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Theorem xnegdi 10783
Description: Extended real version of xnegdi 10783. (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 10672 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 elxr 10672 . . . 4  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = 
+oo  \/  B  =  -oo ) )
3 recn 9036 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 9036 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  CC )
5 negdi 9314 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  +  B )  =  (
-u A  +  -u B ) )
63, 4, 5syl2an 464 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u ( A  +  B )  =  (
-u A  +  -u B ) )
7 readdcl 9029 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
8 rexneg 10753 . . . . . . . 8  |-  ( ( A  +  B )  e.  RR  ->  - e
( A  +  B
)  =  -u ( A  +  B )
)
97, 8syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
- e ( A  +  B )  = 
-u ( A  +  B ) )
10 renegcl 9320 . . . . . . . 8  |-  ( A  e.  RR  ->  -u A  e.  RR )
11 renegcl 9320 . . . . . . . 8  |-  ( B  e.  RR  ->  -u B  e.  RR )
12 rexadd 10774 . . . . . . . 8  |-  ( (
-u A  e.  RR  /\  -u B  e.  RR )  ->  ( -u A + e -u B )  =  ( -u A  +  -u B ) )
1310, 11, 12syl2an 464 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A + e -u B )  =  ( -u A  +  -u B ) )
146, 9, 133eqtr4d 2446 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
- e ( A  +  B )  =  ( -u A + e -u B ) )
15 rexadd 10774 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A + e B )  =  ( A  +  B ) )
16 xnegeq 10749 . . . . . . 7  |-  ( ( A + e B )  =  ( A  +  B )  ->  - e ( A + e B )  =  - e ( A  +  B ) )
1715, 16syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
- e ( A + e B )  =  - e ( A  +  B ) )
18 rexneg 10753 . . . . . . 7  |-  ( A  e.  RR  ->  - e A  =  -u A )
19 rexneg 10753 . . . . . . 7  |-  ( B  e.  RR  ->  - e B  =  -u B )
2018, 19oveqan12d 6059 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (  - e A + e  - e B )  =  (
-u A + e -u B ) )
2114, 17, 203eqtr4d 2446 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
- e ( A + e B )  =  (  - e A + e  - e B ) )
22 xnegpnf 10751 . . . . . 6  |-  - e  +oo  =  -oo
23 oveq2 6048 . . . . . . . 8  |-  ( B  =  +oo  ->  ( A + e B )  =  ( A + e  +oo ) )
24 rexr 9086 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  RR* )
25 renemnf 9089 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/=  -oo )
26 xaddpnf1 10768 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  ->  ( A + e  +oo )  =  +oo )
2724, 25, 26syl2anc 643 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A + e  +oo )  =  +oo )
2823, 27sylan9eqr 2458 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A + e B )  =  +oo )
29 xnegeq 10749 . . . . . . 7  |-  ( ( A + e B )  =  +oo  ->  - e ( A + e B )  =  - e  +oo )
3028, 29syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  - e ( A + e B )  =  - e  +oo )
31 xnegeq 10749 . . . . . . . . 9  |-  ( B  =  +oo  ->  - e B  =  - e  +oo )
3231, 22syl6eq 2452 . . . . . . . 8  |-  ( B  =  +oo  ->  - e B  =  -oo )
3332oveq2d 6056 . . . . . . 7  |-  ( B  =  +oo  ->  (  - e A + e  - e B )  =  (  - e A + e  -oo )
)
3418, 10eqeltrd 2478 . . . . . . . 8  |-  ( A  e.  RR  ->  - e A  e.  RR )
35 rexr 9086 . . . . . . . . 9  |-  (  - e A  e.  RR  -> 
- e A  e. 
RR* )
36 renepnf 9088 . . . . . . . . 9  |-  (  - e A  e.  RR  -> 
- e A  =/= 
+oo )
37 xaddmnf1 10770 . . . . . . . . 9  |-  ( ( 
- e A  e. 
