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Theorem xnegdi 9855
Description: Extended real version of negdi 8204. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegdi  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )

Proof of Theorem xnegdi
StepHypRef Expression
1 elxr 9763 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9763 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 recn 7935 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 7935 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  CC )
5 negdi 8204 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  +  B )  =  (
-u A  +  -u B ) )
63, 4, 5syl2an 289 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u ( A  +  B )  =  (
-u A  +  -u B ) )
7 readdcl 7928 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
8 rexneg 9817 . . . . . . . 8  |-  ( ( A  +  B )  e.  RR  ->  -e
( A  +  B
)  =  -u ( A  +  B )
)
97, 8syl 14 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e ( A  +  B )  = 
-u ( A  +  B ) )
10 renegcl 8208 . . . . . . . 8  |-  ( A  e.  RR  ->  -u A  e.  RR )
11 renegcl 8208 . . . . . . . 8  |-  ( B  e.  RR  ->  -u B  e.  RR )
12 rexadd 9839 . . . . . . . 8  |-  ( (
-u A  e.  RR  /\  -u B  e.  RR )  ->  ( -u A +e -u B
)  =  ( -u A  +  -u B ) )
1310, 11, 12syl2an 289 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A +e -u B )  =  ( -u A  +  -u B ) )
146, 9, 133eqtr4d 2220 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e ( A  +  B )  =  ( -u A +e -u B ) )
15 rexadd 9839 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A +e
B )  =  ( A  +  B ) )
16 xnegeq 9814 . . . . . . 7  |-  ( ( A +e B )  =  ( A  +  B )  ->  -e ( A +e B )  = 
-e ( A  +  B ) )
1715, 16syl 14 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e ( A +e B )  =  -e ( A  +  B ) )
18 rexneg 9817 . . . . . . 7  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
19 rexneg 9817 . . . . . . 7  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
2018, 19oveqan12d 5888 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (  -e A +e  -e
B )  =  (
-u A +e -u B ) )
2114, 17, 203eqtr4d 2220 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-e ( A +e B )  =  (  -e
A +e  -e B ) )
22 xnegpnf 9815 . . . . . 6  |-  -e +oo  = -oo
23 oveq2 5877 . . . . . . . 8  |-  ( B  = +oo  ->  ( A +e B )  =  ( A +e +oo ) )
24 rexr 7993 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  RR* )
25 renemnf 7996 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= -oo )
26 xaddpnf1 9833 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  ->  ( A +e +oo )  = +oo )
2724, 25, 26syl2anc 411 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A +e +oo )  = +oo )
2823, 27sylan9eqr 2232 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A +e
B )  = +oo )
29 xnegeq 9814 . . . . . . 7  |-  ( ( A +e B )  = +oo  ->  -e ( A +e B )  = 
-e +oo )
3028, 29syl 14 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e ( A +e B )  =  -e +oo )
31 xnegeq 9814 . . . . . . . . 9  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
3231, 22eqtrdi 2226 . . . . . . . 8  |-  ( B  = +oo  ->  -e
B  = -oo )
3332oveq2d 5885 . . . . . . 7  |-  ( B  = +oo  ->  (  -e A +e  -e B )  =  (  -e A +e -oo )
)
3418, 10eqeltrd 2254 . . . . . . . 8  |-  ( A  e.  RR  ->  -e
A  e.  RR )
35 rexr 7993 . . . . . . . . 9  |-  (  -e A  e.  RR  -> 
-e A  e. 
RR* )
36 renepnf 7995 . . . . . . . . 9  |-  (  -e A  e.  RR  -> 
-e A  =/= +oo )
37 xaddmnf1 9835 . . . . . . . . 9  |-  ( ( 
-e A  e. 
