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Theorem oddcomabszz 30711
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 2539 . . . . . 6  |-  ( a  =  D  ->  (
a  e.  ZZ  <->  D  e.  ZZ ) )
21anbi2d 703 . . . . 5  |-  ( a  =  D  ->  (
( ph  /\  a  e.  ZZ )  <->  ( ph  /\  D  e.  ZZ ) ) )
3 csbeq1 3438 . . . . . . 7  |-  ( a  =  D  ->  [_ a  /  x ]_ A  = 
[_ D  /  x ]_ A )
43fveq2d 5870 . . . . . 6  |-  ( a  =  D  ->  ( abs `  [_ a  /  x ]_ A )  =  ( abs `  [_ D  /  x ]_ A ) )
5 fveq2 5866 . . . . . . 7  |-  ( a  =  D  ->  ( abs `  a )  =  ( abs `  D
) )
65csbeq1d 3442 . . . . . 6  |-  ( a  =  D  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ ( abs `  D )  /  x ]_ A
)
74, 6eqeq12d 2489 . . . . 5  |-  ( a  =  D  ->  (
( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A  <->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A ) )
82, 7imbi12d 320 . . . 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 1683 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ )
10 nfcsb1v 3451 . . . . . . . . . . 11  |-  F/_ x [_ a  /  x ]_ A
1110nfel1 2645 . . . . . . . . . 10  |-  F/ x [_ a  /  x ]_ A  e.  RR
129, 11nfim 1867 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
13 eleq1 2539 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
x  e.  ZZ  <->  a  e.  ZZ ) )
1413anbi2d 703 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
15 csbeq1a 3444 . . . . . . . . . . 11  |-  ( x  =  a  ->  A  =  [_ a  /  x ]_ A )
1615eleq1d 2536 . . . . . . . . . 10  |-  ( x  =  a  ->  ( A  e.  RR  <->  [_ a  /  x ]_ A  e.  RR ) )
1714, 16imbi12d 320 . . . . . . . . 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 1982 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ a  /  x ]_ A  e.  RR )
2019adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ a  /  x ]_ A  e.  RR )
21 nfv 1683 . . . . . . . . . 10  |-  F/ x
( ph  /\  a  e.  ZZ  /\  0  <_ 
a )
22 nfcv 2629 . . . . . . . . . . 11  |-  F/_ x
0
23 nfcv 2629 . . . . . . . . . . 11  |-  F/_ x  <_
2422, 23, 10nfbr 4491 . . . . . . . . . 10  |-  F/ x
0  <_  [_ a  /  x ]_ A
2521, 24nfim 1867 . . . . . . . . 9  |-  F/ x
( ( ph  /\  a  e.  ZZ  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
26 breq2 4451 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
0  <_  x  <->  0  <_  a ) )
2713, 263anbi23d 1302 . . . . . . . . . 10  |-  ( x  =  a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  a  e.  ZZ  /\  0  <_  a ) ) )
2815breq2d 4459 . . . . . . . . . 10  |-  ( x  =  a  ->  (
0  <_  A  <->  0  <_  [_ a  /  x ]_ A ) )
2927, 28imbi12d 320 . . . . . . . . 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 1982 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ  /\  0  <_  a
)  ->  0  <_  [_ a  /  x ]_ A )
32313expa 1196 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  0  <_  [_ a  /  x ]_ A )
3320, 32absidd 13220 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ a  /  x ]_ A )
34 zre 10869 . . . . . . . . 9  |-  ( a  e.  ZZ  ->  a  e.  RR )
3534ad2antlr 726 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  a  e.  RR )
36 absid 13095 . . . . . . . 8  |-  ( ( a  e.  RR  /\  0  <_  a )  -> 
( abs `  a
)  =  a )
3735, 36sylancom 667 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  a )  =  a )
3837csbeq1d 3442 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ a  /  x ]_ A
)
3933, 38eqtr4d 2511 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  0  <_  a )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
40 nfv 1683 . . . . . . . 8  |-  F/ y ( ( ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  =  -u [_ a  /  x ]_ A )
41 eleq1 2539 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  e.  ZZ  <->  a  e.  ZZ ) )
4241anbi2d 703 . . . . . . . . 9  |-  ( y  =  a  ->  (
( ph  /\  y  e.  ZZ )  <->  ( ph  /\  a  e.  ZZ ) ) )
43 negex 9819 . . . . . . . . . . . 12  |-  -u y  e.  _V
44 nfcv 2629 . . . . . . . . . . . 12  |-  F/_ x C
45 oddcomabszz.5 . . . . . . . . . . . 12  |-  ( x  =  -u y  ->  A  =  C )
4643, 44, 45csbief 3460 . . . . . . . . . . 11  |-  [_ -u y  /  x ]_ A  =  C
47 negeq 9813 . . . . . . . . . . . 12  |-  ( y  =  a  ->  -u y  =  -u a )
4847csbeq1d 3442 . . . . . . . . . . 11  |-  ( y  =  a  ->  [_ -u y  /  x ]_ A  = 
[_ -u a  /  x ]_ A )
4946, 48syl5eqr 2522 . . . . . . . . . 10  |-  ( y  =  a  ->  C  =  [_ -u a  /  x ]_ A )
50 vex 3116 . . . . . . . . . . . . 13  |-  y  e. 
