Theorem oddcomabszz 35792

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 2517	. . . . . 6 |- ( a = D -> ( a e. ZZ <-> D e. ZZ ) )
2	1	anbi2d 710	. . . . 5 |- ( a = D -> ( ( ph /\ a e. ZZ ) <-> ( ph /\ D e. ZZ ) ) )
3		csbeq1 3366	. . . . . . 7 |- ( a = D -> [_ a / x ]_ A = [_ D / x ]_ A )
4	3	fveq2d 5869	. . . . . 6 |- ( a = D -> ( abs ` [_ a / x ]_ A ) = ( abs ` [_ D / x ]_ A ) )
5		fveq2 5865	. . . . . . 7 |- ( a = D -> ( abs ` a ) = ( abs ` D ) )
6	5	csbeq1d 3370	. . . . . 6 |- ( a = D -> [_ ( abs ` a ) / x ]_ A = [_ ( abs ` D ) / x ]_ A )
7	4, 6	eqeq12d 2466	. . . . 5 |- ( a = D -> ( ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A <-> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A ) )
8	2, 7	imbi12d 322	. . . 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 1761	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ )
10		nfcsb1v 3379	. . . . . . . . . . 11 |- F/_ x [_ a / x ]_ A
11	10	nfel1 2606	. . . . . . . . . 10 |- F/ x [_ a / x ]_ A e. RR
12	9, 11	nfim 2003	. . . . . . . . 9 |- F/ x ( ( ph /\ a e. ZZ ) -> [_ a / x ]_ A e. RR )
13		eleq1 2517	. . . . . . . . . . 11 |- ( x = a -> ( x e. ZZ <-> a e. ZZ ) )
14	13	anbi2d 710	. . . . . . . . . 10 |- ( x = a -> ( ( ph /\ x e. ZZ ) <-> ( ph /\ a e. ZZ ) ) )
15		csbeq1a 3372	. . . . . . . . . . 11 |- ( x = a -> A = [_ a / x ]_ A )
16	15	eleq1d 2513	. . . . . . . . . 10 |- ( x = a -> ( A e. RR <-> [_ a / x ]_ A e. RR ) )
17	14, 16	imbi12d 322	. . . . . . . . 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 2106	. . . . . . . 8 |- ( ( ph /\ a e. ZZ ) -> [_ a / x ]_ A e. RR )
20	19	adantr 467	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> [_ a / x ]_ A e. RR )
21		nfv 1761	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ /\ 0 <_ a )
22		nfcv 2592	. . . . . . . . . . 11 |- F/_ x 0
23		nfcv 2592	. . . . . . . . . . 11 |- F/_ x <_
24	22, 23, 10	nfbr 4447	. . . . . . . . . 10 |- F/ x 0 <_ [_ a / x ]_ A
25	21, 24	nfim 2003	. . . . . . . . 9 |- F/ x ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
26		breq2 4406	. . . . . . . . . . 11 |- ( x = a -> ( 0 <_ x <-> 0 <_ a ) )
27	13, 26	3anbi23d 1342	. . . . . . . . . 10 |- ( x = a -> ( ( ph /\ x e. ZZ /\ 0 <_ x ) <-> ( ph /\ a e. ZZ /\ 0 <_ a ) ) )
28	15	breq2d 4414	. . . . . . . . . 10 |- ( x = a -> ( 0 <_ A <-> 0 <_ [_ a / x ]_ A ) )
29	27, 28	imbi12d 322	. . . . . . . . 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 2106	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
32	31	3expa 1208	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
33	20, 32	absidd 13484	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` [_ a / x ]_ A ) = [_ a / x ]_ A )
34		zre 10941	. . . . . . . . 9 |- ( a e. ZZ -> a e. RR )
35	34	ad2antlr 733	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> a e. RR )
36		absid 13359	. . . . . . . 8 |- ( ( a e. RR /\ 0 <_ a ) -> ( abs ` a ) = a )
37	35, 36	sylancom 673	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` a ) = a )
38	37	csbeq1d 3370	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> [_ ( abs ` a ) / x ]_ A = [_ a / x ]_ A )