RR*  /\  -e A  =/= +oo )  -> 
(  -e A +e -oo )  = -oo )
3835, 36, 37syl2anc 411 . . . . . . . 8  |-  (  -e A  e.  RR  ->  (  -e A +e -oo )  = -oo )
3934, 38syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  (  -e A +e -oo )  = -oo )
4033, 39sylan9eqr 2232 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  (  -e A +e  -e
B )  = -oo )
4122, 30, 403eqtr4a 2236 . . . . 5  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e ( A +e B )  =  (  -e
A +e  -e B ) )
42 xnegmnf 9816 . . . . . 6  |-  -e -oo  = +oo
43 oveq2 5877 . . . . . . . 8  |-  ( B  = -oo  ->  ( A +e B )  =  ( A +e -oo ) )
44 renepnf 7995 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= +oo )
45 xaddmnf1 9835 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A +e -oo )  = -oo )
4624, 44, 45syl2anc 411 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A +e -oo )  = -oo )
4743, 46sylan9eqr 2232 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A +e
B )  = -oo )
48 xnegeq 9814 . . . . . . 7  |-  ( ( A +e B )  = -oo  ->  -e ( A +e B )  = 
-e -oo )
4947, 48syl 14 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  -> 
-e ( A +e B )  =  -e -oo )
50 xnegeq 9814 . . . . . . . . 9  |-  ( B  = -oo  ->  -e
B  =  -e -oo )
5150, 42eqtrdi 2226 . . . . . . . 8  |-  ( B  = -oo  ->  -e
B  = +oo )
5251oveq2d 5885 . . . . . . 7  |-  ( B  = -oo  ->  (  -e A +e  -e B )  =  (  -e A +e +oo )
)
53 renemnf 7996 . . . . . . . . 9  |-  (  -e A  e.  RR  -> 
-e A  =/= -oo )
54 xaddpnf1 9833 . . . . . . . . 9  |-  ( ( 
-e A  e. 
RR*  /\  -e A  =/= -oo )  -> 
(  -e A +e +oo )  = +oo )
5535, 53, 54syl2anc 411 . . . . . . . 8  |-  (  -e A  e.  RR  ->  (  -e A +e +oo )  = +oo )
5634, 55syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  (  -e A +e +oo )  = +oo )
5752, 56sylan9eqr 2232 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  (  -e A +e  -e
B )  = +oo )
5842, 49, 573eqtr4a 2236 . . . . 5  |-  ( ( A  e.  RR  /\  B  = -oo )  -> 
-e ( A +e B )  =  (  -e
A +e  -e B ) )
5921, 41, 583jaodan 1306 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )
602, 59sylan2b 287 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  -e ( A +e B )  =  (  -e A +e  -e
B ) )
61 xneg0 9818 . . . . . . 7  |-  -e 0  =  0
62 simpr 110 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  B  = -oo )
6362oveq2d 5885 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  ( +oo +e -oo ) )
64 pnfaddmnf 9837 . . . . . . . . 9  |-  ( +oo +e -oo )  =  0
6563, 64eqtrdi 2226 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( +oo +e B )  =  0 )
66 xnegeq 9814 . . . . . . . 8  |-  ( ( +oo +e B )  =  0  ->  -e ( +oo +e B )  = 
-e 0 )
6765, 66syl 14 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  -e
( +oo +e B )  =  -e 0 )
6851adantl 277 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  -e
B  = +oo )
6968oveq2d 5885 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( -oo +e  -e
B )  =  ( -oo +e +oo ) )
70 mnfaddpnf 9838 . . . . . . . 8  |-  ( -oo +e +oo )  =  0
7169, 70eqtrdi 2226 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  ( -oo +e  -e
B )  =  0 )
7261, 67, 713eqtr4a 2236 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = -oo )  ->  -e
( +oo +e B )  =  ( -oo +e  -e B ) )
73 xaddpnf2 9834 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( +oo +e B )  = +oo )
74 xnegeq 9814 . . . . . . . 8  |-  ( ( +oo +e B )  = +oo  ->  -e ( +oo +e B )  = 
-e +oo )
7573, 74syl 14 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  -e
( +oo +e B )  =  -e +oo )
76 xnegcl 9819 . . . . . . . 8  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
77 xnegeq 9814 . . . . . . . . . . . 12  |-  (  -e B  = +oo  -> 
-e  -e
B  =  -e +oo )
7877, 22eqtrdi 2226 . . . . . . . . . . 11  |-  (  -e B  = +oo  -> 
-e  -e
B  = -oo )
79 xnegneg 9820 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  -e  -e B  =  B )
8079eqeq1d 2186 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  (  -e  -e B  = -oo  <->  B  = -oo ) )
8178, 80imbitrid 154 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  (  -e B  = +oo  ->  B  = -oo )
)
8281necon3d 2391 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  =/= -oo  ->  -e
B  =/= +oo )
)
8382imp 124 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  -e
B  =/= +oo )
84 xaddmnf2 9836 . . . . . . . 8  |-  ( ( 
-e B  e. 
