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Theorem abslt 11068
Description: Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
Assertion
Ref Expression
abslt  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u B  <  A  /\  A  < 
B ) ) )

Proof of Theorem abslt
StepHypRef Expression
1 simpll 527 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  RR )
21renegcld 8314 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  e.  RR )
31recnd 7963 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  CC )
4 abscl 11031 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
53, 4syl 14 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  e.  RR )
6 simplr 528 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  B  e.  RR )
7 leabs 11054 . . . . . . 7  |-  ( -u A  e.  RR  ->  -u A  <_  ( abs `  -u A
) )
82, 7syl 14 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  -u A ) )
9 absneg 11030 . . . . . . 7  |-  ( A  e.  CC  ->  ( abs `  -u A )  =  ( abs `  A
) )
103, 9syl 14 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  -u A
)  =  ( abs `  A ) )
118, 10breqtrd 4026 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  A ) )
12 simpr 110 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  <  B )
132, 5, 6, 11, 12lelttrd 8059 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <  B
)
14 leabs 11054 . . . . . 6  |-  ( A  e.  RR  ->  A  <_  ( abs `  A
) )
1514ad2antrr 488 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <_  ( abs `  A ) )
161, 5, 6, 15, 12lelttrd 8059 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <  B )
1713, 16jca 306 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( -u A  < 
B  /\  A  <  B ) )
18 simpll 527 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  A  e.  RR )
19 simpl 109 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  e.  RR )
2019recnd 7963 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  e.  CC )
2120, 9syl 14 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( abs `  -u A )  =  ( abs `  A
) )
2219renegcld 8314 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  -u A  e.  RR )
23 0red 7936 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  0  e.  RR )
24 ltabs 11067 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  <  0 )
2519, 23, 24ltled 8053 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  <_  0 )
2619le0neg1d 8451 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( A  <_  0  <->  0  <_  -u A ) )
2725, 26mpbid 147 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  0  <_ 
-u A )
28 absid 11051 . . . . . . . 8  |-  ( (
-u A  e.  RR  /\  0  <_  -u A )  ->  ( abs `  -u A
)  =  -u A
)
2922, 27, 28syl2anc 411 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( abs `  -u A )  = 
-u A )
3021, 29eqtr3d 2212 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( abs `  A )  = 
-u A )
3118, 30sylan 283 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  A  <  ( abs `  A ) )  ->  ( abs `  A
)  =  -u A
)
32 simplrl 535 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  A  <  ( abs `  A ) )  ->  -u A  <  B
)
3331, 32eqbrtrd 4022 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  A  <  ( abs `  A ) )  ->  ( abs `  A
)  <  B )
34 simpr 110 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  <  B )
35 simprr 531 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  A  <  B )
36 simplr 528 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  B  e.  RR )
3718recnd 7963 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  A  e.  CC )
3837, 4syl 14 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  -> 
( abs `  A
)  e.  RR )
39 axltwlin 8002 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( abs `  A )  e.  RR )  ->  ( A  <  B  ->  ( A  <  ( abs `  A
)  \/  ( abs `  A )  <  B
) ) )
4018, 36, 38, 39syl3anc 1238 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  -> 
( A  <  B  ->  ( A  <  ( abs `  A )  \/  ( abs `  A
)  <  B )
) )
4135, 40mpd 13 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  -> 
( A  <  ( abs `  A )  \/  ( abs `  A
)  <  B )
)
4233, 34, 41mpjaodan 798 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  -> 
( abs `  A
)  <  B )
4317, 42impbida 596 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u A  <  B  /\  A  < 
B ) ) )
44 ltnegcon1 8397 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  < 
B  <->  -u B  <  A
) )
4544anbi1d 465 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( -u A  <  B  /\  A  < 
B )  <->  ( -u B  <  A  /\  A  < 
B ) ) )
4643, 45bitrd 188 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u B  <  A  /\  A  < 
B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   class class class wbr 4000   ` cfv 5211   CCcc 7787   RRcr 7788   0cc0 7789    < clt 7969    <_ cle 7970   -ucneg 8106   abscabs 10977
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908  ax-caucvg 7909
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-n0 9153  df-z 9230  df-uz 9505  df-rp 9628  df-seqfrec 10419  df-exp 10493  df-cj 10822  df-re 10823  df-im 10824  df-rsqrt 10978  df-abs 10979
This theorem is referenced by:  absdiflt  11072  abslti  11118  absltd  11154
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