Theorem mertenslemub 11555

Description: Lemma for mertensabs 11558. An upper bound for T. (Contributed by Jim Kingdon, 3-Dec-2022.)

Hypotheses

Ref	Expression
mertenslemub.gb	|- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )
mertenslemub.b	|- ( ( ph /\ k e. NN0 ) -> B e. CC )
mertenslemub.cvg	|- ( ph -> seq 0 ( + , G ) e. dom ~~> )
mertenslemub.t	|- T = { z | E. n e. ( 0 ... ( S - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) }
mertenslemub.elt	|- ( ph -> X e. T )
mertenslemub.s	|- ( ph -> S e. NN )

Assertion

Ref	Expression
mertenslemub	|- ( ph -> X <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )

Distinct variable groups:   k, G, n, z   S, k, n, z   n, X, z   ph, k, n

Allowed substitution hints:   ph( z)   B( z, k, n)   T( z, k, n)   X( k)

Proof of Theorem mertenslemub

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mertenslemub.elt	. . . 4 |- ( ph -> X e. T )
2		eqeq1 2194	. . . . . . 7 |- ( z = X -> ( z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) <-> X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) ) )
3	2	rexbidv 2488	. . . . . 6 |- ( z = X -> ( E. n e. ( 0 ... ( S - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) <-> E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) ) )
4		mertenslemub.t	. . . . . 6 |- T = { z | E. n e. ( 0 ... ( S - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) }
5	3, 4	elab2g 2896	. . . . 5 |- ( X e. T -> ( X e. T <-> E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) ) )
6	1, 5	syl 14	. . . 4 |- ( ph -> ( X e. T <-> E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) ) )
7	1, 6	mpbid 147	. . 3 |- ( ph -> E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )
8		fvoveq1 5911	. . . . . . 7 |- ( n = a -> ( ZZ>= ` ( n + 1 ) ) = ( ZZ>= ` ( a + 1 ) ) )
9	8	sumeq1d 11387	. . . . . 6 |- ( n = a -> sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) = sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) )
10	9	fveq2d 5531	. . . . 5 |- ( n = a -> ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) )
11	10	eqeq2d 2199	. . . 4 |- ( n = a -> ( X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) <-> X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) )
12	11	cbvrexv 2716	. . 3 |- ( E. n e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) <-> E. a e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) )
13	7, 12	sylib 122	. 2 |- ( ph -> E. a e. ( 0 ... ( S - 1 ) ) X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) )
14		simprr 531	. . 3 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) )
15		0zd 9278	. . . . 5 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> 0 e. ZZ )
16		mertenslemub.s	. . . . . . . 8 |- ( ph -> S e. NN )
17	16	adantr 276	. . . . . . 7 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> S e. NN )
18	17	nnzd 9387	. . . . . 6 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> S e. ZZ )
19		1zzd 9293	. . . . . 6 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> 1 e. ZZ )
20	18, 19	zsubcld 9393	. . . . 5 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> ( S - 1 ) e. ZZ )
21	15, 20	fzfigd 10444	. . . 4 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> ( 0 ... ( S - 1 ) ) e. Fin )
22		eqid 2187	. . . . . . 7 |- ( ZZ>= ` ( n + 1 ) ) = ( ZZ>= ` ( n + 1 ) )
23		elfzelz 10038	. . . . . . . . 9 |- ( n e. ( 0 ... ( S - 1 ) ) -> n e. ZZ )
24	23	adantl 277	. . . . . . . 8 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> n e. ZZ )
25	24	peano2zd 9391	. . . . . . 7 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> ( n + 1 ) e. ZZ )
26		eqidd 2188	. . . . . . 7 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ( G ` k ) = ( G ` k ) )
27		simpll 527	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ph )
28		elfznn0 10127	. . . . . . . . . . 11 |- ( n e. ( 0 ... ( S - 1 ) ) -> n e. NN0 )
29	28	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> n e. NN0 )
30		peano2nn0 9229	. . . . . . . . . 10 |- ( n e. NN0 -> ( n + 1 ) e. NN0 )
31	29, 30	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ( n + 1 ) e. NN0 )
32		eluznn0 9612	. . . . . . . . 9 |- ( ( ( n + 1 ) e. NN0 /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> k e. NN0 )
33	31, 32	sylancom 420	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> k e. NN0 )
34		mertenslemub.gb	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )
35		mertenslemub.b	. . . . . . . . 9 |- ( ( ph /\ k e. NN0 ) -> B e. CC )
36	34, 35	eqeltrd 2264	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) e. CC )
37	27, 33, 36	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. ( ZZ>= ` ( n + 1 ) ) ) -> ( G ` k ) e. CC )
38		mertenslemub.cvg	. . . . . . . . 9 |- ( ph -> seq 0 ( + , G ) e. dom ~~> )
39	38	adantr 276	. . . . . . . 8 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> seq 0 ( + , G ) e. dom ~~> )
40		nn0uz 9575	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
41	28	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> n e. NN0 )
42	41, 30	syl 14	. . . . . . . . 9 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> ( n + 1 ) e. NN0 )
43	36	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) /\ k e. NN0 ) -> ( G ` k ) e. CC )
44	40, 42, 43	iserex 11360	. . . . . . . 8 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> ( seq 0 ( + , G ) e. dom ~~> <-> seq ( n + 1 ) ( + , G ) e. dom ~~> ) )
45	39, 44	mpbid 147	. . . . . . 7 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> seq ( n + 1 ) ( + , G ) e. dom ~~> )
46	22, 25, 26, 37, 45	isumcl 11446	. . . . . 6 |- ( ( ph /\ n e. ( 0 ... ( S - 1 ) ) ) -> sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) e. CC )
47	46	adantlr 477	. . . . 5 |- ( ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) /\ n e. ( 0 ... ( S - 1 ) ) ) -> sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) e. CC )
48	47	abscld 11203	. . . 4 |- ( ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) /\ n e. ( 0 ... ( S - 1 ) ) ) -> ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) e. RR )
49	47	absge0d 11206	. . . 4 |- ( ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) /\ n e. ( 0 ... ( S - 1 ) ) ) -> 0 <_ ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )
50		simprl 529	. . . 4 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> a e. ( 0 ... ( S - 1 ) ) )
51	21, 48, 49, 10, 50	fsumge1 11482	. . 3 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )
52	14, 51	eqbrtrd 4037	. 2 |- ( ( ph /\ ( a e. ( 0 ... ( S - 1 ) ) /\ X = ( abs ` sum_ k e. ( ZZ>= ` ( a + 1 ) ) ( G ` k ) ) ) ) -> X <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )
53	13, 52	rexlimddv 2609	1 |- ( ph -> X <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   {cab 2173   E.wrex 2466   class class class wbr 4015   dom cdm 4638   ` cfv 5228  (class class class)co 5888   CCcc 7822   0cc0 7824   1c1 7825   + caddc 7827   <_ cle 8006   - cmin 8141   NNcn 8932   NN0cn0 9189   ZZcz 9266   ZZ>=cuz 9541   ...cfz 10021   seqcseq 10458   abscabs 11019   ~~> cli 11299   sum_csu 11374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-frec 6405  df-1o 6430  df-oadd 6434  df-er 6548  df-en 6754  df-dom 6755  df-fin 6756  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-ico 9907  df-fz 10022  df-fzo 10156  df-seqfrec 10459  df-exp 10533  df-ihash 10769  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-clim 11300  df-sumdc 11375

This theorem is referenced by:  mertenslem2  11557