_V
51 nfcv 2629 . . . . . . . . . . . . 13  |-  F/_ x B
52 oddcomabszz.4 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  A  =  B )
5350, 51, 52csbief 3460 . . . . . . . . . . . 12  |-  [_ y  /  x ]_ A  =  B
54 csbeq1 3438 . . . . . . . . . . . 12  |-  ( y  =  a  ->  [_ y  /  x ]_ A  = 
[_ a  /  x ]_ A )
5553, 54syl5eqr 2522 . . . . . . . . . . 11  |-  ( y  =  a  ->  B  =  [_ a  /  x ]_ A )
5655negeqd 9815 . . . . . . . . . 10  |-  ( y  =  a  ->  -u B  =  -u [_ a  /  x ]_ A )
5749, 56eqeq12d 2489 . . . . . . . . 9  |-  ( y  =  a  ->  ( C  =  -u B  <->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A ) )
5842, 57imbi12d 320 . . . . . . . 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 1982 . . . . . . 7  |-  ( (
ph  /\  a  e.  ZZ )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
6160adantr 465 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ -u a  /  x ]_ A  = 
-u [_ a  /  x ]_ A )
6234ad2antlr 726 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  a  e.  RR )
63 absnid 13097 . . . . . . . 8  |-  ( ( a  e.  RR  /\  a  <_  0 )  -> 
( abs `  a
)  =  -u a
)
6462, 63sylancom 667 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  a )  = 
-u a )
6564csbeq1d 3442 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ ( abs `  a )  /  x ]_ A  =  [_ -u a  /  x ]_ A )
6619adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  e.  RR )
67 znegcl 10899 . . . . . . . . . . 11  |-  ( a  e.  ZZ  ->  -u a  e.  ZZ )
68 nfv 1683 . . . . . . . . . . . . . 14  |-  F/ x
( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )
69 nfcsb1v 3451 . . . . . . . . . . . . . . 15  |-  F/_ x [_ -u a  /  x ]_ A
7022, 23, 69nfbr 4491 . . . . . . . . . . . . . 14  |-  F/ x
0  <_  [_ -u a  /  x ]_ A
7168, 70nfim 1867 . . . . . . . . . . . . 13  |-  F/ x
( ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  -> 
0  <_  [_ -u a  /  x ]_ A )
72 negex 9819 . . . . . . . . . . . . 13  |-  -u a  e.  _V
73 eleq1 2539 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
x  e.  ZZ  <->  -u a  e.  ZZ ) )
74 breq2 4451 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  (
0  <_  x  <->  0  <_  -u a ) )
7573, 743anbi23d 1302 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
( ph  /\  x  e.  ZZ  /\  0  <_  x )  <->  ( ph  /\  -u a  e.  ZZ  /\  0  <_  -u a ) ) )
76 csbeq1a 3444 . . . . . . . . . . . . . . 15  |-  ( x  =  -u a  ->  A  =  [_ -u a  /  x ]_ A )
7776breq2d 4459 . . . . . . . . . . . . . 14  |-  ( x  =  -u a  ->  (
0  <_  A  <->  0  <_  [_ -u a  /  x ]_ A ) )
7875, 77imbi12d 320 . . . . . . . . . . . . 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 3164 . . . . . . . . . . . 12  |-  ( (
ph  /\  -u a  e.  ZZ  /\  0  <_  -u a )  ->  0  <_  [_ -u a  /  x ]_ A )
80793expia 1198 . . . . . . . . . . 11  |-  ( (
ph  /\  -u a  e.  ZZ )  ->  (
0  <_  -u a  -> 
0  <_  [_ -u a  /  x ]_ A ) )
8167, 80sylan2 474 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_  [_ -u a  /  x ]_ A ) )
