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Theorem abslt 10700
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 499 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. RR )
21renegcld 8009 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A e. RR )
31recnd 7666 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. CC )
4 abscl 10663 . . . . . 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 500 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> B e. RR )
7 leabs 10686 . . . . . . 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 10662 . . . . . . 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 3899 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A <_ ( abs ` A ) )
12 simpr 109 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( abs ` A ) < B )
132, 5, 6, 11, 12lelttrd 7758 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A < B )
14 leabs 10686 . . . . . 6 |- ( A e. RR -> A <_ ( abs ` A ) )
1514ad2antrr 475 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A <_ ( abs ` A ) )
161, 5, 6, 15, 12lelttrd 7758 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A < B )
1713, 16jca 302 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( -u A < B /\ A < B ) )
18 simpll 499 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> A e. RR )
19 simpl 108 . . . . . . . . 9 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A e. RR )
2019recnd 7666 . . . . . . . 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 8009 . . . . . . . 8 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> -u A e. RR )
23 0red 7639 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> 0 e. RR )
24 ltabs 10699 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
2519, 23, 24ltled 7752 . . . . . . . . 9 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A <_ 0 )
2619le0neg1d 8146 . . . . . . . . 9 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> ( A <_ 0 <-> 0 <_ -u A ) )
2725, 26mpbid 146 . . . . . . . 8 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> 0 <_ -u A )
28 absid 10683 . . . . . . . 8 |- ( ( -u A e. RR /\ 0 <_ -u A ) -> ( abs ` -u A ) = -u A )
2922, 27, 28syl2anc 406 . . . . . . 7 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> ( abs ` -u A ) = -u A )
3021, 29eqtr3d 2134 . . . . . 6 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> ( abs ` A ) = -u A )
3118, 30sylan 279 . . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) /\ A < ( abs ` A ) ) -> ( abs ` A ) = -u A )
32 simplrl 505 . . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) /\ A < ( abs ` A ) ) -> -u A < B )
3331, 32eqbrtrd 3895 . . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) /\ A < ( abs ` A ) ) -> ( abs ` A ) < B )
34 simpr 109 . . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) /\ ( abs ` A ) < B ) -> ( abs ` A ) < B )
35 simprr 502 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> A < B )
36 simplr 500 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> B e. RR )
3718recnd 7666 . . . . . . 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 7704 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ ( abs ` A ) e. RR ) -> ( A < B -> ( A < ( abs ` A ) \/ ( abs ` A ) < B ) ) )
4018, 36, 38, 39syl3anc 1184 . . . . 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 753 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> ( abs ` A ) < B )
4317, 42impbida 566 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B <-> ( -u A < B /\ A < B ) ) )
44 ltnegcon1 8092 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) )
4544anbi1d 456 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( -u A < B /\ A < B ) <-> ( -u B < A /\ A < B ) ) )
4643, 45bitrd 187 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 103   <-> wb 104   \/ wo 670   = wceq 1299   e. wcel 1448   class class class wbr 3875   ` cfv 5059   CCcc 7498   RRcr 7499   0cc0 7500   < clt 7672   <_ cle 7673   -ucneg 7805   abscabs 10609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611
This theorem is referenced by:  absdiflt  10704  abslti  10750  absltd  10786
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