Theorem caucvgrelemcau 10544

Description: Lemma for caucvgre 10545. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)

Hypotheses

Ref	Expression
caucvgre.f	⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
caucvgre.cau	⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))

Assertion

Ref	Expression
caucvgrelemcau	⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ ℕ (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))

Distinct variable groups:   𝑘,𝐹,𝑛   𝜑,𝑘,𝑛   𝑘,𝑟,𝑛

Allowed substitution hints:   𝜑(𝑟)   𝐹(𝑟)

Proof of Theorem caucvgrelemcau

Step	Hyp	Ref	Expression
1		simplr 498	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑛 ∈ ℕ)
2	1	nnred 8533	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑛 ∈ ℝ)
3		simpr 109	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
4	3	nnred 8533	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
5		ltle 7669	. . . . . 6 ⊢ ((𝑛 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑛 < 𝑘 → 𝑛 ≤ 𝑘))
6	2, 4, 5	syl2anc 404	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑛 < 𝑘 → 𝑛 ≤ 𝑘))
7		eluznn 9186	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
8	7	ex 114	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (ℤ≥‘𝑛) → 𝑘 ∈ ℕ))
9		nnz 8867	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
10		eluz1 9122	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝑘 ∈ (ℤ≥‘𝑛) ↔ (𝑘 ∈ ℤ ∧ 𝑛 ≤ 𝑘)))
11	9, 10	syl 14	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (ℤ≥‘𝑛) ↔ (𝑘 ∈ ℤ ∧ 𝑛 ≤ 𝑘)))
12		simpr 109	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ 𝑛 ≤ 𝑘) → 𝑛 ≤ 𝑘)
13	11, 12	syl6bi 162	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (ℤ≥‘𝑛) → 𝑛 ≤ 𝑘))
14	8, 13	jcad 302	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (ℤ≥‘𝑛) → (𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘)))
15		nnz 8867	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
16	15	anim1i 334	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘) → (𝑘 ∈ ℤ ∧ 𝑛 ≤ 𝑘))
17	16, 11	syl5ibr 155	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘) → 𝑘 ∈ (ℤ≥‘𝑛)))
18	14, 17	impbid 128	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (ℤ≥‘𝑛) ↔ (𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘)))
19	18	adantl 272	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (ℤ≥‘𝑛) ↔ (𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘)))
20	19	biimpar 292	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘)) → 𝑘 ∈ (ℤ≥‘𝑛))
21		caucvgre.cau	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))
22	21	r19.21bi 2473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))
23	22	r19.21bi 2473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))
24	20, 23	syldan 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘)) → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))
25	24	expr 368	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑛 ≤ 𝑘 → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛)))))
26	6, 25	syld 45	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑛 < 𝑘 → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛)))))
27		ltxrlt 7649	. . . . 5 ⊢ ((𝑛 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑛 < 𝑘 ↔ 𝑛 <ℝ 𝑘))
28	2, 4, 27	syl2anc 404	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑛 < 𝑘 ↔ 𝑛 <ℝ 𝑘))
29		caucvgre.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
30	29	ad2antrr 473	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶ℝ)
31	30, 1	ffvelrnd 5474	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑛) ∈ ℝ)
32	30, 3	ffvelrnd 5474	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
33	1	nnrecred 8567	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
34	32, 33	readdcld 7614	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) + (1 / 𝑛)) ∈ ℝ)
35		ltxrlt 7649	. . . . . . 7 ⊢ (((𝐹‘𝑛) ∈ ℝ ∧ ((𝐹‘𝑘) + (1 / 𝑛)) ∈ ℝ) → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ↔ (𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (1 / 𝑛))))
36	31, 34, 35	syl2anc 404	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ↔ (𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (1 / 𝑛))))
37		nnap0 8549	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 # 0)
38	1, 37	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑛 # 0)
39		caucvgrelemrec 10543	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℝ ∧ 𝑛 # 0) → (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (1 / 𝑛))
40	2, 38, 39	syl2anc 404	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (1 / 𝑛))
41	40	oveq2d 5706	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑘) + (1 / 𝑛)))
42	41	breq2d 3879	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (1 / 𝑛))))
43	36, 42	bitr4d 190	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ↔ (𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))))
44	31, 33	readdcld 7614	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑛) + (1 / 𝑛)) ∈ ℝ)
45		ltxrlt 7649	. . . . . . 7 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ ((𝐹‘𝑛) + (1 / 𝑛)) ∈ ℝ) → ((𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛)) ↔ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (1 / 𝑛))))
46	32, 44, 45	syl2anc 404	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛)) ↔ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (1 / 𝑛))))
47	40	oveq2d 5706	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑛) + (1 / 𝑛)))
48	47	breq2d 3879	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ↔ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (1 / 𝑛))))
49	46, 48	bitr4d 190	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛)) ↔ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1))))
50	43, 49	anbi12d 458	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))) ↔ ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
51	26, 28, 50	3imtr3d 201	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
52	51	ralrimiva 2458	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
53	52	ralrimiva 2458	1 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ ℕ (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  ∀wral 2370   class class class wbr 3867  ⟶wf 5045  ‘cfv 5049  ℩crio 5645  (class class class)co 5690  ℝcr 7446  0cc0 7447  1c1 7448   + caddc 7450   <_ℝ cltrr 7451   · cmul 7452   < clt 7619   ≤ cle 7620   # cap 8155   / cdiv 8236  ℕcn 8520  ℤcz 8848  ℤ_≥cuz 9118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560

This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-z 8849  df-uz 9119

This theorem is referenced by:  caucvgre  10545