ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xposdif Unicode version

Theorem xposdif 9677
Description: Extended real version of posdif 8229. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
Assertion
Ref Expression
xposdif  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )

Proof of Theorem xposdif
StepHypRef Expression
1 elxr 9575 . . 3  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
2 elxr 9575 . . . . 5  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3 posdif 8229 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
4 rexsub 9648 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B +e  -e A )  =  ( B  -  A
) )
54ancoms 266 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B +e  -e A )  =  ( B  -  A
) )
65breq2d 3941 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  ( B +e  -e
A )  <->  0  <  ( B  -  A ) ) )
73, 6bitr4d 190 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
87ex 114 . . . . . 6  |-  ( A  e.  RR  ->  ( B  e.  RR  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
9 rexr 7823 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  RR* )
10 pnfnlt 9585 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  -. +oo  <  B )
1110adantl 275 . . . . . . . . . 10  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
129, 11sylan2 284 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  -. +oo  <  B
)
13 simpl 108 . . . . . . . . . 10  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  A  = +oo )
1413breq1d 3939 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( A  <  B  <-> +oo 
<  B ) )
1512, 14mtbird 662 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  -.  A  <  B
)
16 0xr 7824 . . . . . . . . . 10  |-  0  e.  RR*
17 nltmnf 9586 . . . . . . . . . 10  |-  ( 0  e.  RR*  ->  -.  0  < -oo )
1816, 17ax-mp 5 . . . . . . . . 9  |-  -.  0  < -oo
19 xnegeq 9622 . . . . . . . . . . . . . 14  |-  ( A  = +oo  ->  -e
A  =  -e +oo )
2019adantr 274 . . . . . . . . . . . . 13  |-  ( ( A  = +oo  /\  B  e.  RR )  -> 
-e A  = 
-e +oo )
21 xnegpnf 9623 . . . . . . . . . . . . 13  |-  -e +oo  = -oo
2220, 21syl6eq 2188 . . . . . . . . . . . 12  |-  ( ( A  = +oo  /\  B  e.  RR )  -> 
-e A  = -oo )
2322oveq2d 5790 . . . . . . . . . . 11  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( B +e  -e A )  =  ( B +e -oo ) )
24 renepnf 7825 . . . . . . . . . . . . 13  |-  ( B  e.  RR  ->  B  =/= +oo )
2524adantl 275 . . . . . . . . . . . 12  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  B  =/= +oo )
26 xaddmnf1 9643 . . . . . . . . . . . 12  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( B +e -oo )  = -oo )
279, 25, 26syl2an2 583 . . . . . . . . . . 11  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( B +e -oo )  = -oo )
2823, 27eqtrd 2172 . . . . . . . . . 10  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( B +e  -e A )  = -oo )
2928breq2d 3941 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( 0  <  ( B +e  -e
A )  <->  0  < -oo ) )
3018, 29mtbiri 664 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  -.  0  <  ( B +e  -e
A ) )
3115, 302falsed 691 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
3231ex 114 . . . . . 6  |-  ( A  = +oo  ->  ( B  e.  RR  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
33 simpl 108 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  A  = -oo )
34 mnflt 9581 . . . . . . . . . 10  |-  ( B  e.  RR  -> -oo  <  B )
3534adantl 275 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  -> -oo  <  B )
3633, 35eqbrtrd 3950 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  A  <  B )
37 0ltpnf 9580 . . . . . . . . 9  |-  0  < +oo
38 xnegeq 9622 . . . . . . . . . . . . 13  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
39 xnegmnf 9624 . . . . . . . . . . . . 13  |-  -e -oo  = +oo
4038, 39syl6eq 2188 . . . . . . . . . . . 12  |-  ( A  = -oo  ->  -e
A  = +oo )
4140oveq2d 5790 . . . . . . . . . . 11  |-  ( A  = -oo  ->  ( B +e  -e
