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Theorem oddcomabszz 35792
Description: An odd function which takes nonnegative values on nonnegative arguments commutes with  abs. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Hypotheses
Ref Expression
oddcomabszz.1  |-  ( (
ph  /\  x  e.  ZZ )  ->  A  e.  RR )
oddcomabszz.2  |-  ( (
ph  /\  x  e.  ZZ  /\  0  <_  x
)  ->  0  <_  A )
oddcomabszz.3  |-  ( (
ph  /\  y  e.  ZZ )  ->  C  = 
-u B )
oddcomabszz.4  |-  ( x  =  y  ->  A  =  B )
oddcomabszz.5  |-  ( x  =  -u y  ->  A  =  C )
oddcomabszz.6  |-  ( x  =  D  ->  A  =  E )
oddcomabszz.7  |-  ( x  =  ( abs `  D
)  ->  A  =  F )
Assertion
Ref Expression
oddcomabszz  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  E )  =  F )
Distinct variable groups:    x, B    x, C    x, D, y   
x, E    x, F    y, A    ph, x, y
Allowed substitution hints:    A( x)    B( y)    C( y)    E( y)    F( y)

Proof of Theorem oddcomabszz
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eleq1 2517 . . . . . 6  |-  ( a  =  D  ->  (
a  e.  ZZ  <->  D  e.  ZZ ) )
21anbi2d 710 . . . . 5  |-  ( a  =  D  ->  (
( ph  /\  a  e.  ZZ )  <->  ( ph  /\  D  e.  ZZ ) ) )
3 csbeq1 3366 . . . . . . 7  |-  ( a  =  D  ->  [_ a  /  x ]_ A  = 
[_ D  /  x ]_ A )
43fveq2d 5869 . . . . . 6  |-  ( a  =  D  ->  ( abs `  [_ a  /  x ]_ A )  =  ( abs `  [_ D  /  x ]_ A ) )
5 fveq2 5865 . . . . . . 7  |-  ( a  =  D  ->  ( abs `  a )  =  ( abs `  D
) )
65csbeq1d 3370 . . . . . 6  |-  ( a  =  D  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ ( abs `  D )  /  x ]_ A
)
74, 6eqeq12d 2466 . . . . 5  |-  ( a  =  D  ->  (
( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A  <->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A ) )
82, 7imbi12d 322 . . . 4  |-  ( a  =  D  ->  (
( ( ph  /\  a  e.  ZZ )  ->  ( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A )  <->  ( ( ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A ) ) )
9 nfv 1761 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ )
10 nfcsb1v 3379 . . . . . . . . . . 11  |-  F/_ x [_ a  /  x ]_ A
1110nfel1 2606 . . . . . . . . . 10  |-  F/ x [_ a  /  x ]_ A  e.  RR
129, 11nfim 2003 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
13 eleq1 2517 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
x  e.  ZZ  <->  a  e.  ZZ ) )
1413anbi2d 710 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
15 csbeq1a 3372 . . . . . . . . . . 11  |-  ( x  =  a  ->  A  =  [_ a  /  x ]_ A )
1615eleq1d 2513 . . . . . . . . . 10  |-  ( x  =  a  ->  ( A  e.  RR  <->  [_ a  /  x ]_ A  e.  RR ) )
1714, 16imbi12d 322 . . . . . . . . 9  |-  ( x  =  a  ->  (
( ( ph  /\  x  e.  ZZ )  ->  A  e.  RR )  <-> 
( ( ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR ) ) )
18 oddcomabszz.1 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ZZ )  ->  A  e.  RR )
1912, 17, 18chvar 2106 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
2019adantr 467 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ a  /  x ]_ A  e.  RR )
21 nfv 1761 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ  /\  0  <_ 
a )
22 nfcv 2592 . . . . . . . . . . 11  |-  F/_ x
0
23 nfcv 2592 . . . . . . . . . . 11  |-  F/_ x  <_
2422, 23, 10nfbr 4447 . . . . . . . . . 10  |-  F/ x
0  <_  [_ a  /  x ]_ A
2521, 24nfim 2003 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
26 breq2 4406 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
0  <_  x  <->  0  <_  a ) )
2713, 263anbi23d 1342 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  a  e.  ZZ  /\  0  <_  a ) ) )
2815breq2d 4414 . . . . . . . . . 10  |-  ( x  =  a  ->  (
0  <_  A  <->  0  <_  [_ a  /  x ]_ A ) )
2927, 28imbi12d 322 . . . . . . . . 9  |-  ( x  =  a  ->  (
