Theorem eulerpartlemmf 30437

Description: Lemma for eulerpart 30444. (Contributed by Thierry Arnoux, 30-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)

Hypotheses

Ref	Expression
eulerpart.p	|- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
eulerpart.o	|- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
eulerpart.d	|- D = { g e. P | A. n e. NN ( g ` n ) <_ 1 }
eulerpart.j	|- J = { z e. NN | -. 2 || z }
eulerpart.f	|- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
eulerpart.h	|- H = { r e. ( ( ~P NN0 i^i Fin ) ^m J ) | ( r supp (/) ) e. Fin }
eulerpart.m	|- M = ( r e. H |-> { <. x , y >. | ( x e. J /\ y e. ( r ` x ) ) } )
eulerpart.r	|- R = { f | ( `' f " NN ) e. Fin }
eulerpart.t	|- T = { f e. ( NN0 ^m NN ) | ( `' f " NN ) C_ J }
eulerpart.g	|- G = ( o e. ( T i^i R ) |-> ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( o |` J ) ) ) ) ) )

Assertion

Ref	Expression
eulerpartlemmf	|- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. H )

Distinct variable groups:   f, k, n, x, y, z   f, o, r, A   o, F   H, r   f, J   n, o, r, J, x, y   o, M   f, N   g, n, P   R, o   T, o

Allowed substitution hints:   A( x, y, z, g, k, n)   D( x, y, z, f, g, k, n, o, r)   P( x, y, z, f, k, o, r)   R( x, y, z, f, g, k, n, r)   T( x, y, z, f, g, k, n, r)   F( x, y, z, f, g, k, n, r)   G( x, y, z, f, g, k, n, o, r)   H( x, y, z, f, g, k, n, o)   J( z, g, k)   M( x, y, z, f, g, k, n, r)   N( x, y, z, g, k, n, o, r)   O( x, y, z, f, g, k, n, o, r)