RR*  /\  - e A  =/=  +oo )  ->  (  - e A + e  -oo )  =  -oo )
3835, 36, 37syl2anc 643 . . . . . . . 8  |-  (  - e A  e.  RR  ->  (  - e A + e  -oo )  =  -oo )
3934, 38syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  (  - e A + e  -oo )  =  -oo )
4033, 39sylan9eqr 2458 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
(  - e A + e  - e B )  =  -oo )
4122, 30, 403eqtr4a 2462 . . . . 5  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  - e ( A + e B )  =  ( 
- e A + e  - e B ) )
42 xnegmnf 10752 . . . . . 6  |-  - e  -oo  =  +oo
43 oveq2 6048 . . . . . . . 8  |-  ( B  =  -oo  ->  ( A + e B )  =  ( A + e  -oo ) )
44 renepnf 9088 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/=  +oo )
45 xaddmnf1 10770 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  ( A + e  -oo )  =  -oo )
4624, 44, 45syl2anc 643 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A + e  -oo )  =  -oo )
4743, 46sylan9eqr 2458 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( A + e B )  =  -oo )
48 xnegeq 10749 . . . . . . 7  |-  ( ( A + e B )  =  -oo  ->  - e ( A + e B )  =  - e  -oo )
4947, 48syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  - e ( A + e B )  =  - e  -oo )
50 xnegeq 10749 . . . . . . . . 9  |-  ( B  =  -oo  ->  - e B  =  - e  -oo )
5150, 42syl6eq 2452 . . . . . . . 8  |-  ( B  =  -oo  ->  - e B  =  +oo )
5251oveq2d 6056 . . . . . . 7  |-  ( B  =  -oo  ->  (  - e A + e  - e B )  =  (  - e A + e  +oo )
)
53 renemnf 9089 . . . . . . . . 9  |-  (  - e A  e.  RR  -> 
- e A  =/= 
-oo )
54 xaddpnf1 10768 . . . . . . . . 9  |-  ( ( 
- e A  e. 
RR*  /\  - e A  =/=  -oo )  ->  (  - e A + e  +oo )  =  +oo )
5535, 53, 54syl2anc 643 . . . . . . . 8  |-  (  - e A  e.  RR  ->  (  - e A + e  +oo )  =  +oo )
5634, 55syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  (  - e A + e  +oo )  =  +oo )
5752, 56sylan9eqr 2458 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
(  - e A + e  - e B )  =  +oo )
5842, 49, 573eqtr4a 2462 . . . . 5  |-  ( ( A  e.  RR  /\  B  =  -oo )  ->  - e ( A + e B )  =  ( 
- e A + e  - e B ) )
5921, 41, 583jaodan 1250 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  - e ( A + e B )  =  (  - e A + e  - e B ) )
602, 59sylan2b 462 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  - e ( A + e B )  =  ( 
- e A + e  - e B ) )
61 xneg0 10754 . . . . . . 7  |-  - e
0  =  0
62 simpr 448 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =  -oo )  ->  B  =  -oo )
6362oveq2d 6056 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  =  -oo )  ->  (  +oo + e B )  =  (  +oo + e  -oo ) )
64 pnfaddmnf 10772 . . . . . . . . 9  |-  (  +oo + e  -oo )  =  0
6563, 64syl6eq 2452 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =  -oo )  ->  (  +oo + e B )  =  0 )
66 xnegeq 10749 . . . . . . . 8  |-  ( ( 
+oo + e B )  =  0  ->  - e (  +oo + e B )  =  - e 0 )
6765, 66syl 16 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =  -oo )  ->  - e
(  +oo + e B )  =  - e
0 )
6851adantl 453 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  =  -oo )  ->  - e B  =  +oo )
6968oveq2d 6056 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =  -oo )  ->  (  -oo + e  - e B )  =  ( 
-oo + e  +oo ) )
70 mnfaddpnf 10773 . . . . . . . 8  |-  (  -oo + e  +oo )  =  0
7169, 70syl6eq 2452 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =  -oo )  ->  (  -oo + e  - e B )  =  0 )
7261, 67, 713eqtr4a 2462 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =  -oo )  ->  - e
(  +oo + e B )  =  (  -oo + e  - e B ) )
