Theorem eulerpartlemmf 28511

Description: Lemma for eulerpart 28518 (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 14107	. . . . 5 |- (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin )
2		f1of 5822	. . . . 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 28505	. . . . . . . 8 |- ( A e. ( T i^i R ) <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
14	13	biimpi 194	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
15	14	simp1d 1008	. . . . . 6 |- ( A e. ( T i^i R ) -> A e. ( NN0 ^m NN ) )
16		nn0ex 10822	. . . . . . 7 |- NN0 e. _V
17		nnex 10562	. . . . . . 7 |- NN e. _V
18	16, 17	elmap 7466	. . . . . 6 |- ( A e. ( NN0 ^m NN ) <-> A : NN --> NN0 )
19	15, 18	sylib 196	. . . . 5 |- ( A e. ( T i^i R ) -> A : NN --> NN0 )
20		ssrab2 3581	. . . . . 6 |- { z e. NN | -. 2 || z } C_ NN
21	7, 20	eqsstri 3529	. . . . 5 |- J C_ NN
22		fssres 5757	. . . . 5 |- ( ( A : NN --> NN0 /\ J C_ NN ) -> ( A |` J ) : J --> NN0 )
23	19, 21, 22	sylancl 662	. . . 4 |- ( A e. ( T i^i R ) -> ( A |` J ) : J --> NN0 )
24		fco2 5748	. . . 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 663	. . 3 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) : J --> ( ~P NN0 i^i Fin ) )
26	16	pwex 4639	. . . . 5 |- ~P NN0 e. _V
27	26	inex1 4597	. . . 4 |- ( ~P NN0 i^i Fin ) e. _V
28	17, 21	ssexi 4601	. . . 4 |- J e. _V
29	27, 28	elmap 7466	. . 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 212	. 2 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) )
31	14	simp2d 1009	. . . . 5 |- ( A e. ( T i^i R ) -> ( `' A " NN ) e. Fin )
32		0nn0 10831	. . . . . . . . 9 |- 0 e. NN0
33		suppimacnv 6928	. . . . . . . . 9 |- ( ( A e. ( T i^i R ) /\ 0 e. NN0 ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
34	32, 33	mpan2 671	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
35		frnsuppeq 6929	. . . . . . . . . 10 |- ( ( NN e. _V /\ 0 e. NN0 ) -> ( A : NN --> NN0 -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) ) )
36	17, 32, 35	mp2an 672	. . . . . . . . 9 |- ( A : NN --> NN0 -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) )
37	19, 36	syl 16	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) )
38	34, 37	eqtr3d 2500	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) = ( `' A " ( NN0 \ { 0 } ) ) )
39	38	eleq1d 2526	. . . . . 6 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin ) )
40		dfn2 10829	. . . . . . . 8 |- NN = ( NN0 \ { 0 } )
41	40	imaeq2i 5345	. . . . . . 7 |- ( `' A " NN ) = ( `' A " ( NN0 \ { 0 } ) )
42	41	eleq1i 2534	. . . . . 6 |- ( ( `' A " NN ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin )
43	39, 42	syl6bbr 263	. . . . 5 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " NN ) e. Fin ) )
44	31, 43	mpbird 232	. . . 4 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) e. Fin )
45		resss 5307	. . . . 5 |- ( A |` J ) C_ A
46		cnvss 5185	. . . . 5 |- ( ( A |` J ) C_ A -> `' ( A |` J ) C_ `' A )
47		imass1 5381	. . . . 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 7759	. . . 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 662	. . 3 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin )
51		cnvco 5198	. . . . . 6 |- `' (bits o. ( A |` J ) ) = ( `' ( A |` J ) o. `'bits )
52	51	imaeq1i 5344	. . . . 5 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) )
53		imaco 5518	. . . . 5 |- ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
54	52, 53	eqtri 2486	. . . 4 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
55		ffun 5739	. . . . . 6 |- ( A : NN --> NN0 -> Fun A )
56		funres 5633	. . . . . 6 |- ( Fun A -> Fun ( A |` J ) )
57	19, 55, 56	3syl 20	. . . . 5 |- ( A e. ( T i^i R ) -> Fun ( A |` J ) )
58		ssv 3519	. . . . . . 7 |- ( `'bits " _V ) C_ _V
59		ssdif 3635	. . . . . . 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 14089	. . . . . . 7 |- bits : ZZ --> ~P NN0
62		ffun 5739	. . . . . . 7 |- (bits : ZZ --> ~P NN0 -> Fun bits )
63		difpreima 6016	. . . . . . 7 |- ( Fun bits -> ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) )
64	61, 62, 63	mp2b 10	. . . . . 6 |- ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) )
