Theorem caucvgrelemcau 11661

Description: Lemma for caucvgre 11662. 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 529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑛 ∈ ℕ)
2	1	nnred 9249	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑛 ∈ ℝ)
3		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
4	3	nnred 9249	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
5		ltle 8360	. . . . . 6 ⊢ ((𝑛 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑛 < 𝑘 → 𝑛 ≤ 𝑘))
6	2, 4, 5	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑛 < 𝑘 → 𝑛 ≤ 𝑘))
7		eluznn 9931	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
8	7	ex 115	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (ℤ≥‘𝑛) → 𝑘 ∈ ℕ))
9		nnz 9595	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
10		eluz1 9856	. . . . . . . . . . . . 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 110	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ 𝑛 ≤ 𝑘) → 𝑛 ≤ 𝑘)
13	11, 12	biimtrdi 163	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (ℤ≥‘𝑛) → 𝑛 ≤ 𝑘))
14	8, 13	jcad 307	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (ℤ≥‘𝑛) → (𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘)))
15		nnz 9595	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
16	15	anim1i 340	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘) → (𝑘 ∈ ℤ ∧ 𝑛 ≤ 𝑘))
17	16, 11	imbitrrid 156	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘) → 𝑘 ∈ (ℤ≥‘𝑛)))
18	14, 17	impbid 129	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (ℤ≥‘𝑛) ↔ (𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘)))
19	18	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (ℤ≥‘𝑛) ↔ (𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘)))
20	19	biimpar 297	. . . . . . 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 2630	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))
23	22	r19.21bi 2630	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))
24	20, 23	syldan 282	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑘 ∈ ℕ ∧ 𝑛 ≤ 𝑘)) → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))
25	24	expr 375	. . . . 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 8338	. . . . 5 ⊢ ((𝑛 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑛 < 𝑘 ↔ 𝑛 <ℝ 𝑘))
28	2, 4, 27	syl2anc 411	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑛 < 𝑘 ↔ 𝑛 <ℝ 𝑘))
29		caucvgre.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
30	29	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶ℝ)
31	30, 1	ffvelcdmd 5812	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑛) ∈ ℝ)
32	30, 3	ffvelcdmd 5812	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
33	1	nnrecred 9283	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
34	32, 33	readdcld 8302	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) + (1 / 𝑛)) ∈ ℝ)
35		ltxrlt 8338	. . . . . . 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 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ↔ (𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (1 / 𝑛))))
37		nnap0 9265	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 # 0)
38	1, 37	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → 𝑛 # 0)
39		caucvgrelemrec 11660	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℝ ∧ 𝑛 # 0) → (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (1 / 𝑛))
40	2, 38, 39	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1) = (1 / 𝑛))
41	40	oveq2d 6065	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑘) + (1 / 𝑛)))
42	41	breq2d 4120	. . . . . 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 191	. . . . 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 8302	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑛) + (1 / 𝑛)) ∈ ℝ)
45		ltxrlt 8338	. . . . . . 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 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛)) ↔ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (1 / 𝑛))))
47	40	oveq2d 6065	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) = ((𝐹‘𝑛) + (1 / 𝑛)))
48	47	breq2d 4120	. . . . . 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 191	. . . . 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 473	. . . 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 202	. . 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 2615	. 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 2615	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 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520   class class class wbr 4108  ⟶wf 5347  ‘cfv 5351  ℩crio 6001  (class class class)co 6049  ℝcr 8125  0cc0 8126  1c1 8127   + caddc 8129   <_ℝ cltrr 8130   · cmul 8131   < clt 8307   ≤ cle 8308   # cap 8854   / cdiv 8945  ℕcn 9236  ℤcz 9576  ℤ_≥cuz 9852

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-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244

This theorem depends on definitions:  df-bi 117  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 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-z 9577  df-uz 9853

This theorem is referenced by:  caucvgre  11662