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Theorem xltadd1 9500
Description: Extended real version of ltadd1 8058. (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 500 . . . 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 990 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  e.  RR )  ->  C  e.  RR )
4 ltadd1 8058 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( A  +  C )  <  ( B  +  C )
) )
5 simp1 949 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
6 simp3 951 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
75, 6rexaddd 9478 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A +e C )  =  ( A  +  C ) )
8 simp2 950 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
98, 6rexaddd 9478 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B +e C )  =  ( B  +  C ) )
107, 9breq12d 3888 . . . . 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 1184 . . 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 9408 . . . . . 6  |-  ( A  e.  RR  ->  A  < +oo )
1413ad2antlr 476 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  A  < +oo )
15 breq2 3879 . . . . . 6  |-  ( B  = +oo  ->  ( A  <  B  <->  A  < +oo ) )
1615adantl 273 . . . . 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 500 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  A  e.  RR )
19 simpll3 990 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  C  e.  RR )
20 rexadd 9476 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A +e
C )  =  ( A  +  C ) )
21 readdcl 7618 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
2220, 21eqeltrd 2176 . . . . . . 7  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A +e
C )  e.  RR )
2318, 19, 22syl2anc 406 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( A +e C )  e.  RR )
24 ltpnf 9408 . . . . . 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 5713 . . . . . . 7  |-  ( B  = +oo  ->  ( B +e C )  =  ( +oo +e C ) )
2726adantl 273 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( B +e C )  =  ( +oo +e
C ) )
28 rexr 7683 . . . . . . . 8  |-  ( C  e.  RR  ->  C  e.  RR* )
29 renemnf 7686 . . . . . . . 8  |-  ( C  e.  RR  ->  C  =/= -oo )
30 xaddpnf2 9471 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( +oo +e C )  = +oo )
3128, 29, 30syl2anc 406 . . . . . . 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 2132 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = +oo )  ->  ( B +e C )  = +oo )
3425, 33breqtrrd 3901 . . . 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 9419 . . . . . . . 8  |-  ( A  e.  RR*  -> -oo  <_  A )
37363ad2ant1 970 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  -> -oo  <_  A )
3837ad2antrr 475 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  -> -oo  <_  A )
39 mnfxr 7694 . . . . . . 7  |- -oo  e.  RR*
40 simpll1 988 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  A  e.  RR* )
41 xrlenlt 7701 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo  <_  A  <->  -.  A  < -oo ) )
4239, 40, 41sylancr 408 . . . . . 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 3879 . . . . . 6  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
4544adantl 273 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( A  < 
B  <->  A  < -oo )
)
4643, 45mtbird 639 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  -.  A  <  B )
47283ad2ant3 972 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  C  e.  RR* )
4847ad2antrr 475 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  C  e.  RR* )
49 xaddcl 9484 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A +e C )  e.  RR* )
5040, 48, 49syl2anc 406 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( A +e C )  e. 
RR* )
51 mnfle 9419 . . . . . . 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 7701 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  ( A +e C )  e.  RR* )  ->  ( -oo  <_  ( A +e C )  <->  -.  ( A +e C )  < -oo ) )
5439, 50, 53sylancr 408 . . . . . 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 5721 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( B +e C )  =  ( -oo +e
C ) )
58 renepnf 7685 . . . . . . . . . 10  |-  ( C  e.  RR  ->  C  =/= +oo )
59583ad2ant3 972 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  C  =/= +oo )
6059ad2antrr 475 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  C  =/= +oo )
61 xaddmnf2 9473 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= +oo )  ->  ( -oo +e C )  = -oo )
6248, 60, 61syl2anc 406 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( -oo +e C )  = -oo )
6357, 62eqtrd 2132 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  ( B +e C )  = -oo )
6463breq2d 3887 . . . . 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 639 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  e.  RR )  /\  B  = -oo )  ->  -.  ( A +e C )  <  ( B +e C ) )
6646, 652falsed 659 . . 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 9404 . . . . . 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 971 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
7069adantr 272 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  e.  RR )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
7112, 35, 66, 70mpjao3dan 1253 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  e.  RR )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
72 simpl2 953 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  B  e.  RR* )
73 pnfge 9416 . . . . . 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 7690 . . . . . . 7  |- +oo  e.  RR*
7675a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  -> +oo  e.  RR* )
77 xrlenlt 7701 . . . . . 6  |-  ( ( B  e.  RR*  /\ +oo  e.  RR* )  ->  ( B  <_ +oo  <->  -. +oo  <  B
) )
