Theorem oddcomabszz 29434

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 2526	. . . . . 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 3399	. . . . . . 7 |- ( a = D -> [_ a / x ]_ A = [_ D / x ]_ A )
4	3	fveq2d 5804	. . . . . 6 |- ( a = D -> ( abs ` [_ a / x ]_ A ) = ( abs ` [_ D / x ]_ A ) )
5		fveq2 5800	. . . . . . 7 |- ( a = D -> ( abs ` a ) = ( abs ` D ) )
6	5	csbeq1d 3403	. . . . . 6 |- ( a = D -> [_ ( abs ` a ) / x ]_ A = [_ ( abs ` D ) / x ]_ A )
7	4, 6	eqeq12d 2476	. . . . 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 1674	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ )
10		nfcsb1v 3412	. . . . . . . . . . 11 |- F/_ x [_ a / x ]_ A
11	10	nfel1 2632	. . . . . . . . . 10 |- F/ x [_ a / x ]_ A e. RR
12	9, 11	nfim 1858	. . . . . . . . 9 |- F/ x ( ( ph /\ a e. ZZ ) -> [_ a / x ]_ A e. RR )
13		eleq1 2526	. . . . . . . . . . 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 3405	. . . . . . . . . . 11 |- ( x = a -> A = [_ a / x ]_ A )
16	15	eleq1d 2523	. . . . . . . . . 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 1969	. . . . . . . 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 1674	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ /\ 0 <_ a )
22		nfcv 2616	. . . . . . . . . . 11 |- F/_ x 0
23		nfcv 2616	. . . . . . . . . . 11 |- F/_ x <_
24	22, 23, 10	nfbr 4445	. . . . . . . . . 10 |- F/ x 0 <_ [_ a / x ]_ A
25	21, 24	nfim 1858	. . . . . . . . 9 |- F/ x ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
26		breq2 4405	. . . . . . . . . . 11 |- ( x = a -> ( 0 <_ x <-> 0 <_ a ) )
27	13, 26	3anbi23d 1293	. . . . . . . . . 10 |- ( x = a -> ( ( ph /\ x e. ZZ /\ 0 <_ x ) <-> ( ph /\ a e. ZZ /\ 0 <_ a ) ) )
28	15	breq2d 4413	. . . . . . . . . 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 1969	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
32	31	3expa 1188	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
33	20, 32	absidd 13028	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` [_ a / x ]_ A ) = [_ a / x ]_ A )
34		zre 10762	. . . . . . . . 9 |- ( a e. ZZ -> a e. RR )
35	34	ad2antlr 726	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> a e. RR )
36		absid 12904	. . . . . . . 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 3403	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> [_ ( abs ` a ) / x ]_ A = [_ a / x ]_ A )
39	33, 38	eqtr4d 2498	. . . . 5 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
40		nfv 1674	. . . . . . . 8 |- F/ y ( ( ph /\ a e. ZZ ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
41		eleq1 2526	. . . . . . . . . 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 9720	. . . . . . . . . . . 12 |- -u y e. _V
44		nfcv 2616	. . . . . . . . . . . 12 |- F/_ x C
45		oddcomabszz.5	. . . . . . . . . . . 12 |- ( x = -u y -> A = C )
46	43, 44, 45	csbief 3421	. . . . . . . . . . 11 |- [_ -u y / x ]_ A = C
47		negeq 9714	. . . . . . . . . . . 12 |- ( y = a -> -u y = -u a )
48	47	csbeq1d 3403	. . . . . . . . . . 11 |- ( y = a -> [_ -u y / x ]_ A = [_ -u a / x ]_ A )
49	46, 48	syl5eqr 2509	. . . . . . . . . 10 |- ( y = a -> C = [_ -u a / x ]_ A )
50		vex 3081	. . . . . . . . . . . . 13 |- y e. _V
51		nfcv 2616	. . . . . . . . . . . . 13 |- F/_ x B
52		oddcomabszz.4	. . . . . . . . . . . . 13 |- ( x = y -> A = B )
53	50, 51, 52	csbief 3421	. . . . . . . . . . . 12 |- [_ y / x ]_ A = B
