Theorem oddcomabszz 29238

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.

Step	Hyp	Ref	Expression
1		eleq1 2498	. . . . . 6 |- ( a = D -> ( a e. ZZ <-> D e. ZZ ) )
2	1	anbi2d 703	. . . . 5 |- ( a = D -> ( ( ph /\ a e. ZZ ) <-> ( ph /\ D e. ZZ ) ) )
3		csbeq1 3286	. . . . . . 7 |- ( a = D -> [_ a / x ]_ A = [_ D / x ]_ A )
4	3	fveq2d 5690	. . . . . 6 |- ( a = D -> ( abs ` [_ a / x ]_ A ) = ( abs ` [_ D / x ]_ A ) )
5		fveq2 5686	. . . . . . 7 |- ( a = D -> ( abs ` a ) = ( abs ` D ) )
6	5	csbeq1d 3290	. . . . . 6 |- ( a = D -> [_ ( abs ` a ) / x ]_ A = [_ ( abs ` D ) / x ]_ A )
7	4, 6	eqeq12d 2452	. . . . 5 |- ( a = D -> ( ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A <-> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A ) )
8	2, 7	imbi12d 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 1673	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ )
10		nfcsb1v 3299	. . . . . . . . . . 11 |- F/_ x [_ a / x ]_ A
11	10	nfel1 2584	. . . . . . . . . 10 |- F/ x [_ a / x ]_ A e. RR
12	9, 11	nfim 1852	. . . . . . . . 9 |- F/ x ( ( ph /\ a e. ZZ ) -> [_ a / x ]_ A e. RR )
13		eleq1 2498	. . . . . . . . . . 11 |- ( x = a -> ( x e. ZZ <-> a e. ZZ ) )
14	13	anbi2d 703	. . . . . . . . . 10 |- ( x = a -> ( ( ph /\ x e. ZZ ) <-> ( ph /\ a e. ZZ ) ) )
15		csbeq1a 3292	. . . . . . . . . . 11 |- ( x = a -> A = [_ a / x ]_ A )
16	15	eleq1d 2504	. . . . . . . . . 10 |- ( x = a -> ( A e. RR <-> [_ a / x ]_ A e. RR ) )
17	14, 16	imbi12d 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 )
19	12, 17, 18	chvar 1957	. . . . . . . 8 |- ( ( ph /\ a e. ZZ ) -> [_ a / x ]_ A e. RR )
20	19	adantr 465	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> [_ a / x ]_ A e. RR )
21		nfv 1673	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ /\ 0 <_ a )
22		nfcv 2574	. . . . . . . . . . 11 |- F/_ x 0
23		nfcv 2574	. . . . . . . . . . 11 |- F/_ x <_
24	22, 23, 10	nfbr 4331	. . . . . . . . . 10 |- F/ x 0 <_ [_ a / x ]_ A
25	21, 24	nfim 1852	. . . . . . . . 9 |- F/ x ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
26		breq2 4291	. . . . . . . . . . 11 |- ( x = a -> ( 0 <_ x <-> 0 <_ a ) )
27	13, 26	3anbi23d 1292	. . . . . . . . . 10 |- ( x = a -> ( ( ph /\ x e. ZZ /\ 0 <_ x ) <-> ( ph /\ a e. ZZ /\ 0 <_ a ) ) )
28	15	breq2d 4299	. . . . . . . . . 10 |- ( x = a -> ( 0 <_ A <-> 0 <_ [_ a / x ]_ A ) )
29	27, 28	imbi12d 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 )
31	25, 29, 30	chvar 1957	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
32	31	3expa 1187	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
33	20, 32	absidd 12901	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` [_ a / x ]_ A ) = [_ a / x ]_ A )
34		zre 10642	. . . . . . . . 9 |- ( a e. ZZ -> a e. RR )
35	34	ad2antlr 726	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> a e. RR )
36		absid 12777	. . . . . . . 8 |- ( ( a e. RR /\ 0 <_ a ) -> ( abs ` a ) = a )
37	35, 36	sylancom 667	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` a ) = a )
38	37	csbeq1d 3290	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> [_ ( abs ` a ) / x ]_ A = [_ a / x ]_ A )
