Theorem eulerpartlemmf 29208

Description: Lemma for eulerpart 29215. (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 14419	. . . . 5 |- (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin )
2		f1of 5814	. . . . 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 29202	. . . . . . . 8 |- ( A e. ( T i^i R ) <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
14	13	biimpi 198	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
15	14	simp1d 1020	. . . . . 6 |- ( A e. ( T i^i R ) -> A e. ( NN0 ^m NN ) )
16		nn0ex 10875	. . . . . . 7 |- NN0 e. _V
17		nnex 10615	. . . . . . 7 |- NN e. _V
18	16, 17	elmap 7500	. . . . . 6 |- ( A e. ( NN0 ^m NN ) <-> A : NN --> NN0 )
19	15, 18	sylib 200	. . . . 5 |- ( A e. ( T i^i R ) -> A : NN --> NN0 )
20		ssrab2 3514	. . . . . 6 |- { z e. NN | -. 2 || z } C_ NN
21	7, 20	eqsstri 3462	. . . . 5 |- J C_ NN
22		fssres 5749	. . . . 5 |- ( ( A : NN --> NN0 /\ J C_ NN ) -> ( A |` J ) : J --> NN0 )
23	19, 21, 22	sylancl 668	. . . 4 |- ( A e. ( T i^i R ) -> ( A |` J ) : J --> NN0 )
24		fco2 5740	. . . 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 669	. . 3 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) : J --> ( ~P NN0 i^i Fin ) )
26	16	pwex 4586	. . . . 5 |- ~P NN0 e. _V
27	26	inex1 4544	. . . 4 |- ( ~P NN0 i^i Fin ) e. _V
28	17, 21	ssexi 4548	. . . 4 |- J e. _V
29	27, 28	elmap 7500	. . 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 216	. 2 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. ( ( ~P NN0 i^i Fin ) ^m J ) )
31	14	simp2d 1021	. . . . 5 |- ( A e. ( T i^i R ) -> ( `' A " NN ) e. Fin )
32		0nn0 10884	. . . . . . . . 9 |- 0 e. NN0
33		suppimacnv 6925	. . . . . . . . 9 |- ( ( A e. ( T i^i R ) /\ 0 e. NN0 ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
34	32, 33	mpan2 677	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( A supp 0 ) = ( `' A " ( _V \ { 0 } ) ) )
35		frnsuppeq 6926	. . . . . . . . . 10 |- ( ( NN e. _V /\ 0 e. NN0 ) -> ( A : NN --> NN0 -> ( A supp 0 ) = ( `' A " ( NN0 \ { 0 } ) ) ) )
36	17, 32, 35	mp2an 678	. . . . . . . . 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 2487	. . . . . . 7 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) = ( `' A " ( NN0 \ { 0 } ) ) )
39	38	eleq1d 2513	. . . . . 6 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin ) )
40		dfn2 10882	. . . . . . . 8 |- NN = ( NN0 \ { 0 } )
41	40	imaeq2i 5166	. . . . . . 7 |- ( `' A " NN ) = ( `' A " ( NN0 \ { 0 } ) )
42	41	eleq1i 2520	. . . . . 6 |- ( ( `' A " NN ) e. Fin <-> ( `' A " ( NN0 \ { 0 } ) ) e. Fin )
43	39, 42	syl6bbr 267	. . . . 5 |- ( A e. ( T i^i R ) -> ( ( `' A " ( _V \ { 0 } ) ) e. Fin <-> ( `' A " NN ) e. Fin ) )
44	31, 43	mpbird 236	. . . 4 |- ( A e. ( T i^i R ) -> ( `' A " ( _V \ { 0 } ) ) e. Fin )
45		resss 5128	. . . . 5 |- ( A |` J ) C_ A
46		cnvss 5007	. . . . 5 |- ( ( A |` J ) C_ A -> `' ( A |` J ) C_ `' A )
47		imass1 5203	. . . . 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 7792	. . . 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 668	. . 3 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( _V \ { 0 } ) ) e. Fin )
51		cnvco 5020	. . . . . 6 |- `' (bits o. ( A |` J ) ) = ( `' ( A |` J ) o. `'bits )
52	51	imaeq1i 5165	. . . . 5 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) )
53		imaco 5340	. . . . 5 |- ( ( `' ( A |` J ) o. `'bits ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
54	52, 53	eqtri 2473	. . . 4 |- ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) = ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) )
55		ffun 5731	. . . . . 6 |- ( A : NN --> NN0 -> Fun A )
56		funres 5621	. . . . . 6 |- ( Fun A -> Fun ( A |` J ) )
57	19, 55, 56	3syl 18	. . . . 5 |- ( A e. ( T i^i R ) -> Fun ( A |` J ) )
58		ssv 3452	. . . . . . 7 |- ( `'bits " _V ) C_ _V
59		ssdif 3568	. . . . . . 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 14400	. . . . . . 7 |- bits : ZZ --> ~P NN0
62		ffun 5731	. . . . . . 7 |- (bits : ZZ --> ~P NN0 -> Fun bits )
63		difpreima 6008	. . . . . . 7 |- ( Fun bits -> ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) ) )
64	61, 62, 63	mp2b 10	. . . . . 6 |- ( `'bits " ( _V \ { (/) } ) ) = ( ( `'bits " _V ) \ ( `'bits " { (/) } ) )
65		bitsf1 14420	. . . . . . . . 9 |- bits : ZZ -1-1-> ~P NN0
66		0z 10948	. . . . . . . . . 10 |- 0 e. ZZ
67		snssi 4116	. . . . . . . . . 10 |- ( 0 e. ZZ -> { 0 } C_ ZZ )
68	66, 67	ax-mp 5	. . . . . . . . 9 |- { 0 } C_ ZZ
