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Theorem xltadd1 9688
Description: Extended real version of ltadd1 8214. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)
Assertion
Ref Expression
xltadd1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )

Proof of Theorem xltadd1
StepHypRef Expression
1 simplr 520 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  e.  RR )  ->  A  e.  RR )
2 simpr 109 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  e.  RR )  ->  B  e.  RR )
3 simpll3 1023 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  e.  RR )  ->  C  e.  RR )
4 ltadd1 8214 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A  +  C )  <  ( B  +  C )
) )
5 simp1 982 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
6 simp3 984 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
75, 6rexaddd 9666 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A +e C )  =  ( A  +  C ) )
8 simp2 983 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
98, 6rexaddd 9666 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B +e C )  =  ( B  +  C ) )
107, 9breq12d 3949 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A +e
C )  <  ( B +e C )  <-> 
( A  +  C
)  <  ( B  +  C ) ) )
114, 10bitr4d 190 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
121, 2, 3, 11syl3anc 1217 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  e.  RR )  ->  ( A  < 
B  <->  ( A +e C )  < 
( B +e
C ) ) )
13 ltpnf 9596 . . . . . 6  |-  ( A  e.  RR  ->  A  < +oo )
1413ad2antlr 481 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  A  < +oo )
15 breq2 3940 . . . . . 6  |-  ( B  = +oo  ->  ( A  <  B  <->  A  < +oo ) )
1615adantl 275 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( A  < 
B  <->  A  < +oo )
)
1714, 16mpbird 166 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  A  <  B
)
18 simplr 520 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  A  e.  RR )
19 simpll3 1023 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  C  e.  RR )
20 rexadd 9664 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A +e
C )  =  ( A  +  C ) )
21 readdcl 7769 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
2220, 21eqeltrd 2217 . . . . . . 7  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A +e
C )  e.  RR )
2318, 19, 22syl2anc 409 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( A +e C )  e.  RR )
24 ltpnf 9596 . . . . . 6  |-  ( ( A +e C )  e.  RR  ->  ( A +e C )  < +oo )
2523, 24syl 14 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( A +e C )  < +oo )
26 oveq1 5788 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e C )  =  ( +oo +e C ) )
2726adantl 275 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( B +e C )  =  ( +oo +e
C ) )
28 rexr 7834 . . . . . . . 8  |-  ( C  e.  RR  ->  C  e.  RR* )
29 renemnf 7837 . . . . . . . 8  |-  ( C  e.  RR  ->  C  =/= -oo )
30 xaddpnf2 9659 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
3128, 29, 30syl2anc 409 . . . . . . 7  |-  ( C  e.  RR  ->  ( +oo +e C )  = +oo )
3219, 31syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( +oo +e C )  = +oo )
3327, 32eqtrd 2173 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( B +e C )  = +oo )
3425, 33breqtrrd 3963 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( A +e C )  < 
( B +e
C ) )
3517, 342thd 174 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( A  < 
B  <->  ( A +e C )  < 
( B +e
C ) ) )
36 mnfle 9607 . . . . . . . 8  |-  ( A  e.  RR*  -> -oo  <_  A )
37363ad2ant1 1003 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  -> -oo  <_  A )
3837ad2antrr 480 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  -> -oo  <_  A )
39 mnfxr 7845 . . . . . . 7  |- -oo  e.  RR*
40 simpll1 1021 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  A  e.  RR* )
41 xrlenlt 7852 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo  <_  A  <->  -.  A  < -oo ) )
4239, 40, 41sylancr 411 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( -oo  <_  A  <->  -.  A  < -oo )
)
4338, 42mpbid 146 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  -.  A  < -oo )
44 breq2 3940 . . . . . 6  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
4544adantl 275 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( A  < 
B  <->  A  < -oo )
)
4643, 45mtbird 663 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  -.  A  <  B )
47283ad2ant3 1005 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  C  e.  RR* )
4847ad2antrr 480 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  C  e.  RR* )
49 xaddcl 9672 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A +e C )  e.  RR* )
5040, 48, 49syl2anc 409 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( A +e C )  e. 
