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Theorem xlt2add 11477
Description: Extended real version of lt2add 10058. Note that ltleadd 10056, 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 xaddcl 11461 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
213ad2ant1 1017 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  e.  RR* )
32adantr 465 . . . . . 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 )  e.  RR* )
4 simp1l 1020 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  e.  RR* )
5 simp2r 1023 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  e.  RR* )
6 xaddcl 11461 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  D  e.  RR* )  ->  ( A +e D )  e.  RR* )
74, 5, 6syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e D )  e.  RR* )
87adantr 465 . . . . . 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 D )  e.  RR* )
9 xaddcl 11461 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  D  e.  RR* )  ->  ( C +e D )  e.  RR* )
1093ad2ant2 1018 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e D )  e.  RR* )
1110adantr 465 . . . . . 6  |-  ( ( ( ( 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* )
12 simp3r 1025 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  <  D )
1312adantr 465 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  B  <  D )
14 simp1r 1021 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  e.  RR* )
1514adantr 465 . . . . . . . 8  |-  ( ( ( ( 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* )
165adantr 465 . . . . . . . 8  |-  ( ( ( ( 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* )
17 simprl 756 . . . . . . . 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.  RR )
18 xltadd2 11474 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  D  e.  RR*  /\  A  e.  RR )  ->  ( B  <  D  <->  ( A +e B )  <  ( A +e D ) ) )
1915, 16, 17, 18syl3anc 1228 . . . . . . 7  |-  ( ( ( ( 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 ) ) )
2013, 19mpbid 210 . . . . . 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 )  <  ( A +e D ) )
21 simp3l 1024 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  <  C )
2221adantr 465 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  <  C )
234adantr 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.  RR* )
24 simp2l 1022 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  e.  RR* )
2524adantr 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.  RR* )
26 simprr 757 . . . . . . . 8  |-  ( ( ( ( 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 )
27 xltadd1 11473 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  D  e.  RR )  ->  ( A  <  C  <->  ( A +e D )  <  ( C +e D ) ) )
2823, 25, 26, 27syl3anc 1228 . . . . . . 7  |-  ( ( ( ( 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 ) ) )
2922, 28mpbid 210 . . . . . 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 D )  <  ( C +e D ) )
303, 8, 11, 20, 29xrlttrd 11387 . . . . 5  |-  ( ( ( ( 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 ) )
3130anassrs 648 . . . 4  |-  ( ( ( ( ( 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 ) )
32 pnfxr 11346 . . . . . . . . . . . 12  |- +oo  e.  RR*
3332a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> +oo  e.  RR* )
34 pnfge 11364 . . . . . . . . . . . 12  |-  ( C  e.  RR*  ->  C  <_ +oo )
3524, 34syl 16 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  <_ +oo )
364, 24, 33, 21, 35xrltletrd 11389 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  < +oo )
37 nltpnft 11392 . . . . . . . . . . . 12  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
3837necon2abid 2711 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A  < +oo  <->  A  =/= +oo )
)
394, 38syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A  < +oo  <->  A  =/= +oo )
)
4036, 39mpbid 210 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  =/= +oo )
41 pnfge 11364 . . . . . . . . . . . 12  |-  ( D  e.  RR*  ->  D  <_ +oo )
425, 41syl 16 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  <_ +oo )
4314, 5, 33, 12, 42xrltletrd 11389 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  < +oo )
44 nltpnft 11392 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  ( B  = +oo  <->  -.  B  < +oo ) )
4544necon2abid 2711 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( B  < +oo  <->  B  =/= +oo )
)
4614, 45syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( B  < +oo  <->  B  =/= +oo )
)
4743, 46mpbid 210 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  =/= +oo )
48 xaddnepnf 11459 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )
)  ->  ( A +e B )  =/= +oo )
494, 40, 14, 47, 48syl22anc 1229 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  =/= +oo )
50 nltpnft 11392 . . . . . . . . . 10  |-  ( ( A +e B )  e.  RR*  ->  ( ( A +e
B )  = +oo  <->  -.  ( A +e B )  < +oo )
)
5150necon2abid 2711 . . . . . . . . 9  |-  ( ( A +e B )  e.  RR*  ->  ( ( A +e
B )  < +oo  <->  ( A +e B )  =/= +oo ) )
522, 51syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( ( A +e B )  < +oo  <->  ( A +e B )  =/= +oo ) )
