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Theorem abslt 11097
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 8337 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A e. RR )
31recnd 7986 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. CC )
4 abscl 11060 . . . . . 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 11083 . . . . . . 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 11059 . . . . . . 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 4030 . . . . 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 8082 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A < B )
14 leabs 11083 . . . . . 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 8082 . . . 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 7986 . . . . . . . 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 8337 . . . . . . . 8 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> -u A e. RR )
23 0red 7958 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> 0 e. RR )
24 ltabs 11096 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
2519, 23, 24ltled 8076 . . . . . . . . 9 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A <_ 0 )
2619le0neg1d 8474 . . . . . . . . 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 11080 . . . . . . . 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 4026 . . . 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 7986 . . . . . . 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 8025 . . . . . 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 8420 . . 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 4004   ` cfv 5217   CCcc 7809   RRcr 7810   0cc0 7811   < clt 7992   <_ cle 7993   -ucneg 8129   abscabs 11006
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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-rp 9654  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008
This theorem is referenced by:  absdiflt  11101  abslti  11147  absltd  11183
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