Theorem eulerpartlemgu 30439

Description: Lemma for eulerpart 30444: Rewriting the U set for an odd partition Note that interestingly, this proof reuses marypha2lem2 8342. (Contributed by Thierry Arnoux, 10-Aug-2018.)

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 ) ) ) ) ) )
eulerpartlemgh.1	|- U = U_ t e. ( ( `' A " NN ) i^i J ) ( { t } X. (bits ` ( A ` t ) ) )

Assertion

Ref	Expression
eulerpartlemgu	|- ( A e. ( T i^i R ) -> U = { <. t , n >. | ( t e. ( ( `' A " NN ) i^i J ) /\ n e. ( (bits o. A ) ` t ) ) } )

Distinct variable groups:   z, t   f, g, k, n, t, A   f, J, n, t   f, N, k, n, t   n, O, t   P, g, k   R, f, k, n, t   T, n, t

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

Proof of Theorem eulerpartlemgu

Step	Hyp	Ref	Expression
1		eulerpartlemgh.1	. 2 |- U = U_ t e. ( ( `' A " NN ) i^i J ) ( { t } X. (bits ` ( A ` t ) ) )
2		eqid 2622	. . . 4 |- U_ t e. ( ( `' A " NN ) i^i J ) ( { t } X. ( (bits o. A ) ` t ) ) = U_ t e. ( ( `' A " NN ) i^i J ) ( { t } X. ( (bits o. A ) ` t ) )
3	2	marypha2lem2 8342	. . 3 |- U_ t e. ( ( `' A " NN ) i^i J ) ( { t } X. ( (bits o. A ) ` t ) ) = { <. t , n >. | ( t e. ( ( `' A " NN ) i^i J ) /\ n e. ( (bits o. A ) ` t ) ) }
4		eulerpart.p	. . . . . . . . . . 11 |- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
5		eulerpart.o	. . . . . . . . . . 11 |- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
6		eulerpart.d	. . . . . . . . . . 11 |- D = { g e. P | A. n e. NN ( g ` n ) <_ 1 }
7		eulerpart.j	. . . . . . . . . . 11 |- J = { z e. NN | -. 2 || z }
8		eulerpart.f	. . . . . . . . . . 11 |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
9		eulerpart.h	. . . . . . . . . . 11 |- H = { r e. ( ( ~P NN0 i^i Fin ) ^m J ) | ( r supp (/) ) e. Fin }
10		eulerpart.m	. . . . . . . . . . 11 |- M = ( r e. H |-> { <. x , y >. | ( x e. J /\ y e. ( r ` x ) ) } )
11		eulerpart.r	. . . . . . . . . . 11 |- R = { f | ( `' f " NN ) e. Fin }
12		eulerpart.t	. . . . . . . . . . 11 |- T = { f e. ( NN0 ^m NN ) | ( `' f " NN ) C_ J }
13	4, 5, 6, 7, 8, 9, 10, 11, 12	eulerpartlemt0 30431	. . . . . . . . . 10 |- ( A e. ( T i^i R ) <-> ( A e. ( NN0 ^m NN ) /\ ( `' A " NN ) e. Fin /\ ( `' A " NN ) C_ J ) )
14	13	simp1bi 1076	. . . . . . . . 9 |- ( A e. ( T i^i R ) -> A e. ( NN0 ^m NN ) )
15		elmapi 7879	. . . . . . . . 9 |- ( A e. ( NN0 ^m NN ) -> A : NN --> NN0 )
16	14, 15	syl 17	. . . . . . . 8 |- ( A e. ( T i^i R ) -> A : NN --> NN0 )
17	16	adantr 481	. . . . . . 7 |- ( ( A e. ( T i^i R ) /\ t e. ( ( `' A " NN ) i^i J ) ) -> A : NN --> NN0 )
18		ffun 6048	. . . . . . 7 |- ( A : NN --> NN0 -> Fun A )
19	17, 18	syl 17	. . . . . 6 |- ( ( A e. ( T i^i R ) /\ t e. ( ( `' A " NN ) i^i J ) ) -> Fun A )
20		inss1 3833	. . . . . . . . 9 |- ( ( `' A " NN ) i^i J ) C_ ( `' A " NN )
21		cnvimass 5485	. . . . . . . . . 10 |- ( `' A " NN ) C_ dom A
22		fdm 6051	. . . . . . . . . . 11 |- ( A : NN --> NN0 -> dom A = NN )
23	16, 22	syl 17	. . . . . . . . . 10 |- ( A e. ( T i^i R ) -> dom A = NN )
24	21, 23	syl5sseq 3653	. . . . . . . . 9 |- ( A e. ( T i^i R ) -> ( `' A " NN ) C_ NN )
25	20, 24	syl5ss 3614	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( ( `' A " NN ) i^i J ) C_ NN )
26	25	sselda 3603	. . . . . . 7 |- ( ( A e. ( T i^i R ) /\ t e. ( ( `' A " NN ) i^i J ) ) -> t e. NN )
27	23	eleq2d 2687	. . . . . . . 8 |- ( A e. ( T i^i R ) -> ( t e. dom A <-> t e. NN ) )
28	27	adantr 481	. . . . . . 7 |- ( ( A e. ( T i^i R ) /\ t e. ( ( `' A " NN ) i^i J ) ) -> ( t e. dom A <-> t e. NN ) )
29	26, 28	mpbird 247	. . . . . 6 |- ( ( A e. ( T i^i R ) /\ t e. ( ( `' A " NN ) i^i J ) ) -> t e. dom A )
30		fvco 6274	. . . . . 6 |- ( ( Fun A /\ t e. dom A ) -> ( (bits o. A ) ` t ) = (bits ` ( A ` t ) ) )
31	19, 29, 30	syl2anc 693	. . . . 5 |- ( ( A e. ( T i^i R ) /\ t e. ( ( `' A " NN ) i^i J ) ) -> ( (bits o. A ) ` t ) = (bits ` ( A ` t ) ) )
32	31	xpeq2d 5139	. . . 4 |- ( ( A e. ( T i^i R ) /\ t e. ( ( `' A " NN ) i^i J ) ) -> ( { t } X. ( (bits o. A ) ` t ) ) = ( { t } X. (bits ` ( A ` t ) ) ) )
33	32	iuneq2dv 4542	. . 3 |- ( A e. ( T i^i R ) -> U_ t e. ( ( `' A " NN ) i^i J ) ( { t } X. ( (bits o. A ) ` t ) ) = U_ t e. ( ( `' A " NN ) i^i J ) ( { t } X. (bits ` ( A ` t ) ) ) )
34	3, 33	syl5reqr 2671	. 2 |- ( A e. ( T i^i R ) -> U_ t e. ( ( `' A " NN ) i^i J ) ( { t } X. (bits ` ( A ` t ) ) ) = { <. t , n >. | ( t e. ( ( `' A " NN ) i^i J ) /\ n e. ( (bits o. A ) ` t ) ) } )
35	1, 34	syl5eq 2668	1 |- ( A e. ( T i^i R ) -> U = { <. t , n >. | ( t e. ( ( `' A " NN ) i^i J ) /\ n e. ( (bits o. A ) ` t ) ) } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U_ciun 4520   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   supp csupp 7295   ^m cmap 7857   Fincfn 7955   1c1 9937   x. cmul 9941   <_ cle 10075   NNcn 11020   2c2 11070   NN0cn0 11292   ^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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by: (None)