39	33, 38	eqtr4d 2473	. . . . 5 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
40		nfv 1673	. . . . . . . 8 |- F/ y ( ( ph /\ a e. ZZ ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
41		eleq1 2498	. . . . . . . . . 10 |- ( y = a -> ( y e. ZZ <-> a e. ZZ ) )
42	41	anbi2d 703	. . . . . . . . 9 |- ( y = a -> ( ( ph /\ y e. ZZ ) <-> ( ph /\ a e. ZZ ) ) )
43		negex 9600	. . . . . . . . . . . 12 |- -u y e. _V
44		nfcv 2574	. . . . . . . . . . . 12 |- F/_ x C
45		oddcomabszz.5	. . . . . . . . . . . 12 |- ( x = -u y -> A = C )
46	43, 44, 45	csbief 3308	. . . . . . . . . . 11 |- [_ -u y / x ]_ A = C
47		negeq 9594	. . . . . . . . . . . 12 |- ( y = a -> -u y = -u a )
48	47	csbeq1d 3290	. . . . . . . . . . 11 |- ( y = a -> [_ -u y / x ]_ A = [_ -u a / x ]_ A )
49	46, 48	syl5eqr 2484	. . . . . . . . . 10 |- ( y = a -> C = [_ -u a / x ]_ A )
50		vex 2970	. . . . . . . . . . . . 13 |- y e. _V
51		nfcv 2574	. . . . . . . . . . . . 13 |- F/_ x B
52		oddcomabszz.4	. . . . . . . . . . . . 13 |- ( x = y -> A = B )
53	50, 51, 52	csbief 3308	. . . . . . . . . . . 12 |- [_ y / x ]_ A = B
54		csbeq1 3286	. . . . . . . . . . . 12 |- ( y = a -> [_ y / x ]_ A = [_ a / x ]_ A )
55	53, 54	syl5eqr 2484	. . . . . . . . . . 11 |- ( y = a -> B = [_ a / x ]_ A )
56	55	negeqd 9596	. . . . . . . . . 10 |- ( y = a -> -u B = -u [_ a / x ]_ A )
57	49, 56	eqeq12d 2452	. . . . . . . . 9 |- ( y = a -> ( C = -u B <-> [_ -u a / x ]_ A = -u [_ a / x ]_ A ) )
58	42, 57	imbi12d 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 )
60	40, 58, 59	chvar 1957	. . . . . . 7 |- ( ( ph /\ a e. ZZ ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
61	60	adantr 465	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
62	34	ad2antlr 726	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> a e. RR )
63		absnid 12779	. . . . . . . 8 |- ( ( a e. RR /\ a <_ 0 ) -> ( abs ` a ) = -u a )
64	62, 63	sylancom 667	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` a ) = -u a )
65	64	csbeq1d 3290	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ ( abs ` a ) / x ]_ A = [_ -u a / x ]_ A )
66	19	adantr 465	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ a / x ]_ A e. RR )
67		znegcl 10672	. . . . . . . . . . 11 |- ( a e. ZZ -> -u a e. ZZ )
68		nfv 1673	. . . . . . . . . . . . . 14 |- F/ x ( ph /\ -u a e. ZZ /\ 0 <_ -u a )
69		nfcsb1v 3299	. . . . . . . . . . . . . . 15 |- F/_ x [_ -u a / x ]_ A
70	22, 23, 69	nfbr 4331	. . . . . . . . . . . . . 14 |- F/ x 0 <_ [_ -u a / x ]_ A
71	68, 70	nfim 1852	. . . . . . . . . . . . 13 |- F/ x ( ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) -> 0 <_ [_ -u a / x ]_ A )
72		negex 9600	. . . . . . . . . . . . 13 |- -u a e. _V
73		eleq1 2498	. . . . . . . . . . . . . . 15 |- ( x = -u a -> ( x e. ZZ <-> -u a e. ZZ ) )
74		breq2 4291	. . . . . . . . . . . . . . 15 |- ( x = -u a -> ( 0 <_ x <-> 0 <_ -u a ) )
75	73, 74	3anbi23d 1292	. . . . . . . . . . . . . 14 |- ( x = -u a -> ( ( ph /\ x e. ZZ /\ 0 <_ x ) <-> ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) ) )
76		csbeq1a 3292	. . . . . . . . . . . . . . 15 |- ( x = -u a -> A = [_ -u a / x ]_ A )