A )  =  ( B +e +oo ) )
4241adantr 274 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B +e  -e A )  =  ( B +e +oo ) )
43 renemnf 7826 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  B  =/= -oo )
4443adantl 275 . . . . . . . . . . 11  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  B  =/= -oo )
45 xaddpnf1 9641 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  B  =/= -oo )  ->  ( B +e +oo )  = +oo )
469, 44, 45syl2an2 583 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B +e +oo )  = +oo )
4742, 46eqtrd 2172 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B +e  -e A )  = +oo )
4837, 47breqtrrid 3966 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  0  <  ( B +e  -e
A ) )
4936, 482thd 174 . . . . . . 7  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
5049ex 114 . . . . . 6  |-  ( A  = -oo  ->  ( B  e.  RR  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
518, 32, 503jaoi 1281 . . . . 5  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( B  e.  RR  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
522, 51sylbi 120 . . . 4  |-  ( A  e.  RR*  ->  ( B  e.  RR  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
53 ltpnf 9579 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  < +oo )
5453adantr 274 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  < +oo )
55 simpr 109 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  B  = +oo )
5654, 55breqtrrd 3956 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  <  B )
5755oveq1d 5789 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B +e  -e A )  =  ( +oo +e  -e A ) )
58 rexneg 9625 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
59 renegcl 8035 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  -u A  e.  RR )
6058, 59eqeltrd 2216 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  -e
A  e.  RR )
6160rexrd 7827 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  -e
A  e.  RR* )
6261adantr 274 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e A  e. 
RR* )
6360renemnfd 7829 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  -e
A  =/= -oo )
6463adantr 274 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e A  =/= -oo )
65 xaddpnf2 9642 . . . . . . . . . . 11  |-  ( ( 
-e A  e. 
RR*  /\  -e A  =/= -oo )  -> 
( +oo +e  -e A )  = +oo )
6662, 64, 65syl2anc 408 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( +oo +e  -e A )  = +oo )
6757, 66eqtrd 2172 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( B +e  -e A )  = +oo )
6837, 67breqtrrid 3966 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  0  <  ( B +e  -e
A ) )
6956, 682thd 174 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
7069ex 114 . . . . . 6  |-  ( A  e.  RR  ->  ( B  = +oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
71 pnfxr 7830 . . . . . . . . . 10  |- +oo  e.  RR*
72 xrltnr 9578 . . . . . . . . . 10  |-  ( +oo  e.  RR*  ->  -. +oo  < +oo )
7371, 72ax-mp 5 . . . . . . . . 9  |-  -. +oo  < +oo
74 breq12 3934 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( A  <  B  <-> +oo 
< +oo ) )
7573, 74mtbiri 664 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  = +oo )  ->  -.  A  <  B
)
76 0re 7778 . . . . . . . . . 10  |-  0  e.  RR
7776ltnri 7868 . . . . . . . . 9  |-  -.  0  <  0
78 simpr 109 . . . . . . . . . . . 12  |-  ( ( A  = +oo  /\  B  = +oo )  ->  B  = +oo )
7919, 21syl6eq 2188 . . . . . . . . . . . . 13  |-  ( A  = +oo  ->  -e
A  = -oo )
8079adantr 274 . . . . . . . . . . . 12  |-  ( ( A  = +oo  /\  B  = +oo )  -> 
-e A  = -oo )
8178, 80oveq12d 5792 . . . . . . . . . . 11  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( B +e  -e A )  =  ( +oo +e -oo ) )
82 pnfaddmnf 9645 . . . . . . . . . . 11  |-  ( +oo +e -oo )  =  0
8381, 82syl6eq 2188 . . . . . . . . . 10  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( B +e  -e A )  =  0 )
8483breq2d 3941 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( 0  <  ( B +e  -e