73 xaddpnf2 10769 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  (  +oo + e B )  =  +oo )
74 xnegeq 10749 . . . . . . . 8  |-  ( ( 
+oo + e B )  =  +oo  ->  - e (  +oo + e B )  =  - e  +oo )
7573, 74syl 16 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  - e
(  +oo + e B )  =  - e  +oo )
76 xnegcl 10755 . . . . . . . . 9  |-  ( B  e.  RR*  ->  - e B  e.  RR* )
7776adantr 452 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  - e B  e.  RR* )
78 xnegeq 10749 . . . . . . . . . . . 12  |-  (  - e B  =  +oo  -> 
- e  - e B  =  - e  +oo )
7978, 22syl6eq 2452 . . . . . . . . . . 11  |-  (  - e B  =  +oo  -> 
- e  - e B  =  -oo )
80 xnegneg 10756 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  - e  - e B  =  B )
8180eqeq1d 2412 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  (  - e  - e B  = 
-oo 
<->  B  =  -oo )
)
8279, 81syl5ib 211 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  (  - e B  =  +oo  ->  B  =  -oo )
)
8382necon3d 2605 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  =/=  -oo  ->  - e B  =/=  +oo ) )
8483imp 419 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  - e B  =/=  +oo )
85 xaddmnf2 10771 . . . . . . . 8  |-  ( ( 
- e B  e. 
RR*  /\  - e B  =/=  +oo )  ->  (  -oo + e  - e B )  =  -oo )
8677, 84, 85syl2anc 643 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  (  -oo + e  - e B )  =  -oo )
8722, 75, 863eqtr4a 2462 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/=  -oo )  ->  - e
(  +oo + e B )  =  (  -oo + e  - e B ) )
8872, 87pm2.61dane 2645 . . . . 5  |-  ( B  e.  RR*  ->  - e
(  +oo + e B )  =  (  -oo + e  - e B ) )
8988adantl 453 . . . 4  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  - e (  +oo + e B )  =  ( 
-oo + e  - e B ) )
90 simpl 444 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  A  =  +oo )
9190oveq1d 6055 . . . . 5  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A + e B )  =  ( 
+oo + e B ) )
92 xnegeq 10749 . . . . 5  |-  ( ( A + e B )  =  (  +oo + e B )  ->  - e ( A + e B )  =  - e (  +oo + e B ) )
9391, 92syl 16 . . . 4  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  - e ( A + e B )  =  - e (  +oo + e B ) )
94 xnegeq 10749 . . . . . . 7  |-  ( A  =  +oo  ->  - e A  =  - e  +oo )
9594adantr 452 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  - e A  =  - e  +oo )
9695, 22syl6eq 2452 . . . . 5  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  - e A  =  -oo )
9796oveq1d 6055 . . . 4  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
(  - e A + e  - e B )  =  (  -oo + e  - e B ) )
9889, 93, 973eqtr4d 2446 . . 3  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  - e ( A + e B )  =  ( 
- e A + e  - e B ) )
99 simpr 448 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  B  =  +oo )  ->  B  =  +oo )
10099oveq2d 6056 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  =  +oo )  ->  (  -oo + e B )  =  (  -oo + e  +oo ) )
101100, 70syl6eq 2452 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =  +oo )  ->  (  -oo + e B )  =  0 )
102 xnegeq 10749 . . . . . . . 8  |-  ( ( 
-oo + e B )  =  0  ->  - e (  -oo + e B )  =  - e 0 )
103101, 102syl 16 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =  +oo )  ->  - e
(  -oo + e B )  =  - e
0 )
10432adantl 453 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  B  =  +oo )  ->  - e B  =  -oo )
105104oveq2d 6056 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =  +oo )  ->  (  +oo + e  - e B )  =  ( 
+oo + e  -oo ) )
106105, 64syl6eq 2452 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =  +oo )  ->  (  +oo + e  - e B )  =  0 )
