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Theorem oddcomabszz 27029
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 2343 . . . . . 6  |-  ( a  =  D  ->  (
a  e.  ZZ  <->  D  e.  ZZ ) )
21anbi2d 684 . . . . 5  |-  ( a  =  D  ->  (
( ph  /\  a  e.  ZZ )  <->  ( ph  /\  D  e.  ZZ ) ) )
3 csbeq1 3084 . . . . . . 7  |-  ( a  =  D  ->  [_ a  /  x ]_ A  = 
[_ D  /  x ]_ A )
43fveq2d 5529 . . . . . 6  |-  ( a  =  D  ->  ( abs `  [_ a  /  x ]_ A )  =  ( abs `  [_ D  /  x ]_ A ) )
5 fveq2 5525 . . . . . . 7  |-  ( a  =  D  ->  ( abs `  a )  =  ( abs `  D
) )
65csbeq1d 3087 . . . . . 6  |-  ( a  =  D  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ ( abs `  D )  /  x ]_ A
)
74, 6eqeq12d 2297 . . . . 5  |-  ( a  =  D  ->  (
( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A  <->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A ) )
82, 7imbi12d 311 . . . 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 1605 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ )
10 nfcsb1v 3113 . . . . . . . . . . 11  |-  F/_ x [_ a  /  x ]_ A
1110nfel1 2429 . . . . . . . . . 10  |-  F/ x [_ a  /  x ]_ A  e.  RR
129, 11nfim 1769 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
13 eleq1 2343 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
x  e.  ZZ  <->  a  e.  ZZ ) )
1413anbi2d 684 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
15 csbeq1a 3089 . . . . . . . . . . 11  |-  ( x  =  a  ->  A  =  [_ a  /  x ]_ A )
1615eleq1d 2349 . . . . . . . . . 10  |-  ( x  =  a  ->  ( A  e.  RR  <->  [_ a  /  x ]_ A  e.  RR ) )
1714, 16imbi12d 311 . . . . . . . . 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 1926 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
2019adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ a  /  x ]_ A  e.  RR )
21 nfv 1605 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ  /\  0  <_ 
a )
22 nfcv 2419 . . . . . . . . . . 11  |-  F/_ x
0
23 nfcv 2419 . . . . . . . . . . 11  |-  F/_ x  <_
2422, 23, 10nfbr 4067 . . . . . . . . . 10  |-  F/ x
0  <_  [_ a  /  x ]_ A
2521, 24nfim 1769 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
26 breq2 4027 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
0  <_  x  <->  0  <_  a ) )
2713, 263anbi23d 1255 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  a  e.  ZZ  /\  0  <_  a ) ) )
2815breq2d 4035 . . . . . . . . . 10  |-  ( x  =  a  ->  (
0  <_  A  <->  0  <_  [_ a  /  x ]_ A ) )
2927, 28imbi12d 311 . . . . . . . . 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 1926 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  0  <_  a
)  ->  0  <_  [_ a  /  x ]_ A )
32313expa 1151 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
3320, 32absidd 11905 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ a  /  x ]_ A )
34 zre 10028 . . . . . . . . 9  |-  ( a  e.  ZZ  ->  a  e.  RR )
3534ad2antlr 707 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  a  e.  RR )
36 absid 11781 . . . . . . . 8  |-  ( ( a  e.  RR  /\  0  <_  a )  -> 
( abs `  a
)  =  a )
3735, 36sylancom 648 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  a )  =  a )
3837csbeq1d 3087 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ a  /  x ]_ A
)
3933, 38eqtr4d 2318 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
40 nfv 1605 . . . . . . . 8  |-  F/ y ( ( ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  =  -u [_ a  /  x ]_ A )
41 eleq1 2343 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  e.  ZZ  <->  a  e.  ZZ ) )
4241anbi2d 684 . . . . . . . . 9  |-  ( y  =  a  ->  (
( ph  /\  y  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
43 negex 9050 . . . . . . . . . . . 12  |-  -u y  e.  _V
44 nfcv 2419 . . . . . . . . . . . 12  |-  F/_ x C
45 oddcomabszz.5 . . . . . . . . . . . 12  |-  ( x  =  -u y  ->  A  =  C )
4643, 44, 45csbief 3122 . . . . . . . . . . 11  |-  [_ -u y  /  x ]_ A  =  C
47 negeq 9044 . . . . . . . . . . . 12  |-  ( y  =  a  ->  -u y  =  -u a )
4847csbeq1d 3087 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ -u y  /  x ]_ A  = 
[_ -u a  /  x ]_ A )
4946, 48syl5eqr 2329 . . . . . . . . . 10  |-  ( y  =  a  ->  C  =  [_ -u a  /  x ]_ A )
50 vex 2791 . . . . . . . . . . . . 13  |-  y  e. 