RR*  /\  -e B  =/= +oo )  -> 
( -oo +e  -e B )  = -oo )
8576, 83, 84syl2an2r 595 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( -oo +e  -e
B )  = -oo )
8622, 75, 853eqtr4a 2236 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  -e
( +oo +e B )  =  ( -oo +e  -e B ) )
87 xrmnfdc 9830 . . . . . . . 8  |-  ( B  e.  RR*  -> DECID  B  = -oo )
88 exmiddc 836 . . . . . . . 8  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
8987, 88syl 14 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  -.  B  = -oo )
)
90 df-ne 2348 . . . . . . . 8  |-  ( B  =/= -oo  <->  -.  B  = -oo )
9190orbi2i 762 . . . . . . 7  |-  ( ( B  = -oo  \/  B  =/= -oo )  <->  ( B  = -oo  \/  -.  B  = -oo ) )
9289, 91sylibr 134 . . . . . 6  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  B  =/= -oo ) )
9372, 86, 92mpjaodan 798 . . . . 5  |-  ( B  e.  RR*  ->  -e
( +oo +e B )  =  ( -oo +e  -e B ) )
9493adantl 277 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e ( +oo +e B )  =  ( -oo +e  -e B ) )
95 simpl 109 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
9695oveq1d 5884 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( +oo +e B ) )
97 xnegeq 9814 . . . . 5  |-  ( ( A +e B )  =  ( +oo +e B )  ->  -e ( A +e B )  =  -e ( +oo +e B ) )
9896, 97syl 14 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e ( A +e B )  = 
-e ( +oo +e B ) )
99 xnegeq 9814 . . . . . . 7  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
10099adantr 276 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e A  =  -e +oo )
101100, 22eqtrdi 2226 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e A  = -oo )
102101oveq1d 5884 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
(  -e A +e  -e B )  =  ( -oo +e  -e B ) )
10394, 98, 1023eqtr4d 2220 . . 3  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -e ( A +e B )  =  (  -e A +e  -e
B ) )
104 simpr 110 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  B  = +oo )
105104oveq2d 5885 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  ( -oo +e +oo ) )
106105, 70eqtrdi 2226 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( -oo +e B )  =  0 )
107 xnegeq 9814 . . . . . . . 8  |-  ( ( -oo +e B )  =  0  ->  -e ( -oo +e B )  = 
-e 0 )
108106, 107syl 14 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  -e
( -oo +e B )  =  -e 0 )
10932adantl 277 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  -e
B  = -oo )
110109oveq2d 5885 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( +oo +e  -e
B )  =  ( +oo +e -oo ) )
111110, 64eqtrdi 2226 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  ( +oo +e  -e
B )  =  0 )
11261, 108, 1113eqtr4a 2236 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  = +oo )  ->  -e
( -oo +e B )  =  ( +oo +e  -e B ) )
113 xaddmnf2 9836 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
114 xnegeq 9814 . . . . . . . 8  |-  ( ( -oo +e B )  = -oo  ->  -e ( -oo +e B )  = 
-e -oo )
115113, 114syl 14 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  -e
( -oo +e B )  =  -e -oo )
116 xnegeq 9814 . . . . . . . . . . . 12  |-  (  -e B  = -oo  -> 
-e  -e
B  =  -e -oo )
117116, 42eqtrdi 2226 . . . . . . . . . . 11  |-  (  -e B  = -oo  -> 
-e  -e
B  = +oo )
11879eqeq1d 2186 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  (  -e  -e B  = +oo  <->  B  = +oo ) )
119117, 118imbitrid 154 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  (  -e B  = -oo  ->  B  = +oo )
)
120119necon3d 2391 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  =/= +oo  ->  -e
B  =/= -oo )
)
121120imp 124 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  -e
B  =/= -oo )
122 xaddpnf2 9834 . . . . . . . 8  |-  ( ( 
-e B  e. 