10761, 103, 1063eqtr4a 2462 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =  +oo )  ->  - e
(  -oo + e B )  =  (  +oo + e  - e B ) )
108 xaddmnf2 10771 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/=  +oo )  ->  (  -oo + e B )  =  -oo )
109 xnegeq 10749 . . . . . . . 8  |-  ( ( 
-oo + e B )  =  -oo  ->  - e (  -oo + e B )  =  - e  -oo )
110108, 109syl 16 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/=  +oo )  ->  - e
(  -oo + e B )  =  - e  -oo )
11176adantr 452 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/=  +oo )  ->  - e B  e.  RR* )
112 xnegeq 10749 . . . . . . . . . . . 12  |-  (  - e B  =  -oo  -> 
- e  - e B  =  - e  -oo )
113112, 42syl6eq 2452 . . . . . . . . . . 11  |-  (  - e B  =  -oo  -> 
- e  - e B  =  +oo )
11480eqeq1d 2412 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  (  - e  - e B  = 
+oo 
<->  B  =  +oo )
)
115113, 114syl5ib 211 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  (  - e B  =  -oo  ->  B  =  +oo )
)
116115necon3d 2605 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  =/=  +oo  ->  - e B  =/=  -oo ) )
117116imp 419 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  B  =/=  +oo )  ->  - e B  =/=  -oo )
118 xaddpnf2 10769 . . . . . . . 8  |-  ( ( 
- e B  e. 
RR*  /\  - e B  =/=  -oo )  ->  (  +oo + e  - e B )  =  +oo )
119111, 117, 118syl2anc 643 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/=  +oo )  ->  (  +oo + e  - e B )  =  +oo )
12042, 110, 1193eqtr4a 2462 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/=  +oo )  ->  - e
(  -oo + e B )  =  (  +oo + e  - e B ) )
121107, 120pm2.61dane 2645 . . . . 5  |-  ( B  e.  RR*  ->  - e
(  -oo + e B )  =  (  +oo + e  - e B ) )
122121adantl 453 . . . 4  |-  ( ( A  =  -oo  /\  B  e.  RR* )  ->  - e (  -oo + e B )  =  ( 
+oo + e  - e B ) )
123 simpl 444 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR* )  ->  A  =  -oo )
124123oveq1d 6055 . . . . 5  |-  ( ( A  =  -oo  /\  B  e.  RR* )  -> 
( A + e B )  =  ( 
-oo + e B ) )
125 xnegeq 10749 . . . . 5  |-  ( ( A + e B )  =  (  -oo + e B )  ->  - e ( A + e B )  =  - e (  -oo + e B ) )
126124, 125syl 16 . . . 4  |-  ( ( A  =  -oo  /\  B  e.  RR* )  ->  - e ( A + e B )  =  - e (  -oo + e B ) )
127 xnegeq 10749 . . . . . . 7  |-  ( A  =  -oo  ->  - e A  =  - e  -oo )
128127adantr 452 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR* )  ->  - e A  =  - e  -oo )
129128, 42syl6eq 2452 . . . . 5  |-  ( ( A  =  -oo  /\  B  e.  RR* )  ->  - e A  =  +oo )
130129oveq1d 6055 . . . 4  |-  ( ( A  =  -oo  /\  B  e.  RR* )  -> 
(  - e A + e  - e B )  =  (  +oo + e  - e B ) )
131122, 126, 1303eqtr4d 2446 . . 3  |-  ( ( A  =  -oo  /\  B  e.  RR* )  ->  - e ( A + e B )  =  ( 
- e A + e  - e B ) )
13260, 98, 1313jaoian 1249 . 2  |-  ( ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  /\  B  e.  RR* )  -> 
- e ( A + e B )  =  (  - e A + e  - e B ) )
1331, 132sylanb 459 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  - e
( A + e B )  =  ( 
- e A + e  - e B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1721    =/= wne 2567  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    +oocpnf 9073    -oocmnf 9074   RR*cxr 9075   -ucneg 9248    - ecxne 10663   + ecxad 10664
This theorem is referenced by:  xaddass2  10785  xposdif  10797  xadddi  10830  xrsxmet  18793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-sub 9249  df-neg 9250  df-xneg 10666  df-xadd 10667
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