5349, 52mpbird 232 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  < +oo )
5453adantr 465 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( A +e
B )  < +oo )
55 oveq2 6304 . . . . . . 7  |-  ( D  = +oo  ->  ( C +e D )  =  ( C +e +oo ) )
56 mnfxr 11348 . . . . . . . . . . 11  |- -oo  e.  RR*
5756a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  e.  RR* )
58 mnfle 11367 . . . . . . . . . . 11  |-  ( A  e.  RR*  -> -oo  <_  A )
594, 58syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <_  A
)
6057, 4, 24, 59, 21xrlelttrd 11388 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  C
)
61 ngtmnft 11393 . . . . . . . . . . 11  |-  ( C  e.  RR*  ->  ( C  = -oo  <->  -. -oo  <  C ) )
6261necon2abid 2711 . . . . . . . . . 10  |-  ( C  e.  RR*  ->  ( -oo  <  C  <->  C  =/= -oo )
)
6324, 62syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  C  <->  C  =/= -oo )
)
6460, 63mpbid 210 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  =/= -oo )
65 xaddpnf1 11450 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( C +e +oo )  = +oo )
6624, 64, 65syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e +oo )  = +oo )
6755, 66sylan9eqr 2520 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( C +e
D )  = +oo )
6854, 67breqtrrd 4482 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( A +e
B )  <  ( C +e D ) )
6968adantlr 714 . . . 4  |-  ( ( ( ( ( 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 ) )
70 mnfle 11367 . . . . . . . . . . 11  |-  ( B  e.  RR*  -> -oo  <_  B )
7114, 70syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <_  B
)
7257, 14, 5, 71, 12xrlelttrd 11388 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  D
)
73 ngtmnft 11393 . . . . . . . . . . 11  |-  ( D  e.  RR*  ->  ( D  = -oo  <->  -. -oo  <  D ) )
7473necon2abid 2711 . . . . . . . . . 10  |-  ( D  e.  RR*  ->  ( -oo  <  D  <->  D  =/= -oo )
)
755, 74syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  D  <->  D  =/= -oo )
)
7672, 75mpbid 210 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  =/= -oo )
7776a1d 25 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -.  ( A +e B )  <  ( C +e D )  ->  D  =/= -oo ) )
7877necon4bd 2679 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( D  = -oo  ->  ( A +e B )  <  ( C +e D ) ) )
7978imp 429 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = -oo )  ->  ( A +e
B )  <  ( C +e D ) )
8079adantlr 714 . . . 4  |-  ( ( ( ( ( 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 ) )
81 elxr 11350 . . . . . 6  |-  ( D  e.  RR*  <->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
825, 81sylib 196 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
8382adantr 465 . . . 4  |-  ( ( ( ( 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 ) )
8431, 69, 80, 83mpjao3dan 1295 . . 3  |-  ( ( ( ( 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 ) )
8540a1d 25 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -.  ( A +e B )  <  ( C +e D )  ->  A  =/= +oo ) )
8685necon4bd 2679 . . . 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 ) ) )
8786imp 429 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = +oo )  ->  ( A +e
B )  <  ( C +e D ) )
88 oveq1 6303 . . . . 5  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
89 xaddmnf2 11453 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
9014, 47, 89syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo +e B )  = -oo )
9188, 90sylan9eqr 2520 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  ->  ( A +e
B )  = -oo )
92 xaddnemnf 11458 . . . . . . 7  |-  ( ( ( C  e.  RR*  /\  C  =/= -oo )  /\  ( D  e.  RR*  /\  D  =/= -oo )
)  ->  ( C +e D )  =/= -oo )
9324, 64, 5, 76, 92syl22anc 1229 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e D )  =/= -oo )
94 ngtmnft 11393 . . . . . . . 8  |-  ( ( C +e D )  e.  RR*  ->  ( ( C +e
D )  = -oo  <->  -. -oo 
<  ( C +e D ) ) )
9594necon2abid 2711 . . . . . . 7  |-  ( ( C +e D )  e.  RR*  ->  ( -oo  <  ( C +e D )  <-> 
( C +e
D )  =/= -oo ) )
9610, 95syl 16 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  ( C +e
D )  <->  ( C +e D )  =/= -oo ) )
9793, 96mpbird 232 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  ( C +e D ) )
9897adantr 465 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  -> -oo  <  ( C +e D ) )
9991, 98eqbrtrd 4476 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  ->  ( A +e
B )  <  ( C +e D ) )
100 elxr 11350 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
1014, 100sylib 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 ) )
10284, 87, 99, 101mpjao3dan 1295 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  <  ( C +e D ) )
1031023expia 1198 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 972    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456  (class class class)co 6296   RRcr 9508   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645    <_ cle 9646   +ecxad 11341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-xneg 11343  df-xadd 11344
This theorem is referenced by:  bldisj  21027  iscau3  21843  xrofsup  27742  xrge0addgt0  27841
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