A )  <->  0  <  0 ) )
8577, 84mtbiri 664 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  = +oo )  ->  -.  0  <  ( B +e  -e
A ) )
8675, 852falsed 691 . . . . . . 7  |-  ( ( A  = +oo  /\  B  = +oo )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
8786ex 114 . . . . . 6  |-  ( A  = +oo  ->  ( B  = +oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
88 mnfltpnf 9583 . . . . . . . . 9  |- -oo  < +oo
89 breq12 3934 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  <-> -oo 
< +oo ) )
9088, 89mpbiri 167 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = +oo )  ->  A  <  B )
91 oveq1 5781 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( B +e +oo )  =  ( +oo +e +oo ) )
9241, 91sylan9eq 2192 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( B +e  -e A )  =  ( +oo +e +oo ) )
93 pnfnemnf 7832 . . . . . . . . . . 11  |- +oo  =/= -oo
94 xaddpnf1 9641 . . . . . . . . . . 11  |-  ( ( +oo  e.  RR*  /\ +oo  =/= -oo )  ->  ( +oo +e +oo )  = +oo )
9571, 93, 94mp2an 422 . . . . . . . . . 10  |-  ( +oo +e +oo )  = +oo
9692, 95syl6eq 2188 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( B +e  -e A )  = +oo )
9737, 96breqtrrid 3966 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = +oo )  ->  0  <  ( B +e  -e
A ) )
9890, 972thd 174 . . . . . . 7  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
9998ex 114 . . . . . 6  |-  ( A  = -oo  ->  ( B  = +oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
10070, 87, 993jaoi 1281 . . . . 5  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( B  = +oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
1012, 100sylbi 120 . . . 4  |-  ( A  e.  RR*  ->  ( B  = +oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
102 rexr 7823 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  RR* )
103102adantr 274 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  A  e.  RR* )
104 nltmnf 9586 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  -.  A  < -oo )
105103, 104syl 14 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
106 simpr 109 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  B  = -oo )
107106breq2d 3941 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
108105, 107mtbird 662 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  <  B
)
109106oveq1d 5789 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( B +e  -e A )  =  ( -oo +e  -e A ) )
110 rexr 7823 . . . . . . . . . . . . . 14  |-  (  -e A  e.  RR  -> 
-e A  e. 
RR* )
111 renepnf 7825 . . . . . . . . . . . . . 14  |-  (  -e A  e.  RR  -> 
-e A  =/= +oo )
112 xaddmnf2 9644 . . . . . . . . . . . . . 14  |-  ( ( 
-e A  e. 
RR*  /\  -e A  =/= +oo )  -> 
( -oo +e  -e A )  = -oo )
113110, 111, 112syl2anc 408 . . . . . . . . . . . . 13  |-  (  -e A  e.  RR  ->  ( -oo +e  -e A )  = -oo )
11460, 113syl 14 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  ( -oo +e  -e
A )  = -oo )
115114adantr 274 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( -oo +e  -e A )  = -oo )
116109, 115eqtrd 2172 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( B +e  -e A )  = -oo )
117116breq2d 3941 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( 0  <  ( B +e  -e
A )  <->  0  < -oo ) )
11818, 117mtbiri 664 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  0  <  ( B +e  -e
A ) )
119108, 1182falsed 691 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
120119ex 114 . . . . . 6  |-  ( A  e.  RR  ->  ( B  = -oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
121 eleq1 2202 . . . . . . . . . . . 12  |-  ( A  = +oo  ->  ( A  e.  RR*  <-> +oo  e.  RR* ) )
12271, 121mpbiri 167 . . . . . . . . . . 11  |-  ( A  = +oo  ->  A  e.  RR* )
123122adantr 274 . . . . . . . . . 10  |-  ( ( A  = +oo  /\  B  = -oo )  ->  A  e.  RR* )
124123, 104syl 14 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  = -oo )  ->  -.  A  < -oo )
125 simpr 109 . . . . . . . . . 10  |-  ( ( A  = +oo  /\  B  = -oo )  ->  B  = -oo )
