Theorem mertenslemub 12220

Description: Lemma for mertensabs 12223. An upper bound for 𝑇. (Contributed by Jim Kingdon, 3-Dec-2022.)

Hypotheses

Ref	Expression
mertenslemub.gb	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
mertenslemub.b	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
mertenslemub.cvg	⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
mertenslemub.t	⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑆 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}
mertenslemub.elt	⊢ (𝜑 → 𝑋 ∈ 𝑇)
mertenslemub.s	⊢ (𝜑 → 𝑆 ∈ ℕ)

Assertion

Ref	Expression
mertenslemub	⊢ (𝜑 → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))

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

Allowed substitution hints:   𝜑(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 ⊢ (𝜑 → 𝑋 ∈ 𝑇)
2		eqeq1 2239	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
3	2	rexbidv 2543	. . . . . 6 ⊢ (𝑧 = 𝑋 → (∃𝑛 ∈ (0...(𝑆 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ ∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
4		mertenslemub.t	. . . . . 6 ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑆 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}
5	3, 4	elab2g 2964	. . . . 5 ⊢ (𝑋 ∈ 𝑇 → (𝑋 ∈ 𝑇 ↔ ∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
6	1, 5	syl 14	. . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ ∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
7	1, 6	mpbid 147	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
8		fvoveq1 6073	. . . . . . 7 ⊢ (𝑛 = 𝑎 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑎 + 1)))
9	8	sumeq1d 12051	. . . . . 6 ⊢ (𝑛 = 𝑎 → Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘))
10	9	fveq2d 5674	. . . . 5 ⊢ (𝑛 = 𝑎 → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
11	10	eqeq2d 2244	. . . 4 ⊢ (𝑛 = 𝑎 → (𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘))))
12	11	cbvrexv 2779	. . 3 ⊢ (∃𝑛 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ ∃𝑎 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
13	7, 12	sylib 122	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (0...(𝑆 − 1))𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
14		simprr 533	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
15		0zd 9589	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 0 ∈ ℤ)
16		mertenslemub.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℕ)
17	16	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑆 ∈ ℕ)
18	17	nnzd 9699	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑆 ∈ ℤ)
19		1zzd 9604	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 1 ∈ ℤ)
20	18, 19	zsubcld 9705	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → (𝑆 − 1) ∈ ℤ)
21	15, 20	fzfigd 10793	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → (0...(𝑆 − 1)) ∈ Fin)
22		eqid 2232	. . . . . . 7 ⊢ (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑛 + 1))
23		elfzelz 10359	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(𝑆 − 1)) → 𝑛 ∈ ℤ)
24	23	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → 𝑛 ∈ ℤ)
25	24	peano2zd 9703	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → (𝑛 + 1) ∈ ℤ)
26		eqidd 2233	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝐺‘𝑘) = (𝐺‘𝑘))
27		simpll 527	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝜑)
28		elfznn0 10448	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0...(𝑆 − 1)) → 𝑛 ∈ ℕ0)
29	28	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑛 ∈ ℕ0)
30		peano2nn0 9536	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
31	29, 30	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝑛 + 1) ∈ ℕ0)
32		eluznn0 9931	. . . . . . . . 9 ⊢ (((𝑛 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ0)
33	31, 32	sylancom 420	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ0)
34		mertenslemub.gb	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
35		mertenslemub.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
36	34, 35	eqeltrd 2309	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
37	27, 33, 36	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝐺‘𝑘) ∈ ℂ)
38		mertenslemub.cvg	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
39	38	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → seq0( + , 𝐺) ∈ dom ⇝ )
40		nn0uz 9889	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0)
41	28	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → 𝑛 ∈ ℕ0)
42	41, 30	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → (𝑛 + 1) ∈ ℕ0)
43	36	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
44	40, 42, 43	iserex 12024	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq(𝑛 + 1)( + , 𝐺) ∈ dom ⇝ ))
45	39, 44	mpbid 147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → seq(𝑛 + 1)( + , 𝐺) ∈ dom ⇝ )
46	22, 25, 26, 37, 45	isumcl 12111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) ∈ ℂ)
47	46	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) ∧ 𝑛 ∈ (0...(𝑆 − 1))) → Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) ∈ ℂ)
48	47	abscld 11866	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) ∧ 𝑛 ∈ (0...(𝑆 − 1))) → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ∈ ℝ)
49	47	absge0d 11869	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) ∧ 𝑛 ∈ (0...(𝑆 − 1))) → 0 ≤ (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
50		simprl 531	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑎 ∈ (0...(𝑆 − 1)))
51	21, 48, 49, 10, 50	fsumge1 12147	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)) ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
52	14, 51	eqbrtrd 4131	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
53	13, 52	rexlimddv 2665	1 ⊢ (𝜑 → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {cab 2218  ∃wrex 2521   class class class wbr 4109  dom cdm 4749  ‘cfv 5352  (class class class)co 6050  ℂcc 8125  0cc0 8127  1c1 8128   + caddc 8130   ≤ cle 8309   − cmin 8444  ℕcn 9237  ℕ₀cn0 9496  ℤcz 9577  ℤ_≥cuz 9853  ...cfz 10342  seqcseq 10809  abscabs 11682   ⇝ cli 11963  Σcsu 12038

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-ico 10227  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039

This theorem is referenced by:  mertenslem2  12222