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Theorem xlt2add 11223
Description: Extended real version of lt2add 9824. Note that ltleadd 9822, which has weaker assumptions, is not true for the extended reals (since  0  + +oo  <  1  + +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
xlt2add  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A  <  C  /\  B  <  D )  ->  ( A +e B )  <  ( C +e D ) ) )

Proof of Theorem xlt2add
StepHypRef Expression
1 simp1l 1012 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  e.  RR* )
2 elxr 11096 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
31, 2sylib 196 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
4 simp2r 1015 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  e.  RR* )
5 elxr 11096 . . . . . . 7  |-  ( D  e.  RR*  <->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
64, 5sylib 196 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
76adantr 465 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  ->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
8 xaddcl 11207 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
983ad2ant1 1009 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  e.  RR* )
109adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e B )  e.  RR* )
11 xaddcl 11207 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  D  e.  RR* )  ->  ( A +e D )  e.  RR* )
121, 4, 11syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e D )  e.  RR* )
1312adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e D )  e.  RR* )
14 xaddcl 11207 . . . . . . . . . 10  |-  ( ( C  e.  RR*  /\  D  e.  RR* )  ->  ( C +e D )  e.  RR* )
15143ad2ant2 1010 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e D )  e.  RR* )
1615adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( C +e D )  e.  RR* )
17 simp3r 1017 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  <  D )
1817adantr 465 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  B  <  D )
19 simp1r 1013 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  e.  RR* )
2019adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR* )
214adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR* )
22 simprl 755 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
23 xltadd2 11220 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  D  e.  RR*  /\  A  e.  RR )  ->  ( B  <  D  <->  ( A +e B )  <  ( A +e D ) ) )
2420, 21, 22, 23syl3anc 1218 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( B  <  D  <->  ( A +e B )  < 
( A +e
D ) ) )
2518, 24mpbid 210 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e B )  <  ( A +e D ) )
26 simp3l 1016 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  <  C )
2726adantr 465 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  <  C )
281adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR* )
29 simp2l 1014 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  e.  RR* )
3029adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR* )
31 simprr 756 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
32 xltadd1 11219 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  D  e.  RR )  ->  ( A  <  C  <->  ( A +e D )  <  ( C +e D ) ) )
3328, 30, 31, 32syl3anc 1218 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A  <  C  <->  ( A +e D )  < 
( C +e
D ) ) )
3427, 33mpbid 210 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e D )  <  ( C +e D ) )
3510, 13, 16, 25, 34xrlttrd 11133 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e B )  <  ( C +e D ) )
3635anassrs 648 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  D  e.  RR )  ->  ( A +e B )  < 
( C +e
D ) )
37 pnfxr 11092 . . . . . . . . . . . . . 14  |- +oo  e.  RR*
3837a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> +oo  e.  RR* )
39 pnfge 11110 . . . . . . . . . . . . . 14  |-  ( C  e.  RR*  ->  C  <_ +oo )
4029, 39syl 16 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  <_ +oo )
411, 29, 38, 26, 40xrltletrd 11135 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  < +oo )
42 nltpnft 11138 . . . . . . . . . . . . . 14  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
4342necon2abid 2668 . . . . . . . . . . . . 13  |-  ( A  e.  RR*  ->  ( A  < +oo  <->  A  =/= +oo )
)
441, 43syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A  < +oo  <->  A  =/= +oo )
)
4541, 44mpbid 210 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  =/= +oo )
46 pnfge 11110 . . . . . . . . . . . . . 14  |-  ( D  e.  RR*  ->  D  <_ +oo )
474, 46syl 16 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  <_ +oo )
4819, 4, 38, 17, 47xrltletrd 11135 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  < +oo )
49 nltpnft 11138 . . . . . . . . . . . . . 14  |-  ( B  e.  RR*  ->  ( B  = +oo  <->  -.  B  < +oo ) )
5049necon2abid 2668 . . . . . . . . . . . . 13  |-  ( B  e.  RR*  ->  ( B  < +oo  <->  B  =/= +oo )
)
5119, 50syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( B  < +oo  <->  B  =/= +oo )
)
5248, 51mpbid 210 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  =/= +oo )
53 xaddnepnf 11205 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )
)  ->  ( A +e B )  =/= +oo )
541, 45, 19, 52, 53syl22anc 1219 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  =/= +oo )
55 nltpnft 11138 . . . . . . . . . . . 12  |-  ( ( A +e B )  e.  RR*  ->  ( ( A +e
B )  = +oo  <->  -.  ( A +e B )  < +oo )
)
