Theorem oddcomabszz 37509

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 2689	. . . . . 6 |- ( a = D -> ( a e. ZZ <-> D e. ZZ ) )
2	1	anbi2d 740	. . . . 5 |- ( a = D -> ( ( ph /\ a e. ZZ ) <-> ( ph /\ D e. ZZ ) ) )
3		csbeq1 3536	. . . . . . 7 |- ( a = D -> [_ a / x ]_ A = [_ D / x ]_ A )
4	3	fveq2d 6195	. . . . . 6 |- ( a = D -> ( abs ` [_ a / x ]_ A ) = ( abs ` [_ D / x ]_ A ) )
5		fveq2 6191	. . . . . . 7 |- ( a = D -> ( abs ` a ) = ( abs ` D ) )
6	5	csbeq1d 3540	. . . . . 6 |- ( a = D -> [_ ( abs ` a ) / x ]_ A = [_ ( abs ` D ) / x ]_ A )
7	4, 6	eqeq12d 2637	. . . . 5 |- ( a = D -> ( ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A <-> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A ) )
8	2, 7	imbi12d 334	. . . 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 1843	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ )
10		nfcsb1v 3549	. . . . . . . . . . 11 |- F/_ x [_ a / x ]_ A
11	10	nfel1 2779	. . . . . . . . . 10 |- F/ x [_ a / x ]_ A e. RR
12	9, 11	nfim 1825	. . . . . . . . 9 |- F/ x ( ( ph /\ a e. ZZ ) -> [_ a / x ]_ A e. RR )
13		eleq1 2689	. . . . . . . . . . 11 |- ( x = a -> ( x e. ZZ <-> a e. ZZ ) )
14	13	anbi2d 740	. . . . . . . . . 10 |- ( x = a -> ( ( ph /\ x e. ZZ ) <-> ( ph /\ a e. ZZ ) ) )
15		csbeq1a 3542	. . . . . . . . . . 11 |- ( x = a -> A = [_ a / x ]_ A )
16	15	eleq1d 2686	. . . . . . . . . 10 |- ( x = a -> ( A e. RR <-> [_ a / x ]_ A e. RR ) )
17	14, 16	imbi12d 334	. . . . . . . . 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 2262	. . . . . . . 8 |- ( ( ph /\ a e. ZZ ) -> [_ a / x ]_ A e. RR )
20	19	adantr 481	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> [_ a / x ]_ A e. RR )
21		nfv 1843	. . . . . . . . . 10 |- F/ x ( ph /\ a e. ZZ /\ 0 <_ a )
22		nfcv 2764	. . . . . . . . . . 11 |- F/_ x 0
23		nfcv 2764	. . . . . . . . . . 11 |- F/_ x <_
24	22, 23, 10	nfbr 4699	. . . . . . . . . 10 |- F/ x 0 <_ [_ a / x ]_ A
25	21, 24	nfim 1825	. . . . . . . . 9 |- F/ x ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
26		breq2 4657	. . . . . . . . . . 11 |- ( x = a -> ( 0 <_ x <-> 0 <_ a ) )
27	13, 26	3anbi23d 1402	. . . . . . . . . 10 |- ( x = a -> ( ( ph /\ x e. ZZ /\ 0 <_ x ) <-> ( ph /\ a e. ZZ /\ 0 <_ a ) ) )
28	15	breq2d 4665	. . . . . . . . . 10 |- ( x = a -> ( 0 <_ A <-> 0 <_ [_ a / x ]_ A ) )
29	27, 28	imbi12d 334	. . . . . . . . 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 2262	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
32	31	3expa 1265	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> 0 <_ [_ a / x ]_ A )
33	20, 32	absidd 14161	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` [_ a / x ]_ A ) = [_ a / x ]_ A )
34		zre 11381	. . . . . . . . 9 |- ( a e. ZZ -> a e. RR )
35	34	ad2antlr 763	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> a e. RR )
36		absid 14036	. . . . . . . 8 |- ( ( a e. RR /\ 0 <_ a ) -> ( abs ` a ) = a )
37	35, 36	sylancom 701	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` a ) = a )
38	37	csbeq1d 3540	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> [_ ( abs ` a ) / x ]_ A = [_ a / x ]_ A )
