Theorem eulerpartlemsv2 26693

Description: Lemma for eulerpart 26717. Value of the sum of a finite partition A (Contributed by Thierry Arnoux, 19-Aug-2018.)

Hypotheses

Ref	Expression
eulerpartlems.r	|- R = { f | ( `' f " NN ) e. Fin }
eulerpartlems.s	|- S = ( f e. ( ( NN0 ^m NN ) i^i R ) |-> sum_ k e. NN ( ( f ` k ) x. k ) )

Assertion

Ref	Expression
eulerpartlemsv2	|- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) )

Distinct variable groups:   f, k, A   R, f, k

Allowed substitution hints:   S( f, k)

Proof of Theorem eulerpartlemsv2

Step	Hyp	Ref	Expression
1		eulerpartlems.r	. . 3 |- R = { f | ( `' f " NN ) e. Fin }
2		eulerpartlems.s	. . 3 |- S = ( f e. ( ( NN0 ^m NN ) i^i R ) |-> sum_ k e. NN ( ( f ` k ) x. k ) )
3	1, 2	eulerpartlemsv1 26691	. 2 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. NN ( ( A ` k ) x. k ) )
4		cnvimass 5184	. . . 4 |- ( `' A " NN ) C_ dom A
5	1, 2	eulerpartlemelr 26692	. . . . . 6 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) )
6	5	simpld 459	. . . . 5 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> A : NN --> NN0 )
7		fdm 5558	. . . . 5 |- ( A : NN --> NN0 -> dom A = NN )
8	6, 7	syl 16	. . . 4 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> dom A = NN )
9	4, 8	syl5sseq 3399	. . 3 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( `' A " NN ) C_ NN )
10	6	adantr 465	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> A : NN --> NN0 )
11	9	sselda 3351	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN )
12	10, 11	ffvelrnd 5839	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( A ` k ) e. NN0 )
13	11	nnnn0d 10628	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN0 )
14	12, 13	nn0mulcld 10633	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. NN0 )
15	14	nn0cnd 10630	. . 3 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. CC )
16		simpr 461	. . . . . . . 8 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. ( NN \ ( `' A " NN ) ) )
17	16	eldifad 3335	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. NN )
18	16	eldifbd 3336	. . . . . . . . 9 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. k e. ( `' A " NN ) )
19	6	adantr 465	. . . . . . . . . 10 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> A : NN --> NN0 )
20		ffn 5554	. . . . . . . . . 10 |- ( A : NN --> NN0 -> A Fn NN )
21		elpreima 5818	. . . . . . . . . 10 |- ( A Fn NN -> ( k e. ( `' A " NN ) <-> ( k e. NN /\ ( A ` k ) e. NN ) ) )
22	19, 20, 21	3syl 20	. . . . . . . . 9 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( k e. ( `' A " NN ) <-> ( k e. NN /\ ( A ` k ) e. NN ) ) )
23	18, 22	mtbid 300	. . . . . . . 8 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. ( k e. NN /\ ( A ` k ) e. NN ) )
24		imnan 422	. . . . . . . 8 |- ( ( k e. NN -> -. ( A ` k ) e. NN ) <-> -. ( k e. NN /\ ( A ` k ) e. NN ) )
25	23, 24	sylibr 212	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( k e. NN -> -. ( A ` k ) e. NN ) )
26	17, 25	mpd 15	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. ( A ` k ) e. NN )
27	19, 17	ffvelrnd 5839	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) e. NN0 )
28		elnn0 10573	. . . . . . 7 |- ( ( A ` k ) e. NN0 <-> ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) )
29	27, 28	sylib 196	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) )
30		orel1 382	. . . . . 6 |- ( -. ( A ` k ) e. NN -> ( ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) -> ( A ` k ) = 0 ) )
31	26, 29, 30	sylc 60	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) = 0 )
32	31	oveq1d 6101	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = ( 0 x. k ) )
33	17	nncnd 10330	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. CC )
34	33	mul02d 9559	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( 0 x. k ) = 0 )
35	32, 34	eqtrd 2470	. . 3 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = 0 )
36		nnuz 10888	. . . . 5 |- NN = ( ZZ>= ` 1 )
37	36	eqimssi 3405	. . . 4 |- NN C_ ( ZZ>= ` 1 )
38	37	a1i 11	. . 3 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> NN C_ ( ZZ>= ` 1 ) )
39	9, 15, 35, 38	sumss 13193	. 2 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) = sum_ k e. NN ( ( A ` k ) x. k ) )
40	3, 39	eqtr4d 2473	1 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. ( `' A " NN ) ( ( A ` k ) x. k ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   {cab 2424   \ cdif 3320   i^i cin 3322   C_ wss 3323   e. cmpt 4345   `'ccnv 4834   dom cdm 4835   "cima 4838   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   ^m cmap 7206   Fincfn 7302   0cc0 9274   1c1 9275   x. cmul 9279   NNcn 10314   NN0cn0 10571   ZZ>=cuz 10853   sum_csu 13155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156

This theorem is referenced by:  eulerpartlemsf  26694  eulerpartlemgs2  26715