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Theorem leabs 11304
Description: A real number is less than or equal to its absolute value. (Contributed by NM, 27-Feb-2005.)
Assertion
Ref Expression
leabs |- ( A e. RR -> A <_ ( abs ` A ) )
Proof of Theorem leabs
StepHypRef Expression
1 simpr 110 . . . . 5 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ ( abs ` A ) < 0 ) -> ( abs ` A ) < 0 )
2 recn 8040 . . . . . . . 8 |- ( A e. RR -> A e. CC )
3 absge0 11290 . . . . . . . 8 |- ( A e. CC -> 0 <_ ( abs ` A ) )
42, 3syl 14 . . . . . . 7 |- ( A e. RR -> 0 <_ ( abs ` A ) )
54ad2antrr 488 . . . . . 6 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ ( abs ` A ) < 0 ) -> 0 <_ ( abs ` A ) )
6 0red 8055 . . . . . . 7 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ ( abs ` A ) < 0 ) -> 0 e. RR )
7 abscl 11281 . . . . . . . . 9 |- ( A e. CC -> ( abs ` A ) e. RR )
82, 7syl 14 . . . . . . . 8 |- ( A e. RR -> ( abs ` A ) e. RR )
98ad2antrr 488 . . . . . . 7 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ ( abs ` A ) < 0 ) -> ( abs ` A ) e. RR )
106, 9lenltd 8172 . . . . . 6 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ ( abs ` A ) < 0 ) -> ( 0 <_ ( abs ` A ) <-> -. ( abs ` A ) < 0 ) )
115, 10mpbid 147 . . . . 5 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ ( abs ` A ) < 0 ) -> -. ( abs ` A ) < 0 )
121, 11pm2.21fal 1392 . . . 4 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ ( abs ` A ) < 0 ) -> F. )
13 simpll 527 . . . . . . 7 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ 0 < A ) -> A e. RR )
14 0red 8055 . . . . . . . 8 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ 0 < A ) -> 0 e. RR )
15 simpr 110 . . . . . . . 8 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ 0 < A ) -> 0 < A )
1614, 13, 15ltled 8173 . . . . . . 7 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ 0 < A ) -> 0 <_ A )
17 absid 11301 . . . . . . 7 |- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )
1813, 16, 17syl2anc 411 . . . . . 6 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ 0 < A ) -> ( abs ` A ) = A )
19 simplr 528 . . . . . 6 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ 0 < A ) -> ( abs ` A ) < A )
2018, 19eqbrtrrd 4067 . . . . 5 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ 0 < A ) -> A < A )
2113ltnrd 8166 . . . . 5 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ 0 < A ) -> -. A < A )
2220, 21pm2.21fal 1392 . . . 4 |- ( ( ( A e. RR /\ ( abs ` A ) < A ) /\ 0 < A ) -> F. )
23 0re 8054 . . . . . . 7 |- 0 e. RR
24 axltwlin 8122 . . . . . . 7 |- ( ( ( abs ` A ) e. RR /\ A e. RR /\ 0 e. RR ) -> ( ( abs ` A ) < A -> ( ( abs ` A ) < 0 \/ 0 < A ) ) )
2523, 24mp3an3 1338 . . . . . 6 |- ( ( ( abs ` A ) e. RR /\ A e. RR ) -> ( ( abs ` A ) < A -> ( ( abs ` A ) < 0 \/ 0 < A ) ) )
268, 25mpancom 422 . . . . 5 |- ( A e. RR -> ( ( abs ` A ) < A -> ( ( abs ` A ) < 0 \/ 0 < A ) ) )
2726imp 124 . . . 4 |- ( ( A e. RR /\ ( abs ` A ) < A ) -> ( ( abs ` A ) < 0 \/ 0 < A ) )
2812, 22, 27mpjaodan 799 . . 3 |- ( ( A e. RR /\ ( abs ` A ) < A ) -> F. )
2928inegd 1391 . 2 |- ( A e. RR -> -. ( abs ` A ) < A )
30 id 19 . . 3 |- ( A e. RR -> A e. RR )
3130, 8lenltd 8172 . 2 |- ( A e. RR -> ( A <_ ( abs ` A ) <-> -. ( abs ` A ) < A ) )
3229, 31mpbird 167 1 |- ( A e. RR -> A <_ ( abs ` A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1372   F. wfal 1377   e. wcel 2175   class class class wbr 4043   ` cfv 5268   CCcc 7905   RRcr 7906   0cc0 7907   < clt 8089   <_ cle 8090   abscabs 11227
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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025  ax-arch 8026  ax-caucvg 8027
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-frec 6467  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-2 9077  df-3 9078  df-4 9079  df-n0 9278  df-z 9355  df-uz 9631  df-rp 9758  df-seqfrec 10574  df-exp 10665  df-cj 11072  df-re 11073  df-im 11074  df-rsqrt 11228  df-abs 11229
This theorem is referenced by:  abslt  11318  absle  11319  abssubap0  11320  releabs  11326  leabsi  11358  leabsd  11391  dfabsmax  11447
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