ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abslt Unicode version
Theorem abslt 11832
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 531 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. RR )
21renegcld 8697 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A e. RR )
31recnd 8344 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. CC )
4 abscl 11795 . . . . . 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 533 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> B e. RR )
7 leabs 11818 . . . . . . 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 11794 . . . . . . 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 4151 . . . . 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 8441 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A < B )
14 leabs 11818 . . . . . 6 |- ( A e. RR -> A <_ ( abs ` A ) )
1514ad2antrr 492 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A <_ ( abs ` A ) )
161, 5, 6, 15, 12lelttrd 8441 . . . 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 531 . . . . . 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 8344 . . . . . . . 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 8697 . . . . . . . 8 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> -u A e. RR )
23 0red 8317 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> 0 e. RR )
24 ltabs 11831 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
2519, 23, 24ltled 8435 . . . . . . . . 9 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A <_ 0 )
2619le0neg1d 8835 . . . . . . . . 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 11815 . . . . . . . 8 |- ( ( -u A e. RR /\ 0 <_ -u A ) -> ( abs ` -u A ) = -u A )
2922, 27, 28syl2anc 415 . . . . . . 7 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> ( abs ` -u A ) = -u A )
3021, 29eqtr3d 2273 . . . . . 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 541 . . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) /\ A < ( abs ` A ) ) -> -u A < B )
3331, 32eqbrtrd 4147 . . . 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 537 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> A < B )
36 simplr 533 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> B e. RR )
3718recnd 8344 . . . . . . 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 8383 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ ( abs ` A ) e. RR ) -> ( A < B -> ( A < ( abs ` A ) \/ ( abs ` A ) < B ) ) )
4018, 36, 38, 39syl3anc 1278 . . . . 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 810 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> ( abs ` A ) < B )
4317, 42impbida 604 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B <-> ( -u A < B /\ A < B ) ) )
44 ltnegcon1 8781 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) )
4544anbi1d 469 . 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 720   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372   CCcc 8167   RRcr 8168   0cc0 8169   < clt 8350   <_ cle 8351   -ucneg 8488   abscabs 11741
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-rp 10034  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743
This theorem is referenced by:  absdiflt  11836  abslti  11882  absltd  11918
  Copyright terms: Public domain W3C validator