Theorem eulerpartlemsv2 29008

Description: Lemma for eulerpart 29032. 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 29006	. 2 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. NN ( ( A ` k ) x. k ) )
4		cnvimass 5208	. . . 4 |- ( `' A " NN ) C_ dom A
5	1, 2	eulerpartlemelr 29007	. . . . . 6 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( A : NN --> NN0 /\ ( `' A " NN ) e. Fin ) )
6	5	simpld 460	. . . . 5 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> A : NN --> NN0 )
7		fdm 5750	. . . . 5 |- ( A : NN --> NN0 -> dom A = NN )
8	6, 7	syl 17	. . . 4 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> dom A = NN )
9	4, 8	syl5sseq 3518	. . 3 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( `' A " NN ) C_ NN )
10	6	adantr 466	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> A : NN --> NN0 )
11	9	sselda 3470	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN )
12	10, 11	ffvelrnd 6038	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( A ` k ) e. NN0 )
13	11	nnnn0d 10925	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN0 )
14	12, 13	nn0mulcld 10930	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. NN0 )
15	14	nn0cnd 10927	. . 3 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. CC )
16		simpr 462	. . . . . . . 8 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. ( NN \ ( `' A " NN ) ) )
17	16	eldifad 3454	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. NN )
18	16	eldifbd 3455	. . . . . . . . 9 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. k e. ( `' A " NN ) )
19	6	adantr 466	. . . . . . . . . 10 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> A : NN --> NN0 )
20		ffn 5746	. . . . . . . . . 10 |- ( A : NN --> NN0 -> A Fn NN )
21		elpreima 6017	. . . . . . . . . 10 |- ( A Fn NN -> ( k e. ( `' A " NN ) <-> ( k e. NN /\ ( A ` k ) e. NN ) ) )
22	19, 20, 21	3syl 18	. . . . . . . . 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 301	. . . . . . . 8 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> -. ( k e. NN /\ ( A ` k ) e. NN ) )
24		imnan 423	. . . . . . . 8 |- ( ( k e. NN -> -. ( A ` k ) e. NN ) <-> -. ( k e. NN /\ ( A ` k ) e. NN ) )
25	23, 24	sylibr 215	. . . . . . 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 6038	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) e. NN0 )
28		elnn0 10871	. . . . . . 7 |- ( ( A ` k ) e. NN0 <-> ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) )
29	27, 28	sylib 199	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) )
30		orel1 383	. . . . . 6 |- ( -. ( A ` k ) e. NN -> ( ( ( A ` k ) e. NN \/ ( A ` k ) = 0 ) -> ( A ` k ) = 0 ) )
31	26, 29, 30	sylc 62	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) = 0 )
32	31	oveq1d 6320	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = ( 0 x. k ) )
33	17	nncnd 10625	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. CC )
34	33	mul02d 9830	. . . 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 11194	. . . . 5 |- NN = ( ZZ>= ` 1 )
37	36	eqimssi 3524	. . . 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 13768	. 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 187   \/ wo 369   /\ wa 370   = wceq 1437   e. wcel 1870   {cab 2414   \ cdif 3439   i^i cin 3441   C_ wss 3442   |-> cmpt 4484   `'ccnv 4853   dom cdm 4854   "cima 4857   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   ^m cmap 7480   Fincfn 7577   0cc0 9538   1c1 9539   x. cmul 9543   NNcn 10609   NN0cn0 10869   ZZ>=cuz 11159   sum_csu 13730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731

This theorem is referenced by:  eulerpartlemsf  29009  eulerpartlemgs2  29030