Proof of Theorem eulerpartlemmf

Step	Hyp	Ref	Expression
1		bitsf1o 15167	. . . . 5 |- (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin )
2		f1of 6137	. . . . 5 |- ( (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) -> (bits |` NN0 ) : NN0 --> ( ~P NN0 i^i Fin ) )
3	1, 2	ax-mp 5	. . . 4 |- (bits |` NN0 ) : NN0 --> ( ~P NN0 i^i Fin )
4		eulerpart.p	. . . . . . . . 9 |- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
5		eulerpart.o	. . . . . . . . 9 |- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
6		eulerpart.d	. . . . . . . . 9 |- D = { g e. P | A. n e. NN ( g ` n ) <_ 1 }
7		eulerpart.j	. . . . . . . . 9 |- J = { z e. NN | -. 2 || z }
8		eulerpart.f	. . . . . . . . 9 |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
9		eulerpart.h	. . . . . . . . 9 |- H = { r e. ( ( ~P NN0 i^i Fin ) ^m J ) | ( r supp (/) ) e. Fin }
10		eulerpart.m	. . . . . . . . 9 |- M = ( r e. H |-> { <. x , y >. | ( x e. J /\ y e. ( r ` x ) ) } )
11		eulerpart.r	. . . . . . . . 9 |- R = { f | ( `' f " NN ) e. Fin }
12		eulerpart.t	. . . . . . . . 9 |- T = { f e. ( NN0 ^m NN ) | ( `' f " NN ) C_ J }
13	4, 5, 6, 7, 8, 9, 10, 11, 12	eulerpartlemt0 30431	. . . . . . . 8 |- ( A e. ( T i^i R ) <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
14	13	biimpi 206	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
15	14	simp1d 1073	. . . . . 6 |- ( A e. ( T i^i R ) -> A e. ( NN0 ^m NN ) )
16		nn0ex 11298	. . . . . . 7 |- NN0 e. _V
17		nnex 11026	. . . . . . 7 |- NN e. _V
18	16, 17	elmap 7886	. . . . . 6 |- ( A e. ( NN0 ^m NN ) <-> A : NN --> NN0 )
19	15, 18	sylib 208	. . . . 5 |- ( A e. ( T i^i R ) -> A : NN --> NN0 )
20		ssrab2 3687	. . . . . 6 |- { z e. NN | -. 2 || z } C_ NN
21	7, 20	eqsstri 3635	. . . . 5 |- J C_ NN
22		fssres 6070	. . . . 5 |- ( ( A : NN --> NN0 /\ J C_ NN ) -> ( A |` J ) : J --> NN0 )
23	19, 21, 22	sylancl 694	. . . 4 |- ( A e. ( T i^i R ) -> ( A |` J ) : J --> NN0 )
24		fco2 6059	. . . 4 |- ( ( (bits |` NN0 ) : NN0 --> ( ~P NN0 i^i Fin ) /\ ( A |` J ) : J --> NN0 ) -> (bits o. ( A |` J ) ) : J --> ( ~P NN0 i^i Fin ) )
25	3, 23, 24	sylancr 695	. . 3 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) : J --> ( ~P NN0 i^i Fin ) )
26	16	pwex 4848	. . . . 5 |- ~P NN0 e. _V
27	26	inex1 4799	. . . 4 |- ( ~P NN0 i^i Fin ) e. _V
28	17, 21	ssexi 4803	. . . 4 |- J e. _V
29	27, 28	elmap 7886	. . 3 |- ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) <-> (bits o. ( A |` J ) ) : J --> ( ~P NN0 i^i Fin ) )
30	25, 29	sylibr 224	. 2 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) )
31	14	simp2d 1074	. . . . 5 |- ( A e. ( T i^i R ) -> ( `' A " NN ) e. Fin )
32		0nn0 11307	. . . . . . . . 9 |- 0 e. NN0
33		suppimacnv 7306	. . . . . . . . 9 |- ( ( A e. ( T i^i R ) /\ 0 e. NN0 ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
34	32, 33	mpan2 707	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
35		frnsuppeq 7307	. . . . . . . . . 10 |- ( ( NN e. _V /\ 0 e. NN0 ) -> ( A : NN --> NN0 -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) ) )
36	17, 32, 35	mp2an 708	. . . . . . . . 9 |- ( A : NN --> NN0 -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) )
37	19, 36	syl 17	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) )
38	34, 37	eqtr3d 2658	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) = ( `' A " ( NN0 \ { 0 } ) ) )
39	38	eleq1d 2686	. . . . . 6 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin ) )
40		dfn2 11305	. . . . . . . 8 |- NN = ( NN0 \ { 0 } )
41	40	imaeq2i 5464	. . . . . . 7 |- ( `' A " NN ) = ( `' A " ( NN0 \ { 0 } ) )
42	41	eleq1i 2692	. . . . . 6 |- ( ( `' A " NN ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin )
43	39, 42	syl6bbr 278	. . . . 5 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " NN ) e. Fin ) )
44	31, 43	mpbird 247	. . . 4 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) e. Fin )
45		resss 5422	. . . . 5 |- ( A |` J ) C_ A
46		cnvss 5294	. . . . 5 |- ( ( A |` J ) C_ A -> `' ( A |` J ) C_ `' A )
47		imass1 5500	. . . . 5 |- ( `' ( A |` J ) C_ `' A -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) C_ ( `' A " ( _V \ { 0 } ) ) )
48	45, 46, 47	mp2b 10	. . . 4 |- ( `' ( A |` J ) " ( _V \ { 0 } ) ) C_ ( `' A " ( _V \ { 0 } ) )
49		ssfi 8180	. . . 4 |- ( ( ( `' A " ( _V \ { 0 } ) ) e. Fin /\ ( `' ( A |` J ) " ( _V \ { 0 } ) ) C_ ( `' A " ( _V \ { 0 } ) ) ) -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin )
50	44, 48, 49	sylancl 694	. . 3 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin )
51		cnvco 5308	. . . . . 6 |- `' (bits o. ( A |` J ) ) = ( `' ( A |` J ) o. `'bits )
52	51	imaeq1i 5463	. . . . 5 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) )
53		imaco 5640	. . . . 5 |- ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
54	52, 53	eqtri 2644	. . . 4 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
55		ffun 6048	. . . . . 6 |- ( A : NN --> NN0 -> Fun A )
56		funres 5929	. . . . . 6 |- ( Fun A -> Fun ( A |` J ) )
57	19, 55, 56	3syl 18	. . . . 5 |- ( A e. ( T i^i R ) -> Fun ( A |` J ) )
