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Theorem oddcomabszz 26699
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 2448 . . . . . 6  |-  ( a  =  D  ->  (
a  e.  ZZ  <->  D  e.  ZZ ) )
21anbi2d 685 . . . . 5  |-  ( a  =  D  ->  (
( ph  /\  a  e.  ZZ )  <->  ( ph  /\  D  e.  ZZ ) ) )
3 csbeq1 3198 . . . . . . 7  |-  ( a  =  D  ->  [_ a  /  x ]_ A  = 
[_ D  /  x ]_ A )
43fveq2d 5673 . . . . . 6  |-  ( a  =  D  ->  ( abs `  [_ a  /  x ]_ A )  =  ( abs `  [_ D  /  x ]_ A ) )
5 fveq2 5669 . . . . . . 7  |-  ( a  =  D  ->  ( abs `  a )  =  ( abs `  D
) )
65csbeq1d 3201 . . . . . 6  |-  ( a  =  D  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ ( abs `  D )  /  x ]_ A
)
74, 6eqeq12d 2402 . . . . 5  |-  ( a  =  D  ->  (
( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A  <->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A ) )
82, 7imbi12d 312 . . . 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 1626 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ )
10 nfcsb1v 3227 . . . . . . . . . . 11  |-  F/_ x [_ a  /  x ]_ A
1110nfel1 2534 . . . . . . . . . 10  |-  F/ x [_ a  /  x ]_ A  e.  RR
129, 11nfim 1822 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
13 eleq1 2448 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
x  e.  ZZ  <->  a  e.  ZZ ) )
1413anbi2d 685 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
15 csbeq1a 3203 . . . . . . . . . . 11  |-  ( x  =  a  ->  A  =  [_ a  /  x ]_ A )
1615eleq1d 2454 . . . . . . . . . 10  |-  ( x  =  a  ->  ( A  e.  RR  <->  [_ a  /  x ]_ A  e.  RR ) )
1714, 16imbi12d 312 . . . . . . . . 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 2023 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
2019adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ a  /  x ]_ A  e.  RR )
21 nfv 1626 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ  /\  0  <_ 
a )
22 nfcv 2524 . . . . . . . . . . 11  |-  F/_ x
0
23 nfcv 2524 . . . . . . . . . . 11  |-  F/_ x  <_
2422, 23, 10nfbr 4198 . . . . . . . . . 10  |-  F/ x
0  <_  [_ a  /  x ]_ A
2521, 24nfim 1822 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
26 breq2 4158 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
0  <_  x  <->  0  <_  a ) )
2713, 263anbi23d 1257 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  a  e.  ZZ  /\  0  <_  a ) ) )
2815breq2d 4166 . . . . . . . . . 10  |-  ( x  =  a  ->  (
0  <_  A  <->  0  <_  [_ a  /  x ]_ A ) )
2927, 28imbi12d 312 . . . . . . . . 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 2023 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  0  <_  a
)  ->  0  <_  [_ a  /  x ]_ A )
32313expa 1153 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
3320, 32absidd 12153 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ a  /  x ]_ A )
34 zre 10219 . . . . . . . . 9  |-  ( a  e.  ZZ  ->  a  e.  RR )
3534ad2antlr 708 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  a  e.  RR )
36 absid 12029 . . . . . . . 8  |-  ( ( a  e.  RR  /\  0  <_  a )  -> 
( abs `  a
)  =  a )
3735, 36sylancom 649 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  a )  =  a )
3837csbeq1d 3201 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ a  /  x ]_ A
)
3933, 38eqtr4d 2423 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
40 nfv 1626 . . . . . . . 8  |-  F/ y ( ( ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  =  -u [_ a  /  x ]_ A )
41 eleq1 2448 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  e.  ZZ  <->  a  e.  ZZ ) )
4241anbi2d 685 . . . . . . . . 9  |-  ( y  =  a  ->  (
( ph  /\  y  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
43 negex 9237 . . . . . . . . . . . 12  |-  -u y  e.  _V
44 nfcv 2524 . . . . . . . . . . . 12  |-  F/_ x C
45 oddcomabszz.5 . . . . . . . . . . . 12  |-  ( x  =  -u y  ->  A  =  C )
4643, 44, 45csbief 3236 . . . . . . . . . . 11  |-  [_ -u y  /  x ]_ A  =  C
47 negeq 9231 . . . . . . . . . . . 12  |-  ( y  =  a  ->  -u y  =  -u a )
4847csbeq1d 3201 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ -u y  /  x ]_ A  = 
[_ -u a  /  x ]_ A )
4946, 48syl5eqr 2434 . . . . . . . . . 10  |-  ( y  =  a  ->  C  =  [_ -u a  /  x ]_ A )
50 vex 2903 . . . . . . . . . . . . 13  |-  y  e. 