( ( ph  /\  x  e.  ZZ  /\  0  <_  x )  ->  0  <_  A )  <->  ( ( ph  /\  a  e.  ZZ  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A ) ) )
30 oddcomabszz.2 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ZZ  /\  0  <_  x
)  ->  0  <_  A )
3125, 29, 30chvar 2106 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  0  <_  a
)  ->  0  <_  [_ a  /  x ]_ A )
32313expa 1208 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
3320, 32absidd 13484 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ a  /  x ]_ A )
34 zre 10941 . . . . . . . . 9  |-  ( a  e.  ZZ  ->  a  e.  RR )
3534ad2antlr 733 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  a  e.  RR )
36 absid 13359 . . . . . . . 8  |-  ( ( a  e.  RR  /\  0  <_  a )  -> 
( abs `  a
)  =  a )
3735, 36sylancom 673 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  a )  =  a )
3837csbeq1d 3370 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ a  /  x ]_ A
)
3933, 38eqtr4d 2488 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
40 nfv 1761 . . . . . . . 8  |-  F/ y ( ( ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  =  -u [_ a  /  x ]_ A )
41 eleq1 2517 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  e.  ZZ  <->  a  e.  ZZ ) )
4241anbi2d 710 . . . . . . . . 9  |-  ( y  =  a  ->  (
( ph  /\  y  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
43 negex 9873 . . . . . . . . . . . 12  |-  -u y  e.  _V
44 oddcomabszz.5 . . . . . . . . . . . 12  |-  ( x  =  -u y  ->  A  =  C )
4543, 44csbie 3389 . . . . . . . . . . 11  |-  [_ -u y  /  x ]_ A  =  C
46 negeq 9867 . . . . . . . . . . . 12  |-  ( y  =  a  ->  -u y  =  -u a )
4746csbeq1d 3370 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ -u y  /  x ]_ A  = 
[_ -u a  /  x ]_ A )
4845, 47syl5eqr 2499 . . . . . . . . . 10  |-  ( y  =  a  ->  C  =  [_ -u a  /  x ]_ A )
49 vex 3048 . . . . . . . . . . . . 13  |-  y  e. 
_V
50 oddcomabszz.4 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  A  =  B )
5149, 50csbie 3389 . . . . . . . . . . . 12  |-  [_ y  /  x ]_ A  =  B
52 csbeq1 3366 . . . . . . . . . . . 12  |-  ( y  =  a  ->  [_ y  /  x ]_ A  = 
[_ a  /  x ]_ A )
5351, 52syl5eqr 2499 . . . . . . . . . . 11  |-  ( y  =  a  ->  B  =  [_ a  /  x ]_ A )
5453negeqd 9869 . . . . . . . . . 10  |-  ( y  =  a  ->  -u B  =  -u [_ a  /  x ]_ A )
5548, 54eqeq12d 2466 . . . . . . . . 9  |-  ( y  =  a  ->  ( C  =  -u B  <->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A ) )
5642, 55imbi12d 322 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( ph  /\  y  e.  ZZ )  ->  C  =  -u B
)  <->  ( ( ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  =  -u [_ a  /  x ]_ A ) ) )
57 oddcomabszz.3 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ZZ )  ->  C  = 
-u B )
5840, 56, 57chvar 2106 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
5958adantr 467 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
6034ad2antlr 733 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  a  e.  RR )
61 absnid 13361 . . . . . . . 8  |-  ( ( a  e.  RR  /\  a  <_  0 )  -> 
( abs `  a
)  =  -u a
)
6260, 61sylancom 673 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  a )  = 
-u a )
6362csbeq1d 3370 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ -u a  /  x ]_ A )
6419adantr 467 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  e.  RR )
65 znegcl 10972 . . . . . . . . . . 11  |-  ( a  e.  ZZ  ->  -u a  e.  ZZ )
66 nfv 1761 . . . . . . . . . . . . . 14  |-  F/ x
( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )
67 nfcsb1v 3379 . . . . . . . . . . . . . . 15  |-  F/_ x [_ -u a  /  x ]_ A
6822, 23, 67nfbr 4447 . . . . . . . . . . . . . 14  |-  F/ x
0  <_  [_ -u a  /  x ]_ A
6966, 68nfim 2003 . . . . . . . . . . . . 13  |-  F/ x
( ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  -> 
0  <_  [_ -u a  /  x ]_ A )
