Theorem mertenslemub 11680

Description: Lemma for mertensabs 11683. 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 2200	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
3	2	rexbidv 2495	. . . . . 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 2908	. . . . 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 5942	. . . . . . 7 ⊢ (𝑛 = 𝑎 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑎 + 1)))
9	8	sumeq1d 11512	. . . . . 6 ⊢ (𝑛 = 𝑎 → Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘))
10	9	fveq2d 5559	. . . . 5 ⊢ (𝑛 = 𝑎 → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
11	10	eqeq2d 2205	. . . 4 ⊢ (𝑛 = 𝑎 → (𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘))))
12	11	cbvrexv 2727	. . 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 531	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))
15		0zd 9332	. . . . 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 9441	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 𝑆 ∈ ℤ)
19		1zzd 9347	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → 1 ∈ ℤ)
20	18, 19	zsubcld 9447	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → (𝑆 − 1) ∈ ℤ)
21	15, 20	fzfigd 10505	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (0...(𝑆 − 1)) ∧ 𝑋 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑎 + 1))(𝐺‘𝑘)))) → (0...(𝑆 − 1)) ∈ Fin)
22		eqid 2193	. . . . . . 7 ⊢ (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑛 + 1))
23		elfzelz 10094	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(𝑆 − 1)) → 𝑛 ∈ ℤ)
24	23	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → 𝑛 ∈ ℤ)
25	24	peano2zd 9445	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...(𝑆 − 1))) → (𝑛 + 1) ∈ ℤ)
26		eqidd 2194	. . . . . . 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 10183	. . . . . . . . . . 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 9283	. . . . . . . . . 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 9667	. . . . . . . . 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 2270	. . . . . . . 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 9630	. . . . . . . . 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 11485	. . . . . . . 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 11571	. . . . . 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 11328	. . . 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 11331	. . . 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 529	. . . 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 11607	. . 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 4052	. 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 2616	1 ⊢ (𝜑 → 𝑋 ≤ Σ𝑛 ∈ (0...(𝑆 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  ∃wrex 2473   class class class wbr 4030  dom cdm 4660  ‘cfv 5255  (class class class)co 5919  ℂcc 7872  0cc0 7874  1c1 7875   + caddc 7877   ≤ cle 8057   − cmin 8192  ℕcn 8984  ℕ₀cn0 9243  ℤcz 9320  ℤ_≥cuz 9595  ...cfz 10077  seqcseq 10521  abscabs 11144   ⇝ cli 11424  Σcsu 11499

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-ico 9963  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500

This theorem is referenced by:  mertenslem2  11682