39	33, 38	eqtr4d 2488	. . . . 5 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
40		nfv 1761	. . . . . . . 8 |- F/ y ( ( ph /\ a e. ZZ ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
41		eleq1 2517	. . . . . . . . . 10 |- ( y = a -> ( y e. ZZ <-> a e. ZZ ) )
42	41	anbi2d 710	. . . . . . . . 9 |- ( y = a -> ( ( ph /\ y e. ZZ ) <-> ( ph /\ a e. ZZ ) ) )
43		negex 9873	. . . . . . . . . . . 12 |- -u y e. _V
44		oddcomabszz.5	. . . . . . . . . . . 12 |- ( x = -u y -> A = C )
45	43, 44	csbie 3389	. . . . . . . . . . 11 |- [_ -u y / x ]_ A = C
46		negeq 9867	. . . . . . . . . . . 12 |- ( y = a -> -u y = -u a )
47	46	csbeq1d 3370	. . . . . . . . . . 11 |- ( y = a -> [_ -u y / x ]_ A = [_ -u a / x ]_ A )
48	45, 47	syl5eqr 2499	. . . . . . . . . 10 |- ( y = a -> C = [_ -u a / x ]_ A )
49		vex 3048	. . . . . . . . . . . . 13 |- y e. _V
50		oddcomabszz.4	. . . . . . . . . . . . 13 |- ( x = y -> A = B )
51	49, 50	csbie 3389	. . . . . . . . . . . 12 |- [_ y / x ]_ A = B
52		csbeq1 3366	. . . . . . . . . . . 12 |- ( y = a -> [_ y / x ]_ A = [_ a / x ]_ A )
53	51, 52	syl5eqr 2499	. . . . . . . . . . 11 |- ( y = a -> B = [_ a / x ]_ A )
54	53	negeqd 9869	. . . . . . . . . 10 |- ( y = a -> -u B = -u [_ a / x ]_ A )
55	48, 54	eqeq12d 2466	. . . . . . . . 9 |- ( y = a -> ( C = -u B <-> [_ -u a / x ]_ A = -u [_ a / x ]_ A ) )
56	42, 55	imbi12d 322	. . . . . . . 8 |- ( y = a -> ( ( ( ph /\ y e. ZZ ) -> C = -u B ) <-> ( ( ph /\ a e. ZZ ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A ) ) )
57		oddcomabszz.3	. . . . . . . 8 |- ( ( ph /\ y e. ZZ ) -> C = -u B )
58	40, 56, 57	chvar 2106	. . . . . . 7 |- ( ( ph /\ a e. ZZ ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
59	58	adantr 467	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
60	34	ad2antlr 733	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> a e. RR )
61		absnid 13361	. . . . . . . 8 |- ( ( a e. RR /\ a <_ 0 ) -> ( abs ` a ) = -u a )
62	60, 61	sylancom 673	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` a ) = -u a )
63	62	csbeq1d 3370	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ ( abs ` a ) / x ]_ A = [_ -u a / x ]_ A )
64	19	adantr 467	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ a / x ]_ A e. RR )
65		znegcl 10972	. . . . . . . . . . 11 |- ( a e. ZZ -> -u a e. ZZ )
66		nfv 1761	. . . . . . . . . . . . . 14 |- F/ x ( ph /\ -u a e. ZZ /\ 0 <_ -u a )
67		nfcsb1v 3379	. . . . . . . . . . . . . . 15 |- F/_ x [_ -u a / x ]_ A
68	22, 23, 67	nfbr 4447	. . . . . . . . . . . . . 14 |- F/ x 0 <_ [_ -u a / x ]_ A
69	66, 68	nfim 2003	. . . . . . . . . . . . 13 |- F/ x ( ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) -> 0 <_ [_ -u a / x ]_ A )
70		negex 9873	. . . . . . . . . . . . 13 |- -u a e. _V
71		eleq1 2517	. . . . . . . . . . . . . . 15 |- ( x = -u a -> ( x e. ZZ <-> -u a e. ZZ ) )
72		breq2 4406	. . . . . . . . . . . . . . 15 |- ( x = -u a -> ( 0 <_ x <-> 0 <_ -u a ) )
73	71, 72	3anbi23d 1342	. . . . . . . . . . . . . 14 |- ( x = -u a -> ( ( ph /\ x e. ZZ /\ 0 <_ x ) <-> ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) ) )
74		csbeq1a 3372	. . . . . . . . . . . . . . 15 |- ( x = -u a -> A = [_ -u a / x ]_ A )
