Theorem infcvglem3 7223

Description: Lemma for infcvg 7224. Using climsqueeze 7140, show that sequence F, constructed from f, converges to the supremum.

Hypotheses

Ref	Expression
infcvg.1	⊢ R = {x∣∃y ∈ X x = -A}
infcvg.2	⊢ (y ∈ X → A ∈ ℝ)
infcvg.3	⊢ Z ∈ X
infcvg.4	⊢ ∃z ∈ ℝ ∀w ∈ R w ≤ z
infcvg.5c	⊢ S = -sup(R, ℝ, < )
infcvg.9	⊢ G ∈ V
infcvg.10	⊢ (k ∈ ℕ → (G ‘k) = (S + (1 / k)))
infcvg.11	⊢ H ∈ V
infcvg.12	⊢ (k ∈ ℕ → (H ‘k) = (1 / k))
infcvg.6a	⊢ F ∈ V
infcvg.7a	⊢ (y = (f ‘k) → A = B)
infcvg.8a	⊢ (k ∈ ℕ → (F ‘k) = B)

Assertion

Ref	Expression
infcvglem3	⊢ ∃f(f:ℕ–→X ⋀ F ⇝ S)

Distinct variable groups:   x,f,A   x,y,B   k,F   k,G   k,H   z,w,R   f,k,S,y   f,X   x,k,y,X   x,Z,y   y,S