_V
51 nfcv 2419 . . . . . . . . . . . . 13  |-  F/_ x B
52 oddcomabszz.4 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  A  =  B )
5350, 51, 52csbief 3122 . . . . . . . . . . . 12  |-  [_ y  /  x ]_ A  =  B
54 csbeq1 3084 . . . . . . . . . . . 12  |-  ( y  =  a  ->  [_ y  /  x ]_ A  = 
[_ a  /  x ]_ A )
5553, 54syl5eqr 2329 . . . . . . . . . . 11  |-  ( y  =  a  ->  B  =  [_ a  /  x ]_ A )
5655negeqd 9046 . . . . . . . . . 10  |-  ( y  =  a  ->  -u B  =  -u [_ a  /  x ]_ A )
5749, 56eqeq12d 2297 . . . . . . . . 9  |-  ( y  =  a  ->  ( C  =  -u B  <->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A ) )
5842, 57imbi12d 311 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( ph  /\  y  e.  ZZ )  ->  C  =  -u B
)  <->  ( ( ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  =  -u [_ a  /  x ]_ A ) ) )
59 oddcomabszz.3 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ZZ )  ->  C  = 
-u B )
6040, 58, 59chvar 1926 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
6160adantr 451 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
6234ad2antlr 707 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  a  e.  RR )
63 absnid 11783 . . . . . . . 8  |-  ( ( a  e.  RR  /\  a  <_  0 )  -> 
( abs `  a
)  =  -u a
)
6462, 63sylancom 648 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  a )  = 
-u a )
6564csbeq1d 3087 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ -u a  /  x ]_ A )
6619adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  e.  RR )
67 znegcl 10055 . . . . . . . . . . 11  |-  ( a  e.  ZZ  ->  -u a  e.  ZZ )
68 nfv 1605 . . . . . . . . . . . . . 14  |-  F/ x
( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )
69 nfcsb1v 3113 . . . . . . . . . . . . . . 15  |-  F/_ x [_ -u a  /  x ]_ A
7022, 23, 69nfbr 4067 . . . . . . . . . . . . . 14  |-  F/ x
0  <_  [_ -u a  /  x ]_ A
7168, 70nfim 1769 . . . . . . . . . . . . 13  |-  F/ x
( ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  -> 
0  <_  [_ -u a  /  x ]_ A )
72 negex 9050 . . . . . . . . . . . . 13  |-  -u a  e.  _V
73 eleq1 2343 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
x  e.  ZZ  <->  -u a  e.  ZZ ) )
74 breq2 4027 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
0  <_  x  <->  0  <_  -u a ) )
7573, 743anbi23d 1255 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a ) ) )
76 csbeq1a 3089 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  A  =  [_ -u a  /  x ]_ A )
7776breq2d 4035 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
0  <_  A  <->  0  <_  [_ -u a  /  x ]_ A ) )
7875, 77imbi12d 311 . . . . . . . . . . . . 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 ) ) )
7971, 72, 78, 30vtoclf 2837 . . . . . . . . . . . 12  |-  ( (
ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  ->  0  <_  [_ -u a  /  x ]_ A )
80793expia 1153 . . . . . . . . . . 11  |-  ( (
ph  /\  -u a  e.  ZZ )  ->  (
0  <_  -u a  -> 
0  <_  [_ -u a  /  x ]_ A ) )
8167, 80sylan2 460 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_  [_ -u a  /  x ]_ A ) )
8260breq2d 4035 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  [_ -u a  /  x ]_ A  <->  0  <_  -u [_ a  /  x ]_ A ) )
8381, 82sylibd 205 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_ 
-u [_ a  /  x ]_ A ) )
8434adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  a  e.  RR )
8584le0neg1d 9344 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  <->  0  <_  -u a ) )
8619le0neg1d 9344 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( [_ a  /  x ]_ A  <_  0  <->  0  <_  -u [_ a  /  x ]_ A ) )
8783, 85, 863imtr4d 259 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  ->  [_ a  /  x ]_ A  <_ 
0 ) )
8887imp 418 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  <_ 
0 )
8966, 88absnidd 11896 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
-u [_ a  /  x ]_ A )
9061, 65, 893eqtr4rd 2326 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
91 0re 8838 . . . . . . 7  |-  0  e.  RR
92 letric 8921 . . . . . . 7  |-  ( ( 0  e.  RR  /\  a  e.  RR )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9391, 34, 92sylancr 644 . . . . . 6  |-  ( a  e.  ZZ  ->  (
0  <_  a  \/  a  <_  0 ) )
9493adantl 452 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9539, 90, 94mpjaodan 761 . . . 4  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A
)
968, 95vtoclg 2843 . . 3  |-  ( D  e.  ZZ  ->  (
( ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  = 
[_ ( abs `  D
)  /  x ]_ A ) )
9796anabsi7 792 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A
)
98 nfcvd 2420 . . . . 5  |-  ( D  e.  ZZ  ->  F/_ x E )
99 oddcomabszz.6 . . . . 5  |-  ( x  =  D  ->  A  =  E )
10098, 99csbiegf 3121 . . . 4  |-  ( D  e.  ZZ  ->  [_ D  /  x ]_ A  =  E )
101100fveq2d 5529 . . 3  |-  ( D  e.  ZZ  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E
) )
102101adantl 452 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E ) )
103 fvex 5539 . . . 4  |-  ( abs `  D )  e.  _V
104 nfcv 2419 . . . 4  |-  F/_ x F
105 oddcomabszz.7 . . . 4  |-  ( x  =  ( abs `  D
)  ->  A  =  F )
106103, 104, 105csbief 3122 . . 3  |-  [_ ( abs `  D )  /  x ]_ A  =  F
107106a1i 10 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  [_ ( abs `  D )  /  x ]_ A  =  F )
10897, 102, 1073eqtr3d 2323 1  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  E )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   [_csb 3081   class class class wbr 4023   ` cfv 5255   RRcr 8736   0cc0 8737    <_ cle 8868   -ucneg 9038   ZZcz 10024   abscabs 11719
This theorem is referenced by:  rmyabs  27045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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