75	74	breq2d 4414	. . . . . . . . . . . . . 14 |- ( x = -u a -> ( 0 <_ A <-> 0 <_ [_ -u a / x ]_ A ) )
76	73, 75	imbi12d 322	. . . . . . . . . . . . 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 ) ) )
77	69, 70, 76, 30	vtoclf 3099	. . . . . . . . . . . 12 |- ( ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) -> 0 <_ [_ -u a / x ]_ A )
78	77	3expia 1210	. . . . . . . . . . 11 |- ( ( ph /\ -u a e. ZZ ) -> ( 0 <_ -u a -> 0 <_ [_ -u a / x ]_ A ) )
79	65, 78	sylan2 477	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ -u a -> 0 <_ [_ -u a / x ]_ A ) )
80	58	breq2d 4414	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ [_ -u a / x ]_ A <-> 0 <_ -u [_ a / x ]_ A ) )
81	79, 80	sylibd 218	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ -u a -> 0 <_ -u [_ a / x ]_ A ) )
82	34	adantl 468	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ ) -> a e. RR )
83	82	le0neg1d 10185	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( a <_ 0 <-> 0 <_ -u a ) )
84	19	le0neg1d 10185	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( [_ a / x ]_ A <_ 0 <-> 0 <_ -u [_ a / x ]_ A ) )
85	81, 83, 84	3imtr4d 272	. . . . . . . 8 |- ( ( ph /\ a e. ZZ ) -> ( a <_ 0 -> [_ a / x ]_ A <_ 0 ) )
86	85	imp 431	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ a / x ]_ A <_ 0 )
87	64, 86	absnidd 13475	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` [_ a / x ]_ A ) = -u [_ a / x ]_ A )
88	59, 63, 87	3eqtr4rd 2496	. . . . 5 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
89		0re 9643	. . . . . . 7 |- 0 e. RR
90		letric 9734	. . . . . . 7 |- ( ( 0 e. RR /\ a e. RR ) -> ( 0 <_ a \/ a <_ 0 ) )
91	89, 34, 90	sylancr 669	. . . . . 6 |- ( a e. ZZ -> ( 0 <_ a \/ a <_ 0 ) )
92	91	adantl 468	. . . . 5 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ a \/ a <_ 0 ) )
93	39, 88, 92	mpjaodan 795	. . . 4 |- ( ( ph /\ a e. ZZ ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
94	8, 93	vtoclg 3107	. . 3 |- ( D e. ZZ -> ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A ) )
95	94	anabsi7 828	. 2 |- ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A )
96		nfcvd 2593	. . . . 5 |- ( D e. ZZ -> F/_ x E )
97		oddcomabszz.6	. . . . 5 |- ( x = D -> A = E )
98	96, 97	csbiegf 3387	. . . 4 |- ( D e. ZZ -> [_ D / x ]_ A = E )
99	98	fveq2d 5869	. . 3 |- ( D e. ZZ -> ( abs ` [_ D / x ]_ A ) = ( abs ` E ) )
100	99	adantl 468	. 2 |- ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = ( abs ` E ) )
101		fvex 5875	. . . 4 |- ( abs ` D ) e. _V
102		oddcomabszz.7	. . . 4 |- ( x = ( abs ` D ) -> A = F )
103	101, 102	csbie 3389	. . 3 |- [_ ( abs ` D ) / x ]_ A = F
104	103	a1i 11	. 2 |- ( ( ph /\ D e. ZZ ) -> [_ ( abs ` D ) / x ]_ A = F )
105	95, 100, 104	3eqtr3d 2493	1 |- ( ( ph /\ D e. ZZ ) -> ( abs ` E ) = F )

Syntax hints:   -> wi 4   \/ wo 370   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   [_csb 3363   class class class wbr 4402   ` cfv 5582   RRcr 9538   0cc0 9539   <_ cle 9676   -ucneg 9861   ZZcz 10937   abscabs 13297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299

This theorem is referenced by:  rmyabs  35808