Proof of Theorem infcvglem3

Step	Hyp	Ref	Expression
1		infcvg.1	. . 3 ⊢ R = {x∣∃y ∈ X x = -A}
2		infcvg.2	. . 3 ⊢ (y ∈ X → A ∈ ℝ)
3		infcvg.3	. . 3 ⊢ Z ∈ X
4		infcvg.4	. . 3 ⊢ ∃z ∈ ℝ ∀w ∈ R w ≤ z
5		infcvg.5c	. . 3 ⊢ S = -sup(R, ℝ, < )
6		infcvg.7a	. . 3 ⊢ (y = (f ‘k) → A = B)
7	1, 2, 3, 4, 5, 6	infcvglem1 7221	. 2 ⊢ ∃f(f:ℕ–→X ⋀ ∀k ∈ ℕ B < (S + (1 / k)))
8		pm3.26 319	. . . 4 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ B < (S + (1 / k))) → f:ℕ–→X)
9		infcvg.8a	. . . . . . . . . . . 12 ⊢ (k ∈ ℕ → (F ‘k) = B)
10		infcvg.10	. . . . . . . . . . . 12 ⊢ (k ∈ ℕ → (G ‘k) = (S + (1 / k)))
11	9, 10	breq12d 2636	. . . . . . . . . . 11 ⊢ (k ∈ ℕ → ((F ‘k) < (G ‘k) ↔ B < (S + (1 / k))))
12	11	adantl 390	. . . . . . . . . 10 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → ((F ‘k) < (G ‘k) ↔ B < (S + (1 / k))))
13		ltlet 5532	. . . . . . . . . . 11 ⊢ (((F ‘k) ∈ ℝ ⋀ (G ‘k) ∈ ℝ) → ((F ‘k) < (G ‘k) → (F ‘k) ≤ (G ‘k)))
14	9	adantl 390	. . . . . . . . . . . 12 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → (F ‘k) = B)
15		ffvelrn 3820	. . . . . . . . . . . . 13 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → (f ‘k) ∈ X)
16	6	eleq1d 1543	. . . . . . . . . . . . . 14 ⊢ (y = (f ‘k) → (A ∈ ℝ ↔ B ∈ ℝ))
17	16, 2	vtoclga 1855	. . . . . . . . . . . . 13 ⊢ ((f ‘k) ∈ X → B ∈ ℝ)
18	15, 17	syl 10	. . . . . . . . . . . 12 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → B ∈ ℝ)
19	14, 18	eqeltrd 1551	. . . . . . . . . . 11 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → (F ‘k) ∈ ℝ)
20		nnrecret 5954	. . . . . . . . . . . . . 14 ⊢ (k ∈ ℕ → (1 / k) ∈ ℝ)
21	1, 2, 3, 4	infcvgaux1 7219	. . . . . . . . . . . . . . . . . 18 ⊢ (R ⊆ ℝ ⋀ R ≠ ∅ ⋀ ∃z ∈ ℝ ∀w ∈ R w ≤ z)
22	21	suprcli 6063	. . . . . . . . . . . . . . . . 17 ⊢ sup(R, ℝ, < ) ∈ ℝ
23	22	renegcl 5428	. . . . . . . . . . . . . . . 16 ⊢ -sup(R, ℝ, < ) ∈ ℝ
24	5, 23	eqeltr 1547	. . . . . . . . . . . . . . 15 ⊢ S ∈ ℝ
25		axaddrcl 5284	. . . . . . . . . . . . . . 15 ⊢ ((S ∈ ℝ ⋀ (1 / k) ∈ ℝ) → (S + (1 / k)) ∈ ℝ)
26	24, 25	mpan 697	. . . . . . . . . . . . . 14 ⊢ ((1 / k) ∈ ℝ → (S + (1 / k)) ∈ ℝ)
27	20, 26	syl 10	. . . . . . . . . . . . 13 ⊢ (k ∈ ℕ → (S + (1 / k)) ∈ ℝ)
28	10, 27	eqeltrd 1551	. . . . . . . . . . . 12 ⊢ (k ∈ ℕ → (G ‘k) ∈ ℝ)
29	28	adantl 390	. . . . . . . . . . 11 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → (G ‘k) ∈ ℝ)
30	13, 19, 29	sylanc 473	. . . . . . . . . 10 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → ((F ‘k) < (G ‘k) → (F ‘k) ≤ (G ‘k)))
31	12, 30	sylbird 205	. . . . . . . . 9 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → (B < (S + (1 / k)) → (F ‘k) ≤ (G ‘k)))
32	1, 2, 3, 4, 5, 6	infcvgaux2 7220	. . . . . . . . . . . . . 14 ⊢ ((f ‘k) ∈ X → S ≤ B)
33	15, 32	syl 10	. . . . . . . . . . . . 13 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → S ≤ B)
34	33, 14	breqtrrd 2646	. . . . . . . . . . . 12 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → S ≤ (F ‘k))
35	34	a1d 12	. . . . . . . . . . 11 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → ((F ‘k) ≤ (G ‘k) → S ≤ (F ‘k)))
36	35	ancrd 299	. . . . . . . . . 10 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → ((F ‘k) ≤ (G ‘k) → (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k))))
37	29, 19	jca 288	. . . . . . . . . 10 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → ((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ))
38	36, 37	jctild 603	. . . . . . . . 9 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → ((F ‘k) ≤ (G ‘k) → (((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k)))))
39	31, 38	syld 27	. . . . . . . 8 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → (B < (S + (1 / k)) → (((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k)))))
40	39	r19.20dva 1712	. . . . . . 7 ⊢ (f:ℕ–→X → (∀k ∈ ℕ B < (S + (1 / k)) → ∀k ∈ ℕ (((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k)))))
41	40	imp 350	. . . . . 6 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ B < (S + (1 / k))) → ∀k ∈ ℕ (((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k))))
42		nnuz 6440	. . . . . . 7 ⊢ ℕ = (ℤ≥ ‘1)
43		raleq1 1789	. . . . . . 7 ⊢ (ℕ = (ℤ≥ ‘1) → (∀k ∈ ℕ (((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k))) ↔ ∀k ∈ (ℤ≥ ‘1)(((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k)))))
44	42, 43	ax-mp 7	. . . . . 6 ⊢ (∀k ∈ ℕ (((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k))) ↔ ∀k ∈ (ℤ≥ ‘1)(((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k))))
45	41, 44	sylib 198	. . . . 5 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ B < (S + (1 / k))) → ∀k ∈ (ℤ≥ ‘1)(((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k))))
46		infcvg.9	. . . . . . 7 ⊢ G ∈ V
47		infcvg.11	. . . . . . 7 ⊢ H ∈ V
48		infcvg.12	. . . . . . 7 ⊢ (k ∈ ℕ → (H ‘k) = (1 / k))
49	1, 2, 3, 4, 5, 46, 10, 47, 48	infcvglem2 7222	. . . . . 6 ⊢ G ⇝ S
50		1z 6161	. . . . . 6 ⊢ 1 ∈ ℤ
51		infcvg.6a	. . . . . . 7 ⊢ F ∈ V
52	24	elisseti 1821	. . . . . . 7 ⊢ S ∈ V
53	46, 51, 52	climsqueeze2 7141	. . . . . 6 ⊢ ((G ⇝ S ⋀ 1 ∈ ℤ ⋀ ∀k ∈ (ℤ≥ ‘1)(((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k)))) → F ⇝ S)
54	49, 50, 53	mp3an12 908	. . . . 5 ⊢ (∀k ∈ (ℤ≥ ‘1)(((G ‘k) ∈ ℝ ⋀ (F ‘k) ∈ ℝ) ⋀ (S ≤ (F ‘k) ⋀ (F ‘k) ≤ (G ‘k))) → F ⇝ S)
55	45, 54	syl 10	. . . 4 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ B < (S + (1 / k))) → F ⇝ S)
56	8, 55	jca 288	. . 3 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ B < (S + (1 / k))) → (f:ℕ–→X ⋀ F ⇝ S))
57	56	19.22i 1042	. 2 ⊢ (∃f(f:ℕ–→X ⋀ ∀k ∈ ℕ B < (S + (1 / k))) → ∃f(f:ℕ–→X ⋀ F ⇝ S))
58	7, 57	ax-mp 7	1 ⊢ ∃f(f:ℕ–→X ⋀ F ⇝ S)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃wex 982  {cab 1466  ∀wral 1648  ∃wrex 1649  Vcvv 1814   class class class wbr 2624  –→wf 3184   ‘cfv 3188  (class class class)co 3969  supcsup 4582  ℝcr 5245  1c1 5247   + caddc 5249  -cneg 5305   / cdiv 5306   ≤ cle 5307  ℕcn 5308  ℤcz 5310   < clt 5498  ℤ_≥cuz 6418   ⇝ cli 6974

This theorem is referenced by:  infcvg 7224

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-r1 4653  df-rank 4654  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-n0 6102  df-z 6138  df-seq1 6309  df-uz 6419  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-clim 6975

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