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.

Step	Hyp	Ref	Expression
1		eleq1 2539	. . . . . 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 3438	. . . . . . 7 |- ( a = D -> [_ a / x ]_ A = [_ D / x ]_ A )
4	3	fveq2d 5870	. . . . . 6 |- ( a = D -> ( abs ` [_ a / x ]_ A ) = ( abs ` [_ D / x ]_ A ) )
5		fveq2 5866	. . . . . . 7 |- ( a = D -> ( abs ` a ) = ( abs ` D ) )
6	5	csbeq1d 3442	. . . . . 6 |- ( a = D -> [_ ( abs ` a ) / x ]_ A = [_ ( abs ` D ) / x ]_ A )
7	4, 6	eqeq12d 2489	. . . . 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 1683	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ )
10		nfcsb1v 3451	. . . . . . . . . . 11 |- F/_ x [_ a / x ]_ A
11	10	nfel1 2645	. . . . . . . . . 10 |- F/ x [_ a / x ]_ A e. RR
12	9, 11	nfim 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 ) )
14	13	anbi2d 703	. . . . . . . . . 10 |- ( x = a -> ( ( ph /\ x e. ZZ ) <-> ( ph /\ a e. ZZ ) ) )
15		csbeq1a 3444	. . . . . . . . . . 11 |- ( x = a -> A = [_ a / x ]_ A )
16	15	eleq1d 2536	. . . . . . . . . 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 1982	. . . . . . . 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 1683	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ /\ 0 <_ a )
22		nfcv 2629	. . . . . . . . . . 11 |- F/_ x 0
23		nfcv 2629	. . . . . . . . . . 11 |- F/_ x <_
24	22, 23, 10	nfbr 4491	. . . . . . . . . 10 |- F/ x 0 <_ [_ a / x ]_ A
25	21, 24	nfim 1867	. . . . . . . . 9 |- F/ x ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
26		breq2 4451	. . . . . . . . . . 11 |- ( x = a -> ( 0 <_ x <-> 0 <_ a ) )
27	13, 26	3anbi23d 1302	. . . . . . . . . 10 |- ( x = a -> ( ( ph /\ x e. ZZ /\ 0 <_ x ) <-> ( ph /\ a e. ZZ /\ 0 <_ a ) ) )
28	15	breq2d 4459	. . . . . . . . . 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 1982	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
32	31	3expa 1196	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
33	20, 32	absidd 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 )
35	34	ad2antlr 726	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> a e. RR )
36		absid 13095	. . . . . . . 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 3442	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> [_ ( abs ` a ) / x ]_ A = [_ a / x ]_ A )
39	33, 38	eqtr4d 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 ) )
42	41	anbi2d 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 )
46	43, 44, 45	csbief 3460	. . . . . . . . . . 11 |- [_ -u y / x ]_ A = C
47		negeq 9813	. . . . . . . . . . . 12 |- ( y = a -> -u y = -u a )
48	47	csbeq1d 3442	. . . . . . . . . . 11 |- ( y = a -> [_ -u y / x ]_ A = [_ -u a / x ]_ A )
49	46, 48	syl5eqr 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 )
53	50, 51, 52	csbief 3460	. . . . . . . . . . . 12 |- [_ y / x ]_ A = B
54		csbeq1 3438	. . . . . . . . . . . 12 |- ( y = a -> [_ y / x ]_ A = [_ a / x ]_ A )
55	53, 54	syl5eqr 2522	. . . . . . . . . . 11 |- ( y = a -> B = [_ a / x ]_ A )
56	55	negeqd 9815	. . . . . . . . . 10 |- ( y = a -> -u B = -u [_ a / x ]_ A )
57	49, 56	eqeq12d 2489	. . . . . . . . 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 1982	. . . . . . 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 13097	. . . . . . . 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 3442	. . . . . 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 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
70	22, 23, 69	nfbr 4491	. . . . . . . . . . . . . 14 |- F/ x 0 <_ [_ -u a / x ]_ A
71	68, 70	nfim 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 ) )
75	73, 74	3anbi23d 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 )
77	76	breq2d 4459	. . . . . . . . . . . . . 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 3164	. . . . . . . . . . . 12 |- ( ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) -> 0 <_ [_ -u a / x ]_ A )
80	79	3expia 1198	. . . . . . . . . . 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 4459	. . . . . . . . . 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 10125	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( a <_ 0 <-> 0 <_ -u a ) )
86	19	le0neg1d 10125	. . . . . . . . 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 13211	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` [_ a / x ]_ A ) = -u [_ a / x ]_ A )
90	61, 65, 89	3eqtr4rd 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 ) )
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 3171	. . 3 |- ( D e. ZZ -> ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A ) )
97	96	anabsi7 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 )
100	98, 99	csbiegf 3459	. . . 4 |- ( D e. ZZ -> [_ D / x ]_ A = E )
101	100	fveq2d 5870	. . 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 5876	. . . 4 |- ( abs ` D ) e. _V
104		nfcv 2629	. . . 4 |- F/_ x F
105		oddcomabszz.7	. . . 4 |- ( x = ( abs ` D ) -> A = F )
106	103, 104, 105	csbief 3460	. . 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 2516	1 |- ( ( ph /\ D e. ZZ ) -> ( abs ` E ) = F )

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