65		bitsf1 14108	. . . . . . . . 9 |- bits : ZZ -1-1-> ~P NN0
66		0z 10896	. . . . . . . . . 10 |- 0 e. ZZ
67		snssi 4176	. . . . . . . . . 10 |- ( 0 e. ZZ -> { 0 } C_ ZZ )
68	66, 67	ax-mp 5	. . . . . . . . 9 |- { 0 } C_ ZZ
69		f1imacnv 5838	. . . . . . . . 9 |- ( (bits : ZZ -1-1-> ~P NN0 /\ { 0 } C_ ZZ ) -> ( `'bits " (bits " { 0 } ) ) = { 0 } )
70	65, 68, 69	mp2an 672	. . . . . . . 8 |- ( `'bits " (bits " { 0 } ) ) = { 0 }
71		ffn 5737	. . . . . . . . . . . 12 |- (bits : ZZ --> ~P NN0 -> bits Fn ZZ )
72	61, 71	ax-mp 5	. . . . . . . . . . 11 |- bits Fn ZZ
73		fnsnfv 5933	. . . . . . . . . . 11 |- ( (bits Fn ZZ /\ 0 e. ZZ ) -> { (bits ` 0 ) } = (bits " { 0 } ) )
74	72, 66, 73	mp2an 672	. . . . . . . . . 10 |- { (bits ` 0 ) } = (bits " { 0 } )
75		0bits 14101	. . . . . . . . . . 11 |- (bits ` 0 ) = (/)
76	75	sneqi 4043	. . . . . . . . . 10 |- { (bits ` 0 ) } = { (/) }
77	74, 76	eqtr3i 2488	. . . . . . . . 9 |- (bits " { 0 } ) = { (/) }
78	77	imaeq2i 5345	. . . . . . . 8 |- ( `'bits " (bits " { 0 } ) ) = ( `'bits " { (/) } )
79	70, 78	eqtr3i 2488	. . . . . . 7 |- { 0 } = ( `'bits " { (/) } )
80	79	difeq2i 3615	. . . . . 6 |- ( _V \ { 0 } ) = ( _V \ ( `'bits " { (/) } ) )
81	60, 64, 80	3sstr4i 3538	. . . . 5 |- ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } )
82		sspreima 27633	. . . . 5 |- ( ( Fun ( A |` J ) /\ ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } ) ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
83	57, 81, 82	sylancl 662	. . . 4 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
84	54, 83	syl5eqss 3543	. . 3 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
85		ssfi 7759	. . 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 661	. 2 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin )
87		oveq1 6303	. . . . 5 |- ( r = (bits o. ( A |` J ) ) -> ( r supp (/) ) = ( (bits o. ( A |` J ) ) supp (/) ) )
88	87	eleq1d 2526	. . . 4 |- ( r = (bits o. ( A |` J ) ) -> ( ( r supp (/) ) e. Fin <-> ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) )
89	88, 9	elrab2 3259	. . 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 10894	. . . . . 6 |- ZZ e. _V
91		fex 6146	. . . . . 6 |- ( (bits : ZZ --> ~P NN0 /\ ZZ e. _V ) -> bits e. _V )
92	61, 90, 91	mp2an 672	. . . . 5 |- bits e. _V
93		resexg 5326	. . . . 5 |- ( A e. ( T i^i R ) -> ( A |` J ) e. _V )
94		coexg 6750	. . . . 5 |- ( (bits e. _V /\ ( A |` J ) e. _V ) -> (bits o. ( A |` J ) ) e. _V )
95	92, 93, 94	sylancr 663	. . . 4 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. _V )
96		0ex 4587	. . . . . . 7 |- (/) e. _V
97		suppimacnv 6928	. . . . . . 7 |- ( ( (bits o. ( A |` J ) ) e. _V /\ (/) e. _V ) -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
98	96, 97	mpan2 671	. . . . . 6 |- ( (bits o. ( A |` J ) ) e. _V -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
99	98	eleq1d 2526	. . . . 5 |- ( (bits o. ( A |` J ) ) e. _V -> ( ( (bits o. ( A |` J ) ) supp (/) ) e. Fin <-> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) )
100	99	anbi2d 703	. . . 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 16	. . 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 257	. 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 922	1 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. H )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   {cab 2442   A.wral 2807   {crab 2811   _Vcvv 3109   \ cdif 3468   i^i cin 3470   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   class class class wbr 4456   {copab 4514   |-> cmpt 4515   `'ccnv 5007   |` cres 5010   "cima 5011   o. ccom 5012   Fun wfun 5588   Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   supp csupp 6917   ^m cmap 7438   Fincfn 7535   0cc0 9509   1c1 9510   x. cmul 9514   <_ cle 9646   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ^cexp 12169   sum_csu 13520   || cdvds 13998  bitscbits 14081  𝟭cind 28185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-dvds 13999  df-bits 14084

This theorem is referenced by:  eulerpartlemgvv  28512  eulerpartlemgf  28515