7872, 76, 77syl2anc 406 . . . . 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 3885 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( A  <  B  <-> +oo  <  B
) )
8279, 81mtbird 639 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  -.  A  <  B )
8347adantr 272 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  C  e.  RR* )
84 xaddcl 9484 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B +e C )  e.  RR* )
8572, 83, 84syl2anc 406 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( B +e C )  e.  RR* )
86 pnfge 9416 . . . . . . 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 972 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  C  =/= -oo )
8988adantr 272 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  C  =/= -oo )
9083, 89, 30syl2anc 406 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( +oo +e C )  = +oo )
9187, 90breqtrrd 3901 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( B +e C )  <_  ( +oo +e C ) )
92 xaddcl 9484 . . . . . . 7  |-  ( ( +oo  e.  RR*  /\  C  e.  RR* )  ->  ( +oo +e C )  e.  RR* )
9375, 83, 92sylancr 408 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( +oo +e C )  e.  RR* )
94 xrlenlt 7701 . . . . . 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 406 . . . . 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 5721 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( A +e C )  =  ( +oo +e C ) )
9897breq1d 3885 . . . 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 639 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  -.  ( A +e C )  <  ( B +e C ) )
10082, 992falsed 659 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = +oo )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
101 simplr 500 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  A  = -oo )
102 mnflt 9410 . . . . . 6  |-  ( B  e.  RR  -> -oo  <  B )
103102adantl 273 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  -> -oo  <  B )
104101, 103eqbrtrd 3895 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  A  <  B
)
105101oveq1d 5721 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  ( A +e C )  =  ( -oo +e
C ) )
106 simpll3 990 . . . . . . . 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 406 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  ( -oo +e C )  = -oo )
110105, 109eqtrd 2132 . . . . 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 9476 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  =  ( B  +  C ) )
113 readdcl 7618 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
114112, 113eqeltrd 2176 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B +e
C )  e.  RR )
115111, 106, 114syl2anc 406 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  e.  RR )  ->  ( B +e C )  e.  RR )
116 mnflt 9410 . . . . . 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 3895 . . . 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 500 . . . . 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 3888 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A  < 
B  <-> -oo  < +oo )
)
123 oveq1 5713 . . . . . . 7  |-  ( A  = -oo  ->  ( A +e C )  =  ( -oo +e C ) )
12447, 59, 61syl2anc 406 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( -oo +e C )  = -oo )
125123, 124sylan9eqr 2154 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = -oo )  ->  ( A +e C )  = -oo )
126125adantr 272 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( A +e C )  = -oo )
12726adantl 273 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( B +e C )  =  ( +oo +e
C ) )
12847, 88, 30syl2anc 406 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( +oo +e C )  = +oo )
129128ad2antrr 475 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( +oo +e C )  = +oo )
130127, 129eqtrd 2132 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = +oo )  ->  ( B +e C )  = +oo )
131126, 130breq12d 3888 . . . 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 500 . . . . 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 3888 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( A  < 
B  <-> -oo  < -oo )
)
136124ad2antrr 475 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( -oo +e C )  = -oo )
137123eqeq1d 2108 . . . . . . 7  |-  ( A  = -oo  ->  (
( A +e
C )  = -oo  <->  ( -oo +e C )  = -oo ) )
138137ad2antlr 476 . . . . . 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 5721 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( B +e C )  =  ( -oo +e
C ) )
141140, 136eqtrd 2132 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  C  e.  RR )  /\  A  = -oo )  /\  B  = -oo )  ->  ( B +e C )  = -oo )
142139, 141breq12d 3888 . . . 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 272 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = -oo )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
145119, 132, 143, 144mpjao3dan 1253 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  /\  A  = -oo )  ->  ( A  <  B  <->  ( A +e C )  <  ( B +e C ) ) )
146 elxr 9404 . . . 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 970 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
14971, 100, 145, 148mpjao3dan 1253 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 929    /\ w3a 930    = wceq 1299    e. wcel 1448    =/= wne 2267   class class class wbr 3875  (class class class)co 5706   RRcr 7499    + caddc 7503   +oocpnf 7669   -oocmnf 7670   RR*cxr 7671    < clt 7672    <_ cle 7673   +ecxad 9398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-i2m1 7600  ax-0id 7603  ax-rnegex 7604  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-xadd 9401
This theorem is referenced by:  xltadd2  9501  xlt2add  9504  xrmaxaddlem  10868
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