Theorem climinf2lem 41980

Description: A convergent, nonincreasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
climinf2lem.1	⊢ 𝑍 = (ℤ≥‘𝑀)
climinf2lem.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
climinf2lem.3	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
climinf2lem.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
climinf2lem.5	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))

Assertion

Ref	Expression
climinf2lem	⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))

Distinct variable groups:   𝑘,𝐹,𝑥   𝑘,𝑍,𝑥   𝜑,𝑘,𝑥

Allowed substitution hints:   𝑀(𝑥,𝑘)

Proof of Theorem climinf2lem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climinf2lem.1	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		climinf2lem.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		climinf2lem.3	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
4		climinf2lem.4	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
5		climinf2lem.5	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))
6	1, 2, 3, 4, 5	climinf 41880	. 2 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
7	3	frnd 6515	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
8	3	ffnd 6509	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝑍)
9	2, 1	uzidd2 41683	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
10		fnfvelrn 6842	. . . . 5 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
11	8, 9, 10	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
12	11	ne0d 4300	. . 3 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
13		simpr 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
14		fvelrnb 6720	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝑍 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦))
15	8, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦))
16	15	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦))
17	13, 16	mpbid 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦)
18	17	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦)
19		nfv 1911	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝜑
20		nfra1 3219	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)
21	19, 20	nfan 1896	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))
22		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑥 ≤ 𝑦
23		rspa 3206	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ∧ 𝑘 ∈ 𝑍) → 𝑥 ≤ (𝐹‘𝑘))
24		simpl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → 𝑥 ≤ (𝐹‘𝑘))
25		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → (𝐹‘𝑘) = 𝑦)
26	24, 25	breqtrd 5084	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → 𝑥 ≤ 𝑦)
27	26	ex 415	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≤ (𝐹‘𝑘) → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
28	23, 27	syl 17	. . . . . . . . . . . . 13 ⊢ ((∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
29	28	ex 415	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → (𝑘 ∈ 𝑍 → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦)))
30	29	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → (𝑘 ∈ 𝑍 → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦)))
31	21, 22, 30	rexlimd 3317	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → (∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
32	31	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ∧ 𝑦 ∈ ran 𝐹) → (∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
33	18, 32	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ∧ 𝑦 ∈ ran 𝐹) → 𝑥 ≤ 𝑦)
34	33	ralrimiva 3182	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
35	34	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
36	35	ex 415	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
37	36	reximdva 3274	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
38	5, 37	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
39		infxrre 12723	. . 3 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦) → inf(ran 𝐹, ℝ*, < ) = inf(ran 𝐹, ℝ, < ))
40	7, 12, 38, 39	syl3anc 1367	. 2 ⊢ (𝜑 → inf(ran 𝐹, ℝ*, < ) = inf(ran 𝐹, ℝ, < ))
41	6, 40	breqtrrd 5086	1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  ∅c0 4290   class class class wbr 5058  ran crn 5550   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  infcinf 8899  ℝcr 10530  1c1 10532   + caddc 10534  ℝ^*cxr 10668   < clt 10669   ≤ cle 10670  ℤcz 11975  ℤ_≥cuz 12237   ⇝ cli 14835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-fz 12887  df-seq 13364  df-exp 13424  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839

This theorem is referenced by:  climinf2  41981