77	76	breq2d 4299	. . . . . . . . . . . . . 14 |- ( x = -u a -> ( 0 <_ A <-> 0 <_ [_ -u a / x ]_ A ) )
78	75, 77	imbi12d 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 ) ) )
79	71, 72, 78, 30	vtoclf 3018	. . . . . . . . . . . 12 |- ( ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) -> 0 <_ [_ -u a / x ]_ A )
80	79	3expia 1189	. . . . . . . . . . 11 |- ( ( ph /\ -u a e. ZZ ) -> ( 0 <_ -u a -> 0 <_ [_ -u a / x ]_ A ) )
81	67, 80	sylan2 474	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ -u a -> 0 <_ [_ -u a / x ]_ A ) )
82	60	breq2d 4299	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ [_ -u a / x ]_ A <-> 0 <_ -u [_ a / x ]_ A ) )
83	81, 82	sylibd 214	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ -u a -> 0 <_ -u [_ a / x ]_ A ) )
84	34	adantl 466	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ ) -> a e. RR )
85	84	le0neg1d 9903	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( a <_ 0 <-> 0 <_ -u a ) )
86	19	le0neg1d 9903	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( [_ a / x ]_ A <_ 0 <-> 0 <_ -u [_ a / x ]_ A ) )
87	83, 85, 86	3imtr4d 268	. . . . . . . 8 |- ( ( ph /\ a e. ZZ ) -> ( a <_ 0 -> [_ a / x ]_ A <_ 0 ) )
88	87	imp 429	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ a / x ]_ A <_ 0 )
89	66, 88	absnidd 12892	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` [_ a / x ]_ A ) = -u [_ a / x ]_ A )
90	61, 65, 89	3eqtr4rd 2481	. . . . 5 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
91		0re 9378	. . . . . . 7 |- 0 e. RR
92		letric 9467	. . . . . . 7 |- ( ( 0 e. RR /\ a e. RR ) -> ( 0 <_ a \/ a <_ 0 ) )
93	91, 34, 92	sylancr 663	. . . . . 6 |- ( a e. ZZ -> ( 0 <_ a \/ a <_ 0 ) )
94	93	adantl 466	. . . . 5 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ a \/ a <_ 0 ) )
95	39, 90, 94	mpjaodan 784	. . . 4 |- ( ( ph /\ a e. ZZ ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
96	8, 95	vtoclg 3025	. . 3 |- ( D e. ZZ -> ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A ) )
97	96	anabsi7 815	. 2 |- ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A )
98		nfcvd 2575	. . . . 5 |- ( D e. ZZ -> F/_ x E )
99		oddcomabszz.6	. . . . 5 |- ( x = D -> A = E )
100	98, 99	csbiegf 3307	. . . 4 |- ( D e. ZZ -> [_ D / x ]_ A = E )
101	100	fveq2d 5690	. . 3 |- ( D e. ZZ -> ( abs ` [_ D / x ]_ A ) = ( abs ` E ) )
102	101	adantl 466	. 2 |- ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = ( abs ` E ) )
103		fvex 5696	. . . 4 |- ( abs ` D ) e. _V
104		nfcv 2574	. . . 4 |- F/_ x F
105		oddcomabszz.7	. . . 4 |- ( x = ( abs ` D ) -> A = F )
106	103, 104, 105	csbief 3308	. . 3 |- [_ ( abs ` D ) / x ]_ A = F
107	106	a1i 11	. 2 |- ( ( ph /\ D e. ZZ ) -> [_ ( abs ` D ) / x ]_ A = F )
108	97, 102, 107	3eqtr3d 2478	1 |- ( ( ph /\ D e. ZZ ) -> ( abs ` E ) = F )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   [_csb 3283   class class class wbr 4287   ` cfv 5413   RRcr 9273   0cc0 9274   <_ cle 9411   -ucneg 9588   ZZcz 10638   abscabs 12715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717

This theorem is referenced by:  rmyabs  29254