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Theorem abslt 11052
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 524 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. RR )
21renegcld 8299 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A e. RR )
31recnd 7948 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. CC )
4 abscl 11015 . . . . . 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 525 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> B e. RR )
7 leabs 11038 . . . . . . 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 11014 . . . . . . 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 4015 . . . . 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 8044 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A < B )
14 leabs 11038 . . . . . 6 |- ( A e. RR -> A <_ ( abs ` A ) )
1514ad2antrr 485 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A <_ ( abs ` A ) )
161, 5, 6, 15, 12lelttrd 8044 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A < B )
1713, 16jca 304 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( -u A < B /\ A < B ) )
18 simpll 524 . . . . . 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 7948 . . . . . . . 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 8299 . . . . . . . 8 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> -u A e. RR )
23 0red 7921 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> 0 e. RR )
24 ltabs 11051 . . . . . . . . . 10 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
2519, 23, 24ltled 8038 . . . . . . . . 9 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A <_ 0 )
2619le0neg1d 8436 . . . . . . . . 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 11035 . . . . . . . 8 |- ( ( -u A e. RR /\ 0 <_ -u A ) -> ( abs ` -u A ) = -u A )
2922, 27, 28syl2anc 409 . . . . . . 7 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> ( abs ` -u A ) = -u A )
3021, 29eqtr3d 2205 . . . . . 6 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> ( abs ` A ) = -u A )
3118, 30sylan 281 . . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) /\ A < ( abs ` A ) ) -> ( abs ` A ) = -u A )
32 simplrl 530 . . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) /\ A < ( abs ` A ) ) -> -u A < B )
3331, 32eqbrtrd 4011 . . . 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 527 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> A < B )
36 simplr 525 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> B e. RR )
3718recnd 7948 . . . . . . 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 7987 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ ( abs ` A ) e. RR ) -> ( A < B -> ( A < ( abs ` A ) \/ ( abs ` A ) < B ) ) )
4018, 36, 38, 39syl3anc 1233 . . . . 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 793 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -u A < B /\ A < B ) ) -> ( abs ` A ) < B )
4317, 42impbida 591 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B <-> ( -u A < B /\ A < B ) ) )
44 ltnegcon1 8382 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) )
4544anbi1d 462 . 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 703   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198   CCcc 7772   RRcr 7773   0cc0 7774   < clt 7954   <_ cle 7955   -ucneg 8091   abscabs 10961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963
This theorem is referenced by:  absdiflt  11056  abslti  11102  absltd  11138
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