Theorem eulerpartlemsv2 27937

Description: Lemma for eulerpart 27961. 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 27935	. 2 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. NN ( ( A ` k ) x. k ) )
4		cnvimass 5355	. . . 4 |- ( `' A " NN ) C_ dom A
5	1, 2	eulerpartlemelr 27936	. . . . . 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 5733	. . . . 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 3552	. . 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 3504	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN )
12	10, 11	ffvelrnd 6020	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( A ` k ) e. NN0 )
13	11	nnnn0d 10848	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN0 )
14	12, 13	nn0mulcld 10853	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. NN0 )
15	14	nn0cnd 10850	. . 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 3488	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. NN )
18	16	eldifbd 3489	. . . . . . . . 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 5729	. . . . . . . . . 10 |- ( A : NN --> NN0 -> A Fn NN )
21		elpreima 5999	. . . . . . . . . 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 6020	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) e. NN0 )
28		elnn0 10793	. . . . . . 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 6297	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = ( 0 x. k ) )
33	17	nncnd 10548	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. CC )
34	33	mul02d 9773	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( 0 x. k ) = 0 )
35	32, 34	eqtrd 2508	. . 3 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = 0 )
36		nnuz 11113	. . . . 5 |- NN = ( ZZ>= ` 1 )
37	36	eqimssi 3558	. . . 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 13505	. 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 2511	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 1379   e. wcel 1767   {cab 2452   \ cdif 3473   i^i cin 3475   C_ wss 3476   |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   "cima 5002   Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   ^m cmap 7417   Fincfn 7513   0cc0 9488   1c1 9489   x. cmul 9493   NNcn 10532   NN0cn0 10791   ZZ>=cuz 11078   sum_csu 13467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468

This theorem is referenced by:  eulerpartlemsf  27938  eulerpartlemgs2  27959