RR* )
51 mnfle 9607 . . . . . . 7  |-  ( ( A +e C )  e.  RR*  -> -oo 
<_  ( A +e
C ) )
5250, 51syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  -> -oo  <_  ( A +e C ) )
53 xrlenlt 7852 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  ( A +e C )  e.  RR* )  ->  ( -oo  <_  ( A +e C )  <->  -.  ( A +e C )  < -oo ) )
5439, 50, 53sylancr 411 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( -oo  <_  ( A +e C )  <->  -.  ( A +e C )  < -oo ) )
5552, 54mpbid 146 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  -.  ( A +e C )  < -oo )
56 simpr 109 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  B  = -oo )
5756oveq1d 5796 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( B +e C )  =  ( -oo +e
C ) )
58 renepnf 7836 . . . . . . . . . 10  |-  ( C  e.  RR  ->  C  =/= +oo )
59583ad2ant3 1005 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  C  =/= +oo )
6059ad2antrr 480 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  C  =/= +oo )
61 xaddmnf2 9661 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  ->  ( -oo +e C )  = -oo )
6248, 60, 61syl2anc 409 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( -oo +e C )  = -oo )
6357, 62eqtrd 2173 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( B +e C )  = -oo )
6463breq2d 3948 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( ( A +e C )  <  ( B +e C )  <->  ( A +e C )  < -oo ) )
6555, 64mtbird 663 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  -.  ( A +e C )  <  ( B +e C ) )
6646, 652falsed 692 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( A  < 
B  <->  ( A +e C )  < 
( B +e
C ) ) )
67 elxr 9592 . . . . . 6  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
6867biimpi 119 . . . . 5  |-  ( B  e.  RR*  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
69683ad2ant2 1004 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
7069adantr 274 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  e.  RR )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
7112, 35, 66, 70mpjao3dan 1286 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  e.  RR )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
72 simpl2 986 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  B  e.  RR* )
73 pnfge 9604 . . . . . 6  |-  ( B  e.  RR*  ->  B  <_ +oo )
7472, 73syl 14 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  B  <_ +oo )
75 pnfxr 7841 . . . . . . 7  |- +oo  e.  RR*
7675a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  -> +oo  e.  RR* )
77 xrlenlt 7852 . . . . . 6  |-  ( ( B  e.  RR*  /\ +oo  e.  RR* )  ->  ( B  <_ +oo  <->  -. +oo  <  B
) )
7872, 76, 77syl2anc 409 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( B  <_ +oo  <->  -. +oo  <  B
) )
7974, 78mpbid 146 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  -. +oo 
<  B )
80 simpr 109 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  A  = +oo )
8180breq1d 3946 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( A  <  B  <-> +oo  <  B
) )
8279, 81mtbird 663 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  -.  A  <  B )
8347adantr 274 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  C  e.  RR* )
84 xaddcl 9672 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B +e C )  e.  RR* )
8572, 83, 84syl2anc 409 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( B +e C )  e.  RR* )
86 pnfge 9604 . . . . . . 7  |-  ( ( B +e C )  e.  RR*  ->  ( B +e C )  <_ +oo )
8785, 86syl 14 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( B +e C )  <_ +oo )
88293ad2ant3 1005 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  C  =/= -oo )
8988adantr 274 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  C  =/= -oo )
9083, 89, 30syl2anc 409 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( +oo +e C )  = +oo )
9187, 90breqtrrd 3963 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( B +e C )  <_  ( +oo +e C ) )
92 xaddcl 9672 . . . . . . 7  |-  ( ( +oo  e.  RR*  /\  C  e.  RR* )  ->  ( +oo +e C )  e.  RR* )
9375, 83, 92sylancr 411 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( +oo +e C )  e.  RR* )
94 xrlenlt 7852 . . . . . 6  |-  ( ( ( B +e
C )  e.  RR*  /\  ( +oo +e
C )  e.  RR* )  ->  ( ( B +e C )  <_  ( +oo +e C )  <->  -.  ( +oo +e C )  <  ( B +e C ) ) )
9585, 93, 94syl2anc 409 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  (
( B +e
C )  <_  ( +oo +e C )  <->  -.  ( +oo +e
C )  <  ( B +e C ) ) )
9691, 95mpbid 146 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  -.  ( +oo +e C )  <  ( B +e C ) )
9780oveq1d 5796 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( A +e C )  =  ( +oo +e C ) )
9897breq1d 3946 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  (
( A +e
C )  <  ( B +e C )  <-> 
( +oo +e C )  <  ( B +e C ) ) )
9996, 98mtbird 663 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  -.  ( A +e C )  <  ( B +e C ) )
10082, 992falsed 692 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
101 simplr 520 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  A  = -oo )
102 mnflt 9598 . . . . . 6  |-  ( B  e.  RR  -> -oo  <  B )