8260breq2d 4459 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  [_ -u a  /  x ]_ A  <->  0  <_  -u [_ a  /  x ]_ A ) )
8381, 82sylibd 214 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  -u a  ->  0  <_ 
-u [_ a  /  x ]_ A ) )
8434adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ZZ )  ->  a  e.  RR )
8584le0neg1d 10125 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  <->  0  <_  -u a ) )
8619le0neg1d 10125 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( [_ a  /  x ]_ A  <_  0  <->  0  <_  -u [_ a  /  x ]_ A ) )
8783, 85, 863imtr4d 268 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( a  <_  0  ->  [_ a  /  x ]_ A  <_ 
0 ) )
8887imp 429 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  [_ a  /  x ]_ A  <_ 
0 )
8966, 88absnidd 13211 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
-u [_ a  /  x ]_ A )
9061, 65, 893eqtr4rd 2519 . . . . 5  |-  ( ( ( ph  /\  a  e.  ZZ )  /\  a  <_  0 )  ->  ( abs `  [_ a  /  x ]_ A )  = 
[_ ( abs `  a
)  /  x ]_ A )
91 0re 9597 . . . . . . 7  |-  0  e.  RR
92 letric 9686 . . . . . . 7  |-  ( ( 0  e.  RR  /\  a  e.  RR )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9391, 34, 92sylancr 663 . . . . . 6  |-  ( a  e.  ZZ  ->  (
0  <_  a  \/  a  <_  0 ) )
9493adantl 466 . . . . 5  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( 0  <_  a  \/  a  <_  0 ) )
9539, 90, 94mpjaodan 784 . . . 4  |-  ( (
ph  /\  a  e.  ZZ )  ->  ( abs `  [_ a  /  x ]_ A )  =  [_ ( abs `  a )  /  x ]_ A
)
968, 95vtoclg 3171 . . 3  |-  ( D  e.  ZZ  ->  (
( ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  = 
[_ ( abs `  D
)  /  x ]_ A ) )
9796anabsi7 817 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  [_ ( abs `  D )  /  x ]_ A
)
98 nfcvd 2630 . . . . 5  |-  ( D  e.  ZZ  ->  F/_ x E )
99 oddcomabszz.6 . . . . 5  |-  ( x  =  D  ->  A  =  E )
10098, 99csbiegf 3459 . . . 4  |-  ( D  e.  ZZ  ->  [_ D  /  x ]_ A  =  E )
101100fveq2d 5870 . . 3  |-  ( D  e.  ZZ  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E
) )
102101adantl 466 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  [_ D  /  x ]_ A )  =  ( abs `  E ) )
103 fvex 5876 . . . 4  |-  ( abs `  D )  e.  _V
104 nfcv 2629 . . . 4  |-  F/_ x F
105 oddcomabszz.7 . . . 4  |-  ( x  =  ( abs `  D
)  ->  A  =  F )
106103, 104, 105csbief 3460 . . 3  |-  [_ ( abs `  D )  /  x ]_ A  =  F
107106a1i 11 . 2  |-  ( (
ph  /\  D  e.  ZZ )  ->  [_ ( abs `  D )  /  x ]_ A  =  F )
10897, 102, 1073eqtr3d 2516 1  |-  ( (
ph  /\  D  e.  ZZ )  ->  ( abs `  E )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   [_csb 3435   class class class wbr 4447   ` cfv 5588   RRcr 9492   0cc0 9493    <_ cle 9630   -ucneg 9807   ZZcz 10865   abscabs 13033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035
This theorem is referenced by:  rmyabs  30727
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