39	33, 38	eqtr4d 2659	. . . . 5 |- ( ( ( ph /\ a e. ZZ ) /\ 0 <_ a ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
40		nfv 1843	. . . . . . . 8 |- F/ y ( ( ph /\ a e. ZZ ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
41		eleq1 2689	. . . . . . . . . 10 |- ( y = a -> ( y e. ZZ <-> a e. ZZ ) )
42	41	anbi2d 740	. . . . . . . . 9 |- ( y = a -> ( ( ph /\ y e. ZZ ) <-> ( ph /\ a e. ZZ ) ) )
43		negex 10279	. . . . . . . . . . . 12 |- -u y e. _V
44		oddcomabszz.5	. . . . . . . . . . . 12 |- ( x = -u y -> A = C )
45	43, 44	csbie 3559	. . . . . . . . . . 11 |- [_ -u y / x ]_ A = C
46		negeq 10273	. . . . . . . . . . . 12 |- ( y = a -> -u y = -u a )
47	46	csbeq1d 3540	. . . . . . . . . . 11 |- ( y = a -> [_ -u y / x ]_ A = [_ -u a / x ]_ A )
48	45, 47	syl5eqr 2670	. . . . . . . . . 10 |- ( y = a -> C = [_ -u a / x ]_ A )
49		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
50		oddcomabszz.4	. . . . . . . . . . . . 13 |- ( x = y -> A = B )
51	49, 50	csbie 3559	. . . . . . . . . . . 12 |- [_ y / x ]_ A = B
52		csbeq1 3536	. . . . . . . . . . . 12 |- ( y = a -> [_ y / x ]_ A = [_ a / x ]_ A )
53	51, 52	syl5eqr 2670	. . . . . . . . . . 11 |- ( y = a -> B = [_ a / x ]_ A )
54	53	negeqd 10275	. . . . . . . . . 10 |- ( y = a -> -u B = -u [_ a / x ]_ A )
55	48, 54	eqeq12d 2637	. . . . . . . . 9 |- ( y = a -> ( C = -u B <-> [_ -u a / x ]_ A = -u [_ a / x ]_ A ) )
56	42, 55	imbi12d 334	. . . . . . . 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 2262	. . . . . . 7 |- ( ( ph /\ a e. ZZ ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
59	58	adantr 481	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ -u a / x ]_ A = -u [_ a / x ]_ A )
60	34	ad2antlr 763	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> a e. RR )
61		absnid 14038	. . . . . . . 8 |- ( ( a e. RR /\ a <_ 0 ) -> ( abs ` a ) = -u a )
62	60, 61	sylancom 701	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` a ) = -u a )
63	62	csbeq1d 3540	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ ( abs ` a ) / x ]_ A = [_ -u a / x ]_ A )
64	19	adantr 481	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ a / x ]_ A e. RR )
65		znegcl 11412	. . . . . . . . . . 11 |- ( a e. ZZ -> -u a e. ZZ )
66		nfv 1843	. . . . . . . . . . . . . 14 |- F/ x ( ph /\ -u a e. ZZ /\ 0 <_ -u a )
67		nfcsb1v 3549	. . . . . . . . . . . . . . 15 |- F/_ x [_ -u a / x ]_ A
68	22, 23, 67	nfbr 4699	. . . . . . . . . . . . . 14 |- F/ x 0 <_ [_ -u a / x ]_ A
69	66, 68	nfim 1825	. . . . . . . . . . . . 13 |- F/ x ( ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) -> 0 <_ [_ -u a / x ]_ A )
70		negex 10279	. . . . . . . . . . . . 13 |- -u a e. _V
71		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( x = -u a -> ( x e. ZZ <-> -u a e. ZZ ) )
72		breq2 4657	. . . . . . . . . . . . . . 15 |- ( x = -u a -> ( 0 <_ x <-> 0 <_ -u a ) )
73	71, 72	3anbi23d 1402	. . . . . . . . . . . . . 14 |- ( x = -u a -> ( ( ph /\ x e. ZZ /\ 0 <_ x ) <-> ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) ) )
