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Theorem xltnegi 9806
Description: Forward direction of xltneg 9807. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xltnegi  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  -e
B  <  -e A )

Proof of Theorem xltnegi
StepHypRef Expression
1 elxr 9747 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9747 . . . . . 6  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 ltneg 8393 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -u B  <  -u A
) )
4 rexneg 9801 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  =  -u B
)
5 rexneg 9801 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
64, 5breqan12rd 4015 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (  -e B  <  -e A  <->  -u B  <  -u A ) )
73, 6bitr4d 191 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -e B  <  -e
A ) )
87biimpd 144 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
9 xnegeq 9798 . . . . . . . . . . 11  |-  ( B  = +oo  ->  -e
B  =  -e +oo )
10 xnegpnf 9799 . . . . . . . . . . 11  |-  -e +oo  = -oo
119, 10eqtrdi 2224 . . . . . . . . . 10  |-  ( B  = +oo  ->  -e
B  = -oo )
1211adantl 277 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e B  = -oo )
13 renegcl 8192 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  -u A  e.  RR )
145, 13eqeltrd 2252 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  -e
A  e.  RR )
15 mnflt 9754 . . . . . . . . . . 11  |-  (  -e A  e.  RR  -> -oo  <  -e A )
1614, 15syl 14 . . . . . . . . . 10  |-  ( A  e.  RR  -> -oo  <  -e A )
1716adantr 276 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  -> -oo  <  -e A )
1812, 17eqbrtrd 4020 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  -> 
-e B  <  -e A )
1918a1d 22 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
20 simpr 110 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  B  = -oo )
2120breq2d 4010 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
22 rexr 7977 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  RR* )
23 nltmnf 9759 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2422, 23syl 14 . . . . . . . . . 10  |-  ( A  e.  RR  ->  -.  A  < -oo )
2524adantr 276 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  -.  A  < -oo )
2625pm2.21d 619 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  < -oo  -> 
-e B  <  -e A ) )
2721, 26sylbid 150 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  -> 
-e B  <  -e A ) )
288, 19, 273jaodan 1306 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  -e
B  <  -e A ) )
292, 28sylan2b 287 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3029expimpd 363 . . . 4  |-  ( A  e.  RR  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
31 simpl 109 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  A  = +oo )
3231breq1d 4008 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
33 pnfnlt 9758 . . . . . . . 8  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3433adantl 277 . . . . . . 7  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
3534pm2.21d 619 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo  <  B  ->  -e B  <  -e
A ) )
3632, 35sylbid 150 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  -> 
-e B  <  -e A ) )
3736expimpd 363 . . . 4  |-  ( A  = +oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
38 breq1 4001 . . . . . 6  |-  ( A  = -oo  ->  ( A  <  B  <-> -oo  <  B
) )
3938anbi2d 464 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  <-> 
( B  e.  RR*  /\ -oo  <  B ) ) )
40 renegcl 8192 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  -u B  e.  RR )
414, 40eqeltrd 2252 . . . . . . . . . 10  |-  ( B  e.  RR  ->  -e
B  e.  RR )
4241adantr 276 . . . . . . . . 9  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  e.  RR )
43 ltpnf 9751 . . . . . . . . 9  |-  (  -e B  e.  RR  -> 
-e B  < +oo )
4442, 43syl 14 . . . . . . . 8  |-  ( ( B  e.  RR  /\ -oo 
<  B )  ->  -e
B  < +oo )
4511adantr 276 . . . . . . . . 9  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  = -oo )
46 mnfltpnf 9756 . . . . . . . . 9  |- -oo  < +oo
4745, 46eqbrtrdi 4037 . . . . . . . 8  |-  ( ( B  = +oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
48 breq2 4002 . . . . . . . . . 10  |-  ( B  = -oo  ->  ( -oo  <  B  <-> -oo  < -oo ) )
49 mnfxr 7988 . . . . . . . . . . . 12  |- -oo  e.  RR*
50 nltmnf 9759 . . . . . . . . . . . 12  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
5149, 50ax-mp 5 . . . . . . . . . . 11  |-  -. -oo  < -oo
5251pm2.21i 646 . . . . . . . . . 10  |-  ( -oo  < -oo  ->  -e B  < +oo )
5348, 52syl6bi 163 . . . . . . . . 9  |-  ( B  = -oo  ->  ( -oo  <  B  ->  -e
B  < +oo )
)
5453imp 124 . . . . . . . 8  |-  ( ( B  = -oo  /\ -oo 
<  B )  ->  -e
B  < +oo )
5544, 47, 543jaoian 1305 . . . . . . 7  |-  ( ( ( B  e.  RR  \/  B  = +oo  \/  B  = -oo )  /\ -oo  <  B
)  ->  -e B  < +oo )
562, 55sylanb 284 . . . . . 6  |-  ( ( B  e.  RR*  /\ -oo  <  B )  ->  -e
B  < +oo )
57 xnegeq 9798 . . . . . . . 8  |-  ( A  = -oo  ->  -e
A  =  -e -oo )
58 xnegmnf 9800 . . . . . . . 8  |-  -e -oo  = +oo
5957, 58eqtrdi 2224 . . . . . . 7  |-  ( A  = -oo  ->  -e
A  = +oo )
6059breq2d 4010 . . . . . 6  |-  ( A  = -oo  ->  (  -e B  <  -e
A  <->  -e B  < +oo ) )
6156, 60syl5ibr 156 . . . . 5  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\ -oo  <  B )  ->  -e B  <  -e
A ) )
6239, 61sylbid 150 . . . 4  |-  ( A  = -oo  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
6330, 37, 623jaoi 1303 . . 3  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  (
( B  e.  RR*  /\  A  <  B )  ->  -e B  <  -e A ) )
641, 63sylbi 121 . 2  |-  ( A  e.  RR*  ->  ( ( B  e.  RR*  /\  A  <  B )  ->  -e
B  <  -e A ) )
65643impib 1201 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  -e
B  <  -e A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ w3o 977    /\ w3a 978    = wceq 1353    e. wcel 2146   class class class wbr 3998   RRcr 7785   +oocpnf 7963   -oocmnf 7964   RR*cxr 7965    < clt 7966   -ucneg 8103    -ecxne 9740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltadd 7902
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-sub 8104  df-neg 8105  df-xneg 9743
This theorem is referenced by:  xltneg  9807
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