103102adantl 275 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  -> -oo  <  B )
104101, 103eqbrtrd 3957 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  A  <  B
)
105101oveq1d 5796 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  ( A +e C )  =  ( -oo +e
C ) )
106 simpll3 1023 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  C  e.  RR )
107106, 28syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  C  e.  RR* )
108106, 58syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  C  =/= +oo )
109107, 108, 61syl2anc 409 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  ( -oo +e C )  = -oo )
110105, 109eqtrd 2173 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  ( A +e C )  = -oo )
111 simpr 109 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  B  e.  RR )
112 rexadd 9664 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  =  ( B  +  C ) )
113 readdcl 7769 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
114112, 113eqeltrd 2217 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  e.  RR )
115111, 106, 114syl2anc 409 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  ( B +e C )  e.  RR )
116 mnflt 9598 . . . . . 6  |-  ( ( B +e C )  e.  RR  -> -oo 
<  ( B +e C ) )
117115, 116syl 14 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  -> -oo  <  ( B +e C ) )
118110, 117eqbrtrd 3957 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  ( A +e C )  < 
( B +e
C ) )
119104, 1182thd 174 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  ( A  < 
B  <->  ( A +e C )  < 
( B +e
C ) ) )
120 simplr 520 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  A  = -oo )
121 simpr 109 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  B  = +oo )
122120, 121breq12d 3949 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A  < 
B  <-> -oo  < +oo )
)
123 oveq1 5788 . . . . . . 7  |-  ( A  = -oo  ->  ( A +e C )  =  ( -oo +e C ) )
12447, 59, 61syl2anc 409 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( -oo +e C )  = -oo )
125123, 124sylan9eqr 2195 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = -oo )  ->  ( A +e C )  = -oo )
126125adantr 274 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e C )  = -oo )
12726adantl 275 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( B +e C )  =  ( +oo +e
C ) )
12847, 88, 30syl2anc 409 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( +oo +e C )  = +oo )
129128ad2antrr 480 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( +oo +e C )  = +oo )
130127, 129eqtrd 2173 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( B +e C )  = +oo )
131126, 130breq12d 3949 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( ( A +e C )  <  ( B +e C )  <-> -oo  < +oo ) )
132122, 131bitr4d 190 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A  < 
B  <->  ( A +e C )  < 
( B +e
C ) ) )
133 simplr 520 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  A  = -oo )
134 simpr 109 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  B  = -oo )
135133, 134breq12d 3949 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( A  < 
B  <-> -oo  < -oo )
)
136124ad2antrr 480 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( -oo +e C )  = -oo )
137123eqeq1d 2149 . . . . . . 7  |-  ( A  = -oo  ->  (
( A +e
C )  = -oo  <->  ( -oo +e C )  = -oo ) )
138137ad2antlr 481 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( ( A +e C )  = -oo  <->  ( -oo +e C )  = -oo ) )
139136, 138mpbird 166 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( A +e C )  = -oo )
140134oveq1d 5796 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( B +e C )  =  ( -oo +e
C ) )
141140, 136eqtrd 2173 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( B +e C )  = -oo )
142139, 141breq12d 3949 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( ( A +e C )  <  ( B +e C )  <-> -oo  < -oo ) )
143135, 142bitr4d 190 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( A  < 
B  <->  ( A +e C )  < 
( B +e
C ) ) )
14469adantr 274 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = -oo )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
145119, 132, 143, 144mpjao3dan 1286 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = -oo )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
146 elxr 9592 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
147146biimpi 119 . . 3  |-  ( A  e.  RR*  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
1481473ad2ant1 1003 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
14971, 100, 145, 148mpjao3dan 1286 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 962    /\ w3a 963    = wceq 1332    e. wcel 1481    =/= wne 2309   class class class wbr 3936  (class class class)co 5781   RRcr 7642    + caddc 7646   +oocpnf 7820   -oocmnf 7821   RR*cxr 7822    < clt 7823    <_ cle 7824   +ecxad 9586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-i2m1 7748  ax-0id 7751  ax-rnegex 7752  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-xadd 9589
This theorem is referenced by:  xltadd2  9689  xlt2add  9692  xrmaxaddlem  11060
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