126125breq2d 3941 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
127124, 126mtbird 662 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  = -oo )  ->  -.  A  <  B
)
12879oveq2d 5790 . . . . . . . . . . . 12  |-  ( A  = +oo  ->  ( B +e  -e
A )  =  ( B +e -oo ) )
129128adantr 274 . . . . . . . . . . 11  |-  ( ( A  = +oo  /\  B  = -oo )  ->  ( B +e  -e A )  =  ( B +e -oo ) )
130 mnfxr 7834 . . . . . . . . . . . . 13  |- -oo  e.  RR*
131 eleq1 2202 . . . . . . . . . . . . 13  |-  ( B  = -oo  ->  ( B  e.  RR*  <-> -oo  e.  RR* ) )
132130, 131mpbiri 167 . . . . . . . . . . . 12  |-  ( B  = -oo  ->  B  e.  RR* )
133 mnfnepnf 7833 . . . . . . . . . . . . . 14  |- -oo  =/= +oo
134 neeq1 2321 . . . . . . . . . . . . . 14  |-  ( B  = -oo  ->  ( B  =/= +oo  <-> -oo  =/= +oo )
)
135133, 134mpbiri 167 . . . . . . . . . . . . 13  |-  ( B  = -oo  ->  B  =/= +oo )
136135adantl 275 . . . . . . . . . . . 12  |-  ( ( A  = +oo  /\  B  = -oo )  ->  B  =/= +oo )
137132, 136, 26syl2an2 583 . . . . . . . . . . 11  |-  ( ( A  = +oo  /\  B  = -oo )  ->  ( B +e -oo )  = -oo )
138129, 137eqtrd 2172 . . . . . . . . . 10  |-  ( ( A  = +oo  /\  B  = -oo )  ->  ( B +e  -e A )  = -oo )
139138breq2d 3941 . . . . . . . . 9  |-  ( ( A  = +oo  /\  B  = -oo )  ->  ( 0  <  ( B +e  -e
A )  <->  0  < -oo ) )
14018, 139mtbiri 664 . . . . . . . 8  |-  ( ( A  = +oo  /\  B  = -oo )  ->  -.  0  <  ( B +e  -e
A ) )
141127, 1402falsed 691 . . . . . . 7  |-  ( ( A  = +oo  /\  B  = -oo )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
142141ex 114 . . . . . 6  |-  ( A  = +oo  ->  ( B  = -oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
143 xrltnr 9578 . . . . . . . . . 10  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
144130, 143ax-mp 5 . . . . . . . . 9  |-  -. -oo  < -oo
145 breq12 3934 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <-> -oo 
< -oo ) )
146144, 145mtbiri 664 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = -oo )  ->  -.  A  <  B
)
147 oveq1 5781 . . . . . . . . . . . 12  |-  ( B  = -oo  ->  ( B +e +oo )  =  ( -oo +e +oo ) )
14841, 147sylan9eq 2192 . . . . . . . . . . 11  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( B +e  -e A )  =  ( -oo +e +oo ) )
149 mnfaddpnf 9646 . . . . . . . . . . 11  |-  ( -oo +e +oo )  =  0
150148, 149syl6eq 2188 . . . . . . . . . 10  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( B +e  -e A )  =  0 )
151150breq2d 3941 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( 0  <  ( B +e  -e
A )  <->  0  <  0 ) )
15277, 151mtbiri 664 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  = -oo )  ->  -.  0  <  ( B +e  -e
A ) )
153146, 1522falsed 691 . . . . . . 7  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
154153ex 114 . . . . . 6  |-  ( A  = -oo  ->  ( B  = -oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
155120, 142, 1543jaoi 1281 . . . . 5  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( B  = -oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
1562, 155sylbi 120 . . . 4  |-  ( A  e.  RR*  ->  ( B  = -oo  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
15752, 101, 1563jaod 1282 . . 3  |-  ( A  e.  RR*  ->  ( ( B  e.  RR  \/  B  = +oo  \/  B  = -oo )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
1581, 157syl5bi 151 . 2  |-  ( A  e.  RR*  ->  ( B  e.  RR*  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) ) )
159158imp 123 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  0  <  ( B +e  -e A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 961    = wceq 1331    e. wcel 1480    =/= wne 2308   class class class wbr 3929  (class class class)co 5774   RRcr 7631   0cc0 7632   +oocpnf 7809   -oocmnf 7810   RR*cxr 7811    < clt 7812    - cmin 7945   -ucneg 7946    -ecxne 9568   +ecxad 9569
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-sub 7947  df-neg 7948  df-xneg 9571  df-xadd 9572
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator