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Theorem abslt 11599
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 8526 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A e. RR )
31recnd 8175 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. CC )
4 abscl 11562 . . . . . 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 11585 . . . . . . 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 11561 . . . . . . 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 4109 . . . . 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 8271 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A < B )
14 leabs 11585 . . . . . 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 8271 . . . 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 8175 . . . . . . . 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 8526 . . . . . . . 8 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> -u A e. RR )
23 0red 8147 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> 0 e. RR )
24 ltabs 11598 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
2519, 23, 24ltled 8265 . . . . . . . . 9 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A <_ 0 )
2619le0neg1d 8664 . . . . . . . . 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 11582 . . . . . . . 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 2264 . . . . . 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 4105 . . . 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 8175 . . . . . . 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 8214 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ ( abs ` A ) e. RR ) -> ( A < B -> ( A < ( abs ` A ) \/ ( abs ` A ) < B ) ) )
4018, 36, 38, 39syl3anc 1271 . . . . 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 803 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> ( abs ` A ) < B )
4317, 42impbida 598 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B <-> ( -u A < B /\ A < B ) ) )
44 ltnegcon1 8610 . . 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 713   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318   CCcc 7997   RRcr 7998   0cc0 7999   < clt 8181   <_ cle 8182   -ucneg 8318   abscabs 11508
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-rp 9850  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510
This theorem is referenced by:  absdiflt  11603  abslti  11649  absltd  11685
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