54		csbeq1 3399	. . . . . . . . . . . 12 |- ( y = a -> [_ y / x ]_ A = [_ a / x ]_ A )
55	53, 54	syl5eqr 2509	. . . . . . . . . . 11 |- ( y = a -> B = [_ a / x ]_ A )
56	55	negeqd 9716	. . . . . . . . . 10 |- ( y = a -> -u B = -u [_ a / x ]_ A )
57	49, 56	eqeq12d 2476	. . . . . . . . 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 1969	. . . . . . 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 12906	. . . . . . . 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 3403	. . . . . 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 10792	. . . . . . . . . . 11 |- ( a e. ZZ -> -u a e. ZZ )
68		nfv 1674	. . . . . . . . . . . . . 14 |- F/ x ( ph /\ -u a e. ZZ /\ 0 <_ -u a )
69		nfcsb1v 3412	. . . . . . . . . . . . . . 15 |- F/_ x [_ -u a / x ]_ A
70	22, 23, 69	nfbr 4445	. . . . . . . . . . . . . 14 |- F/ x 0 <_ [_ -u a / x ]_ A
71	68, 70	nfim 1858	. . . . . . . . . . . . 13 |- F/ x ( ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) -> 0 <_ [_ -u a / x ]_ A )
72		negex 9720	. . . . . . . . . . . . 13 |- -u a e. _V
73		eleq1 2526	. . . . . . . . . . . . . . 15 |- ( x = -u a -> ( x e. ZZ <-> -u a e. ZZ ) )
74		breq2 4405	. . . . . . . . . . . . . . 15 |- ( x = -u a -> ( 0 <_ x <-> 0 <_ -u a ) )
75	73, 74	3anbi23d 1293	. . . . . . . . . . . . . 14 |- ( x = -u a -> ( ( ph /\ x e. ZZ /\ 0 <_ x ) <-> ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) ) )
76		csbeq1a 3405	. . . . . . . . . . . . . . 15 |- ( x = -u a -> A = [_ -u a / x ]_ A )
77	76	breq2d 4413	. . . . . . . . . . . . . 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 3129	. . . . . . . . . . . 12 |- ( ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) -> 0 <_ [_ -u a / x ]_ A )
80	79	3expia 1190	. . . . . . . . . . 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 4413	. . . . . . . . . 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 10023	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( a <_ 0 <-> 0 <_ -u a ) )
86	19	le0neg1d 10023	. . . . . . . . 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 13019	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` [_ a / x ]_ A ) = -u [_ a / x ]_ A )
90	61, 65, 89	3eqtr4rd 2506	. . . . 5 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
91		0re 9498	. . . . . . 7 |- 0 e. RR
92		letric 9587	. . . . . . 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 3136	. . 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 2617	. . . . 5 |- ( D e. ZZ -> F/_ x E )
99		oddcomabszz.6	. . . . 5 |- ( x = D -> A = E )
100	98, 99	csbiegf 3420	. . . 4 |- ( D e. ZZ -> [_ D / x ]_ A = E )
101	100	fveq2d 5804	. . 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 5810	. . . 4 |- ( abs ` D ) e. _V
104		nfcv 2616	. . . 4 |- F/_ x F
105		oddcomabszz.7	. . . 4 |- ( x = ( abs ` D ) -> A = F )
106	103, 104, 105	csbief 3421	. . 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 2503	1 |- ( ( ph /\ D e. ZZ ) -> ( abs ` E ) = F )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   [_csb 3396   class class class wbr 4401   ` cfv 5527   RRcr 9393   0cc0 9394   <_ cle 9531   -ucneg 9708   ZZcz 10758   abscabs 12842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-seq 11925  df-exp 11984  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844

This theorem is referenced by:  rmyabs  29450