2505 . . . 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 992 . . . 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 11211 . . . 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 784 . 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 993 . . . . 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 1041 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  B  e.  RR* )
69 simpl3l 1043 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  C  e.  RR* )
70 xaddcl 11319 . . . . . 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 992 . . . . . 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 11316 . . . . . 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 11309 . . . . 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 2498 . . 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 6216 . . . . 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 11309 . . . . . 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 2495 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo ) )  /\  A  = +oo )  ->  ( A +e
B )  = +oo )
8382oveq1d 6216 . . 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 6216 . . 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 2505 . 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 988 . . 3  |-  ( ( ( A  e.  RR*  /\  A  =/= -oo )  /\  ( B  e.  RR*  /\  B  =/= -oo )  /\  ( C  e.  RR*  /\  C  =/= -oo )
)  ->  ( A  e.  RR*  /\  A  =/= -oo ) )
87 xrnemnf 11211 . . 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 784 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648  (class class class)co 6201   CCcc 9392   RRcr 9393    + caddc 9397   +oocpnf 9527   -oocmnf 9528   RR*cxr 9529   +ecxad 11199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-addass 9459  ax-i2m1 9462  ax-1ne0 9463  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-xadd 11202
This theorem is referenced by:  xaddass2  11325  xpncan  11326  xadd4d  11378  xrs1mnd  17977  xlt2addrd  26203  xrge0addass  26297  xrge0npcan  26303
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