_V
51 nfcv 2524 . . . . . . . . . . . . 13  |-  F/_ x B
52 oddcomabszz.4 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  A  =  B )
5350, 51, 52csbief 3236 . . . . . . . . . . . 12  |-  [_ y  /  x ]_ A  =  B
54 csbeq1 3198 . . . . . . . . . . . 12  |-  ( y  =  a  ->  [_ y  /  x ]_ A  = 
[_ a  /  x ]_ A )
5553, 54syl5eqr 2434 . . . . . . . . . . 11  |-  ( y  =  a  ->  B  =  [_ a  /  x ]_ A )
5655negeqd 9233 . . . . . . . . . 10  |-  ( y  =  a  ->  -u B  =  -u [_ a  /  x ]_ A )
5749, 56eqeq12d 2402 . . . . . . . . 9  |-  ( y  =  a  ->  ( C  =  -u B  <->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A ) )
5842, 57imbi12d 312 . . . . . . . 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 2023 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
6160adantr 452 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
6234ad2antlr 708 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  a  e.  RR )
63 absnid 12031 . . . . . . . 8  |-  ( ( a  e.  RR  /\  a  <_  0 )  -> 
( abs `  a
)  =  -u a
)
6462, 63sylancom 649 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  a )  = 
-u a )
6564csbeq1d 3201 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ -u a  /  x ]_ A )
6619adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  e.  RR )
67 znegcl 10246 . . . . . . . . . . 11  |-  ( a  e.  ZZ  ->  -u a  e.  ZZ )
68 nfv 1626 . . . . . . . . . . . . . 14  |-  F/ x
( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )
69 nfcsb1v 3227 . . . . . . . . . . . . . . 15  |-  F/_ x [_ -u a  /  x ]_ A
7022, 23, 69nfbr 4198 . . . . . . . . . . . . . 14  |-  F/ x
0  <_  [_ -u a  /  x ]_ A
7168, 70nfim 1822 . . . . . . . . . . . . 13  |-  F/ x
( ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  -> 
0  <_  [_ -u a  /  x ]_ A )
72 negex 9237 . . . . . . . . . . . . 13  |-  -u a  e.  _V
73 eleq1 2448 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
x  e.  ZZ  <->  -u a  e.  ZZ ) )
74 breq2 4158 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
0  <_  x  <->  0  <_  -u a ) )
7573, 743anbi23d 1257 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a ) ) )
76 csbeq1a 3203 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  A  =  [_ -u a  /  x ]_ A )
7776breq2d 4166 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
0  <_  A  <->  0  <_  [_ -u a  /  x ]_ A ) )
7875, 77imbi12d 312 . . . . . . . . . . . . 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 2949 . . . . . . . . . . . 12  |-  ( (
ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  ->  0  <_  [_ -u a  /  x ]_ A )
80793expia 1155 . . . . . . . . . . 11  |-  ( (
ph  /\  -u a  e.  ZZ )  ->  (
0  <_  -u a  -> 
0  <_  [_ -u a  /  x ]_ A ) )
8167, 80sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_  [_ -u a  /  x ]_ A ) )
8260breq2d 4166 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  [_ -u a  /  x ]_ A  <->  0  <_  -u [_ a  /  x ]_ A ) )
8381, 82sylibd 206 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_ 
-u [_ a  /  x ]_ A ) )
8434adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  a  e.  RR )
8584le0neg1d 9531 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  <->  0  <_  -u a ) )
8619le0neg1d 9531 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( [_ a  /  x ]_ A  <_  0  <->  0  <_  -u [_ a  /  x ]_ A ) )
8783, 85, 863imtr4d 260 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  ->  [_ a  /  x ]_ A  <_ 
0 ) )
8887imp 419 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  <_ 
0 )
8966, 88absnidd 12144 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
-u [_ a  /  x ]_ A )
9061, 65, 893eqtr4rd 2431 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
91 0re 9025 . . . . . . 7  |-  0  e.  RR
92 letric 9108 . . . . . . 7  |-  ( ( 0  e.  RR  /\  a  e.  RR )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9391, 34, 92sylancr 645 . . . . . 6  |-  ( a  e.  ZZ  ->  (
0  <_  a  \/  a  <_  0 ) )
9493adantl 453 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9539, 90, 94mpjaodan 762 . . . 4  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A
)
968, 95vtoclg 2955 . . 3  |-  ( D  e.  ZZ  ->  (
( ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  = 
[_ ( abs `  D
)  /  x ]_ A ) )
9796anabsi7 793 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A
)
98 nfcvd 2525 . . . . 5  |-  ( D  e.  ZZ  ->  F/_ x E )
99 oddcomabszz.6 . . . . 5  |-  ( x  =  D  ->  A  =  E )
10098, 99csbiegf 3235 . . . 4  |-  ( D  e.  ZZ  ->  [_ D  /  x ]_ A  =  E )
101100fveq2d 5673 . . 3  |-  ( D  e.  ZZ  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E
) )
102101adantl 453 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E ) )
103 fvex 5683 . . . 4  |-  ( abs `  D )  e.  _V
104 nfcv 2524 . . . 4  |-  F/_ x F
105 oddcomabszz.7 . . . 4  |-  ( x  =  ( abs `  D
)  ->  A  =  F )
106103, 104, 105csbief 3236 . . 3  |-  [_ ( abs `  D )  /  x ]_ A  =  F
107106a1i 11 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  [_ ( abs `  D )  /  x ]_ A  =  F )
10897, 102, 1073eqtr3d 2428 1  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  E )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   [_csb 3195   class class class wbr 4154   ` cfv 5395   RRcr 8923   0cc0 8924    <_ cle 9055   -ucneg 9225   ZZcz 10215   abscabs 11967
This theorem is referenced by:  rmyabs  26715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969
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