Theorem eulerpartlemsv2 29138

Description: Lemma for eulerpart 29162. 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 29136	. 2 |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. NN ( ( A ` k ) x. k ) )
4		cnvimass 5145	. . . 4 |- ( `' A " NN ) C_ dom A
5	1, 2	eulerpartlemelr 29137	. . . . . 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 5688	. . . . 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 3450	. . 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 3402	. . . . . 6 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN )
12	10, 11	ffvelrnd 5977	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( A ` k ) e. NN0 )
13	11	nnnn0d 10871	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> k e. NN0 )
14	12, 13	nn0mulcld 10876	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( `' A " NN ) ) -> ( ( A ` k ) x. k ) e. NN0 )
15	14	nn0cnd 10873	. . 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 3386	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. NN )
18	16	eldifbd 3387	. . . . . . . . 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 5684	. . . . . . . . . 10 |- ( A : NN --> NN0 -> A Fn NN )
21		elpreima 5956	. . . . . . . . . 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 5977	. . . . . . 7 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( A ` k ) e. NN0 )
28		elnn0 10817	. . . . . . 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 6259	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = ( 0 x. k ) )
33	17	nncnd 10571	. . . . 5 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> k e. CC )
34	33	mul02d 9777	. . . 4 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( 0 x. k ) = 0 )
35	32, 34	eqtrd 2457	. . 3 |- ( ( A e. ( ( NN0 ^m NN ) i^i R ) /\ k e. ( NN \ ( `' A " NN ) ) ) -> ( ( A ` k ) x. k ) = 0 )
36		nnuz 11140	. . . . 5 |- NN = ( ZZ>= ` 1 )
37	36	eqimssi 3456	. . . 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 13728	. 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 2460	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 1872   {cab 2409   \ cdif 3371   i^i cin 3373   C_ wss 3374   |-> cmpt 4420   `'ccnv 4790   dom cdm 4791   "cima 4794   Fn wfn 5534   -->wf 5535   ` cfv 5539  (class class class)co 6244   ^m cmap 7422   Fincfn 7519   0cc0 9485   1c1 9486   x. cmul 9490   NNcn 10555   NN0cn0 10815   ZZ>=cuz 11105   sum_csu 13690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562

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 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-se 4751  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-isom 5548  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-oi 7973  df-card 8320  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-n0 10816  df-z 10884  df-uz 11106  df-rp 11249  df-fz 11731  df-fzo 11862  df-seq 12159  df-exp 12218  df-hash 12461  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-clim 13490  df-sum 13691

This theorem is referenced by:  eulerpartlemsf  29139  eulerpartlemgs2  29160