70 negex 9873 . . . . . . . . . . . . 13  |-  -u a  e.  _V
71 eleq1 2517 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
x  e.  ZZ  <->  -u a  e.  ZZ ) )
72 breq2 4406 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
0  <_  x  <->  0  <_  -u a ) )
7371, 723anbi23d 1342 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a ) ) )
74 csbeq1a 3372 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  A  =  [_ -u a  /  x ]_ A )
7574breq2d 4414 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
0  <_  A  <->  0  <_  [_ -u a  /  x ]_ A ) )
7673, 75imbi12d 322 . . . . . . . . . . . . 13  |-  ( x  =  -u a  ->  (
( ( ph  /\  x  e.  ZZ  /\  0  <_  x )  ->  0  <_  A )  <->  ( ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  ->  0  <_  [_ -u a  /  x ]_ A ) ) )
7769, 70, 76, 30vtoclf 3099 . . . . . . . . . . . 12  |-  ( (
ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  ->  0  <_  [_ -u a  /  x ]_ A )
78773expia 1210 . . . . . . . . . . 11  |-  ( (
ph  /\  -u a  e.  ZZ )  ->  (
0  <_  -u a  -> 
0  <_  [_ -u a  /  x ]_ A ) )
7965, 78sylan2 477 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_  [_ -u a  /  x ]_ A ) )
8058breq2d 4414 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  [_ -u a  /  x ]_ A  <->  0  <_  -u [_ a  /  x ]_ A ) )
8179, 80sylibd 218 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_ 
-u [_ a  /  x ]_ A ) )
8234adantl 468 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  a  e.  RR )
8382le0neg1d 10185 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  <->  0  <_  -u a ) )
8419le0neg1d 10185 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( [_ a  /  x ]_ A  <_  0  <->  0  <_  -u [_ a  /  x ]_ A ) )
8581, 83, 843imtr4d 272 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  ->  [_ a  /  x ]_ A  <_ 
0 ) )
8685imp 431 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  <_ 
0 )
8764, 86absnidd 13475 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
-u [_ a  /  x ]_ A )
8859, 63, 873eqtr4rd 2496 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
89 0re 9643 . . . . . . 7  |-  0  e.  RR
90 letric 9734 . . . . . . 7  |-  ( ( 0  e.  RR  /\  a  e.  RR )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9189, 34, 90sylancr 669 . . . . . 6  |-  ( a  e.  ZZ  ->  (
0  <_  a  \/  a  <_  0 ) )
9291adantl 468 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9339, 88, 92mpjaodan 795 . . . 4  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A
)
948, 93vtoclg 3107 . . 3  |-  ( D  e.  ZZ  ->  (
( ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  = 
[_ ( abs `  D
)  /  x ]_ A ) )
9594anabsi7 828 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A
)
96 nfcvd 2593 . . . . 5  |-  ( D  e.  ZZ  ->  F/_ x E )
97 oddcomabszz.6 . . . . 5  |-  ( x  =  D  ->  A  =  E )
9896, 97csbiegf 3387 . . . 4  |-  ( D  e.  ZZ  ->  [_ D  /  x ]_ A  =  E )
9998fveq2d 5869 . . 3  |-  ( D  e.  ZZ  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E
) )
10099adantl 468 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E ) )
101 fvex 5875 . . . 4  |-  ( abs `  D )  e.  _V
102 oddcomabszz.7 . . . 4  |-  ( x  =  ( abs `  D
)  ->  A  =  F )
103101, 102csbie 3389 . . 3  |-  [_ ( abs `  D )  /  x ]_ A  =  F
104103a1i 11 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  [_ ( abs `  D )  /  x ]_ A  =  F )
10595, 100, 1043eqtr3d 2493 1  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  E )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   [_csb 3363   class class class wbr 4402   ` cfv 5582   RRcr 9538   0cc0 9539    <_ cle 9676   -ucneg 9861   ZZcz 10937   abscabs 13297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299
This theorem is referenced by:  rmyabs  35808
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