74		csbeq1a 3542	. . . . . . . . . . . . . . 15 |- ( x = -u a -> A = [_ -u a / x ]_ A )
75	74	breq2d 4665	. . . . . . . . . . . . . 14 |- ( x = -u a -> ( 0 <_ A <-> 0 <_ [_ -u a / x ]_ A ) )
76	73, 75	imbi12d 334	. . . . . . . . . . . . 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 3258	. . . . . . . . . . . 12 |- ( ( ph /\ -u a e. ZZ /\ 0 <_ -u a ) -> 0 <_ [_ -u a / x ]_ A )
78	77	3expia 1267	. . . . . . . . . . 11 |- ( ( ph /\ -u a e. ZZ ) -> ( 0 <_ -u a -> 0 <_ [_ -u a / x ]_ A ) )
79	65, 78	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ -u a -> 0 <_ [_ -u a / x ]_ A ) )
80	58	breq2d 4665	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ [_ -u a / x ]_ A <-> 0 <_ -u [_ a / x ]_ A ) )
81	79, 80	sylibd 229	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ -u a -> 0 <_ -u [_ a / x ]_ A ) )
82	34	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ ) -> a e. RR )
83	82	le0neg1d 10599	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( a <_ 0 <-> 0 <_ -u a ) )
84	19	le0neg1d 10599	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ ) -> ( [_ a / x ]_ A <_ 0 <-> 0 <_ -u [_ a / x ]_ A ) )
85	81, 83, 84	3imtr4d 283	. . . . . . . 8 |- ( ( ph /\ a e. ZZ ) -> ( a <_ 0 -> [_ a / x ]_ A <_ 0 ) )
86	85	imp 445	. . . . . . 7 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> [_ a / x ]_ A <_ 0 )
87	64, 86	absnidd 14152	. . . . . 6 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` [_ a / x ]_ A ) = -u [_ a / x ]_ A )
88	59, 63, 87	3eqtr4rd 2667	. . . . 5 |- ( ( ( ph /\ a e. ZZ ) /\ a <_ 0 ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
89		0re 10040	. . . . . . 7 |- 0 e. RR
90		letric 10137	. . . . . . 7 |- ( ( 0 e. RR /\ a e. RR ) -> ( 0 <_ a \/ a <_ 0 ) )
91	89, 34, 90	sylancr 695	. . . . . 6 |- ( a e. ZZ -> ( 0 <_ a \/ a <_ 0 ) )
92	91	adantl 482	. . . . 5 |- ( ( ph /\ a e. ZZ ) -> ( 0 <_ a \/ a <_ 0 ) )
93	39, 88, 92	mpjaodan 827	. . . 4 |- ( ( ph /\ a e. ZZ ) -> ( abs ` [_ a / x ]_ A ) = [_ ( abs ` a ) / x ]_ A )
94	8, 93	vtoclg 3266	. . 3 |- ( D e. ZZ -> ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A ) )
95	94	anabsi7 860	. 2 |- ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = [_ ( abs ` D ) / x ]_ A )
96		nfcvd 2765	. . . . 5 |- ( D e. ZZ -> F/_ x E )
97		oddcomabszz.6	. . . . 5 |- ( x = D -> A = E )
98	96, 97	csbiegf 3557	. . . 4 |- ( D e. ZZ -> [_ D / x ]_ A = E )
99	98	fveq2d 6195	. . 3 |- ( D e. ZZ -> ( abs ` [_ D / x ]_ A ) = ( abs ` E ) )
100	99	adantl 482	. 2 |- ( ( ph /\ D e. ZZ ) -> ( abs ` [_ D / x ]_ A ) = ( abs ` E ) )
101		fvex 6201	. . . 4 |- ( abs ` D ) e. _V
102		oddcomabszz.7	. . . 4 |- ( x = ( abs ` D ) -> A = F )
103	101, 102	csbie 3559	. . 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 2664	1 |- ( ( ph /\ D e. ZZ ) -> ( abs ` E ) = F )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   [_csb 3533   class class class wbr 4653   ` cfv 5888   RRcr 9935   0cc0 9936   <_ cle 10075   -ucneg 10267   ZZcz 11377   abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  rmyabs  37525