5655necon2abid 2668 . . . . . . . . . . 11  |-  ( ( A +e B )  e.  RR*  ->  ( ( A +e
B )  < +oo  <->  ( A +e B )  =/= +oo ) )
579, 56syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( ( A +e B )  < +oo  <->  ( A +e B )  =/= +oo ) )
5854, 57mpbird 232 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  < +oo )
5958adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( A +e
B )  < +oo )
60 oveq2 6099 . . . . . . . . 9  |-  ( D  = +oo  ->  ( C +e D )  =  ( C +e +oo ) )
61 mnfxr 11094 . . . . . . . . . . . . 13  |- -oo  e.  RR*
6261a1i 11 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  e.  RR* )
63 mnfle 11113 . . . . . . . . . . . . 13  |-  ( A  e.  RR*  -> -oo  <_  A )
641, 63syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <_  A
)
6562, 1, 29, 64, 26xrlelttrd 11134 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  C
)
66 ngtmnft 11139 . . . . . . . . . . . . 13  |-  ( C  e.  RR*  ->  ( C  = -oo  <->  -. -oo  <  C ) )
6766necon2abid 2668 . . . . . . . . . . . 12  |-  ( C  e.  RR*  ->  ( -oo  <  C  <->  C  =/= -oo )
)
6829, 67syl 16 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  C  <->  C  =/= -oo )
)
6965, 68mpbid 210 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  =/= -oo )
70 xaddpnf1 11196 . . . . . . . . . 10  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( C +e +oo )  = +oo )
7129, 69, 70syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e +oo )  = +oo )
7260, 71sylan9eqr 2497 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( C +e
D )  = +oo )
7359, 72breqtrrd 4318 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( A +e
B )  <  ( C +e D ) )
7473adantlr 714 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  D  = +oo )  ->  ( A +e B )  < 
( C +e
D ) )
75 mnfle 11113 . . . . . . . . . . . . 13  |-  ( B  e.  RR*  -> -oo  <_  B )
7619, 75syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <_  B
)
7762, 19, 4, 76, 17xrlelttrd 11134 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  D
)
78 ngtmnft 11139 . . . . . . . . . . . . 13  |-  ( D  e.  RR*  ->  ( D  = -oo  <->  -. -oo  <  D ) )
7978necon2abid 2668 . . . . . . . . . . . 12  |-  ( D  e.  RR*  ->  ( -oo  <  D  <->  D  =/= -oo )
)
804, 79syl 16 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  D  <->  D  =/= -oo )
)
8177, 80mpbid 210 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  =/= -oo )
8281a1d 25 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -.  ( A +e B )  <  ( C +e D )  ->  D  =/= -oo ) )
8382necon4bd 2673 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( D  = -oo  ->  ( A +e B )  <  ( C +e D ) ) )
8483imp 429 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = -oo )  ->  ( A +e
B )  <  ( C +e D ) )
8584adantlr 714 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  D  = -oo )  ->  ( A +e B )  < 
( C +e
D ) )
8636, 74, 853jaodan 1284 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )  ->  ( A +e B )  <  ( C +e D ) )
877, 86mpdan 668 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  ->  ( A +e
B )  <  ( C +e D ) )
8845a1d 25 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -.  ( A +e B )  <  ( C +e D )  ->  A  =/= +oo ) )
8988necon4bd 2673 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A  = +oo  ->  ( A +e B )  <  ( C +e D ) ) )
9089imp 429 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = +oo )  ->  ( A +e
B )  <  ( C +e D ) )
91 oveq1 6098 . . . . . 6  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
92 xaddmnf2 11199 . . . . . . 7  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
9319, 52, 92syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo +e B )  = -oo )
9491, 93sylan9eqr 2497 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  ->  ( A +e
B )  = -oo )
95 xaddnemnf 11204 . . . . . . . 8  |-  ( ( ( C  e.  RR*  /\  C  =/= -oo )  /\  ( D  e.  RR*  /\  D  =/= -oo )
)  ->  ( C +e D )  =/= -oo )
9629, 69, 4, 81, 95syl22anc 1219 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e D )  =/= -oo )
97 ngtmnft 11139 . . . . . . . . 9  |-  ( ( C +e D )  e.  RR*  ->  ( ( C +e
D )  = -oo  <->  -. -oo 
<  ( C +e D ) ) )
9897necon2abid 2668 . . . . . . . 8  |-  ( ( C +e D )  e.  RR*  ->  ( -oo  <  ( C +e D )  <-> 
( C +e
D )  =/= -oo ) )
9915, 98syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  ( C +e
D )  <->  ( C +e D )  =/= -oo ) )
10096, 99mpbird 232 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  ( C +e D ) )
101100adantr 465 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  -> -oo  <  ( C +e D ) )
10294, 101eqbrtrd 4312 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  ->  ( A +e
B )  <  ( C +e D ) )
10387, 90, 1023jaodan 1284 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )  ->  ( A +e B )  <  ( C +e D ) )
1043, 103mpdan 668 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  <  ( C +e D ) )
1051043expia 1189 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A  <  C  /\  B  <  D )  ->  ( A +e B )  <  ( C +e D ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   class class class wbr 4292  (class class class)co 6091   RRcr 9281   +oocpnf 9415   -oocmnf 9416   RR*cxr 9417    < clt 9418    <_ cle 9419   +ecxad 11087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-xneg 11089  df-xadd 11090
This theorem is referenced by:  bldisj  19973  iscau3  20789  xrofsup  26055  xrge0addgt0  26154
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