Theorem eulerpartlemgv 30435

Description: Lemma for eulerpart 30444: value of the function G. (Contributed by Thierry Arnoux, 13-Nov-2017.)

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
eulerpartlemgv	|- ( A e. ( T i^i R ) -> ( G ` A ) = ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( A |` J ) ) ) ) ) )

Distinct variable groups:   A, o   o, F   o, J   o, M   R, o   T, o

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

Proof of Theorem eulerpartlemgv

Step	Hyp	Ref	Expression
1		reseq1 5390	. . . . . 6 |- ( o = A -> ( o |` J ) = ( A |` J ) )
2	1	coeq2d 5284	. . . . 5 |- ( o = A -> (bits o. ( o |` J ) ) = (bits o. ( A |` J ) ) )
3	2	fveq2d 6195	. . . 4 |- ( o = A -> ( M ` (bits o. ( o |` J ) ) ) = ( M ` (bits o. ( A |` J ) ) ) )
4	3	imaeq2d 5466	. . 3 |- ( o = A -> ( F " ( M ` (bits o. ( o |` J ) ) ) ) = ( F " ( M ` (bits o. ( A |` J ) ) ) ) )
5	4	fveq2d 6195	. 2 |- ( o = A -> ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( o |` J ) ) ) ) ) = ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( A |` J ) ) ) ) ) )
6		eulerpart.g	. 2 |- G = ( o e. ( T i^i R ) |-> ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( o |` J ) ) ) ) ) )
7		fvex 6201	. 2 |- ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( A |` J ) ) ) ) ) e. _V
8	5, 6, 7	fvmpt 6282	1 |- ( A e. ( T i^i R ) -> ( G ` A ) = ( (𝟭 ` NN ) ` ( F " ( M ` (bits o. ( A |` J ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ 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   class class class wbr 4653   {copab 4712   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   "cima 5117   o. ccom 5118   ` 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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-fv 5896

This theorem is referenced by:  eulerpartlemgvv  30438  eulerpartlemgf  30441  eulerpartlemn  30443