RR*  /\  -e B  =/= -oo )  -> 
( +oo +e  -e B )  = +oo )
12376, 121, 122syl2an2r 595 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( +oo +e  -e
B )  = +oo )
12442, 115, 1233eqtr4a 2236 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  -e
( -oo +e B )  =  ( +oo +e  -e B ) )
125 xrpnfdc 9829 . . . . . . . 8  |-  ( B  e.  RR*  -> DECID  B  = +oo )
126 exmiddc 836 . . . . . . . 8  |-  (DECID  B  = +oo  ->  ( B  = +oo  \/  -.  B  = +oo ) )
127125, 126syl 14 . . . . . . 7  |-  ( B  e.  RR*  ->  ( B  = +oo  \/  -.  B  = +oo )
)
128 df-ne 2348 . . . . . . . 8  |-  ( B  =/= +oo  <->  -.  B  = +oo )
129128orbi2i 762 . . . . . . 7  |-  ( ( B  = +oo  \/  B  =/= +oo )  <->  ( B  = +oo  \/  -.  B  = +oo ) )
130127, 129sylibr 134 . . . . . 6  |-  ( B  e.  RR*  ->  ( B  = +oo  \/  B  =/= +oo ) )
131112, 124, 130mpjaodan 798 . . . . 5  |-  ( B  e.  RR*  ->  -e
( -oo +e B )  =  ( +oo +e  -e B ) )
132131adantl 277 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e ( -oo +e B )  =  ( +oo +e  -e B ) )
133 simpl 109 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  A  = -oo )
134133oveq1d 5884 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A +e
B )  =  ( -oo +e B ) )
135 xnegeq 9814 . . . . 5  |-  ( ( A +e B )  =  ( -oo +e B )  ->  -e ( A +e B )  =  -e ( -oo +e B ) )
136134, 135syl 14 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e ( A +e B )  = 
-e ( -oo +e B ) )
137 xnegeq 9814 . . . . . . 7  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
138137adantr 276 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e A  =  -e -oo )
139138, 42eqtrdi 2226 . . . . 5  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e A  = +oo )
140139oveq1d 5884 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
(  -e A +e  -e B )  =  ( +oo +e  -e B ) )
141132, 136, 1403eqtr4d 2220 . . 3  |-  ( ( A  = -oo  /\  B  e.  RR* )  ->  -e ( A +e B )  =  (  -e A +e  -e
B ) )
14260, 103, 1413jaoian 1305 . 2  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  -e ( A +e B )  =  (  -e
A +e  -e B ) )
1431, 142sylanb 284 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 708  DECID wdc 834    \/ w3o 977    = wceq 1353    e. wcel 2148    =/= wne 2347  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802    + caddc 7805   +oocpnf 7979   -oocmnf 7980   RR*cxr 7981   -ucneg 8119    -ecxne 9756   +ecxad 9757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-sub 8120  df-neg 8121  df-xneg 9759  df-xadd 9760
This theorem is referenced by:  xaddass2  9857  xrminadd  11267
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