58		ssv 3625	. . . . . . 7 |- ( `'bits " _V ) C_ _V
59		ssdif 3745	. . . . . . 7 |- ( ( `'bits " _V ) C_ _V -> ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) C_ ( _V \ ( `'bits " { (/) } ) ) )
60	58, 59	ax-mp 5	. . . . . 6 |- ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) C_ ( _V \ ( `'bits " { (/) } ) )
61		bitsf 15149	. . . . . . 7 |- bits : ZZ --> ~P NN0
62		ffun 6048	. . . . . . 7 |- (bits : ZZ --> ~P NN0 -> Fun bits )
63		difpreima 6343	. . . . . . 7 |- ( Fun bits -> ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) )
64	61, 62, 63	mp2b 10	. . . . . 6 |- ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) )
65		bitsf1 15168	. . . . . . . . 9 |- bits : ZZ -1-1-> ~P NN0
66		0z 11388	. . . . . . . . . 10 |- 0 e. ZZ
67		snssi 4339	. . . . . . . . . 10 |- ( 0 e. ZZ -> { 0 } C_ ZZ )
68	66, 67	ax-mp 5	. . . . . . . . 9 |- { 0 } C_ ZZ
69		f1imacnv 6153	. . . . . . . . 9 |- ( (bits : ZZ -1-1-> ~P NN0 /\ { 0 } C_ ZZ ) -> ( `'bits " (bits " { 0 } ) ) = { 0 } )
70	65, 68, 69	mp2an 708	. . . . . . . 8 |- ( `'bits " (bits " { 0 } ) ) = { 0 }
71		ffn 6045	. . . . . . . . . . . 12 |- (bits : ZZ --> ~P NN0 -> bits Fn ZZ )
72	61, 71	ax-mp 5	. . . . . . . . . . 11 |- bits Fn ZZ
73		fnsnfv 6258	. . . . . . . . . . 11 |- ( (bits Fn ZZ /\ 0 e. ZZ ) -> { (bits ` 0 ) } = (bits " { 0 } ) )
74	72, 66, 73	mp2an 708	. . . . . . . . . 10 |- { (bits ` 0 ) } = (bits " { 0 } )
75		0bits 15161	. . . . . . . . . . 11 |- (bits ` 0 ) = (/)
76	75	sneqi 4188	. . . . . . . . . 10 |- { (bits ` 0 ) } = { (/) }
77	74, 76	eqtr3i 2646	. . . . . . . . 9 |- (bits " { 0 } ) = { (/) }
78	77	imaeq2i 5464	. . . . . . . 8 |- ( `'bits " (bits " { 0 } ) ) = ( `'bits " { (/) } )
79	70, 78	eqtr3i 2646	. . . . . . 7 |- { 0 } = ( `'bits " { (/) } )
80	79	difeq2i 3725	. . . . . 6 |- ( _V \ { 0 } ) = ( _V \ ( `'bits " { (/) } ) )
81	60, 64, 80	3sstr4i 3644	. . . . 5 |- ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } )
82		sspreima 29447	. . . . 5 |- ( ( Fun ( A |` J ) /\ ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } ) ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
83	57, 81, 82	sylancl 694	. . . 4 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
84	54, 83	syl5eqss 3649	. . 3 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
85		ssfi 8180	. . 3 |- ( ( ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin )
86	50, 84, 85	syl2anc 693	. 2 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin )
87		oveq1 6657	. . . . 5 |- ( r = (bits o. ( A |` J ) ) -> ( r supp (/) ) = ( (bits o. ( A |` J ) ) supp (/) ) )
88	87	eleq1d 2686	. . . 4 |- ( r = (bits o. ( A |` J ) ) -> ( ( r supp (/) ) e. Fin <-> ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) )
89	88, 9	elrab2 3366	. . 3 |- ( (bits o. ( A |` J ) ) e. H <-> ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) )
90		zex 11386	. . . . . 6 |- ZZ e. _V
91		fex 6490	. . . . . 6 |- ( (bits : ZZ --> ~P NN0 /\ ZZ e. _V ) -> bits e. _V )
92	61, 90, 91	mp2an 708	. . . . 5 |- bits e. _V
93		resexg 5442	. . . . 5 |- ( A e. ( T i^i R ) -> ( A |` J ) e. _V )
94		coexg 7117	. . . . 5 |- ( (bits e. _V /\ ( A |` J ) e. _V ) -> (bits o. ( A |` J ) ) e. _V )
95	92, 93, 94	sylancr 695	. . . 4 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. _V )
96		0ex 4790	. . . . . . 7 |- (/) e. _V
97		suppimacnv 7306	. . . . . . 7 |- ( ( (bits o. ( A |` J ) ) e. _V /\ (/) e. _V ) -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
98	96, 97	mpan2 707	. . . . . 6 |- ( (bits o. ( A |` J ) ) e. _V -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
99	98	eleq1d 2686	. . . . 5 |- ( (bits o. ( A |` J ) ) e. _V -> ( ( (bits o. ( A |` J ) ) supp (/) ) e. Fin <-> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) )
100	99	anbi2d 740	. . . 4 |- ( (bits o. ( A |` J ) ) e. _V -> ( ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) <-> ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) ) )
101	95, 100	syl 17	. . 3 |- ( A e. ( T i^i R ) -> ( ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) <-> ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) ) )
102	89, 101	syl5bb 272	. 2 |- ( A e. ( T i^i R ) -> ( (bits o. ( A |` J ) ) e. H <-> ( (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) /\ ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) ) )
103	30, 86, 102	mpbir2and 957	1 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. H )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   {copab 4712   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   supp csupp 7295   ^m cmap 7857   Fincfn 7955   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ^cexp 12860   sum_csu 14416   || cdvds 14983  bitscbits 15141  𝟭cind 30072

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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-fal 1489  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-int 4476  df-iun 4522  df-disj 4621  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-bits 15144

This theorem is referenced by:  eulerpartlemgvv  30438  eulerpartlemgf  30441