69		f1imacnv 5830	. . . . . . . . 9 |- ( (bits : ZZ -1-1-> ~P NN0 /\ { 0 } C_ ZZ ) -> ( `'bits " (bits " { 0 } ) ) = { 0 } )
70	65, 68, 69	mp2an 678	. . . . . . . 8 |- ( `'bits " (bits " { 0 } ) ) = { 0 }
71		ffn 5728	. . . . . . . . . . . 12 |- (bits : ZZ --> ~P NN0 -> bits Fn ZZ )
72	61, 71	ax-mp 5	. . . . . . . . . . 11 |- bits Fn ZZ
73		fnsnfv 5925	. . . . . . . . . . 11 |- ( (bits Fn ZZ /\ 0 e. ZZ ) -> { (bits ` 0 ) } = (bits " { 0 } ) )
74	72, 66, 73	mp2an 678	. . . . . . . . . 10 |- { (bits ` 0 ) } = (bits " { 0 } )
75		0bits 14413	. . . . . . . . . . 11 |- (bits ` 0 ) = (/)
76	75	sneqi 3979	. . . . . . . . . 10 |- { (bits ` 0 ) } = { (/) }
77	74, 76	eqtr3i 2475	. . . . . . . . 9 |- (bits " { 0 } ) = { (/) }
78	77	imaeq2i 5166	. . . . . . . 8 |- ( `'bits " (bits " { 0 } ) ) = ( `'bits " { (/) } )
79	70, 78	eqtr3i 2475	. . . . . . 7 |- { 0 } = ( `'bits " { (/) } )
80	79	difeq2i 3548	. . . . . 6 |- ( _V \ { 0 } ) = ( _V \ ( `'bits " { (/) } ) )
81	60, 64, 80	3sstr4i 3471	. . . . 5 |- ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } )
82		sspreima 28246	. . . . 5 |- ( ( Fun ( A |` J ) /\ ( `'bits " ( _V \ { (/) } ) ) C_ ( _V \ { 0 } ) ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
83	57, 81, 82	sylancl 668	. . . 4 |- ( A e. ( T i^i R ) -> ( `' ( A |` J ) " ( `'bits " ( _V \ { (/) } ) ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
84	54, 83	syl5eqss 3476	. . 3 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) C_ ( `' ( A |` J ) " ( _V \ { 0 } ) ) )
85		ssfi 7792	. . 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 667	. 2 |- ( A e. ( T i^i R ) -> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin )
87		oveq1 6297	. . . . 5 |- ( r = (bits o. ( A |` J ) ) -> ( r supp (/) ) = ( (bits o. ( A |` J ) ) supp (/) ) )
88	87	eleq1d 2513	. . . 4 |- ( r = (bits o. ( A |` J ) ) -> ( ( r supp (/) ) e. Fin <-> ( (bits o. ( A |` J ) ) supp (/) ) e. Fin ) )
89	88, 9	elrab2 3198	. . 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 10946	. . . . . 6 |- ZZ e. _V
91		fex 6138	. . . . . 6 |- ( (bits : ZZ --> ~P NN0 /\ ZZ e. _V ) -> bits e. _V )
92	61, 90, 91	mp2an 678	. . . . 5 |- bits e. _V
93		resexg 5147	. . . . 5 |- ( A e. ( T i^i R ) -> ( A |` J ) e. _V )
94		coexg 6744	. . . . 5 |- ( (bits e. _V /\ ( A |` J ) e. _V ) -> (bits o. ( A |` J ) ) e. _V )
95	92, 93, 94	sylancr 669	. . . 4 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. _V )
96		0ex 4535	. . . . . . 7 |- (/) e. _V
97		suppimacnv 6925	. . . . . . 7 |- ( ( (bits o. ( A |` J ) ) e. _V /\ (/) e. _V ) -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
98	96, 97	mpan2 677	. . . . . 6 |- ( (bits o. ( A |` J ) ) e. _V -> ( (bits o. ( A |` J ) ) supp (/) ) = ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) )
99	98	eleq1d 2513	. . . . 5 |- ( (bits o. ( A |` J ) ) e. _V -> ( ( (bits o. ( A |` J ) ) supp (/) ) e. Fin <-> ( `' (bits o. ( A |` J ) ) " ( _V \ { (/) } ) ) e. Fin ) )
100	99	anbi2d 710	. . . 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 261	. 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 933	1 |- ( A e. ( T i^i R ) -> (bits o. ( A |` J ) ) e. H )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   {cab 2437   A.wral 2737   {crab 2741   _Vcvv 3045   \ cdif 3401   i^i cin 3403   C_ wss 3404   (/)c0 3731   ~Pcpw 3951   {csn 3968   class class class wbr 4402   {copab 4460   |-> cmpt 4461   `'ccnv 4833   |` cres 4836   "cima 4837   o. ccom 4838   Fun wfun 5576   Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   |-> cmpt2 6292   supp csupp 6914   ^m cmap 7472   Fincfn 7569   0cc0 9539   1c1 9540   x. cmul 9544   <_ cle 9676   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ^cexp 12272   sum_csu 13752   || cdvds 14305  bitscbits 14392  𝟭cind 28832

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-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  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-fal 1450  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-int 4235  df-iun 4280  df-disj 4374  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-se 4794  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-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  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-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-dvds 14306  df-bits 14395

This theorem is referenced by:  eulerpartlemgvv  29209  eulerpartlemgf  29212