Theorem climinf2lem 45735

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 45635	. 2 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
7	3	frnd 6714	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
8	3	ffnd 6707	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝑍)
9	2, 1	uzidd2 45443	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
10		fnfvelrn 7070	. . . . 5 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
11	8, 9, 10	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
12	11	ne0d 4317	. . 3 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
13		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
14		fvelrnb 6939	. . . . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦))
17	13, 16	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦)
18	17	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦)
19		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝜑
20		nfra1 3266	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)
21	19, 20	nfan 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))
22		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑥 ≤ 𝑦
23		rspa 3231	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ∧ 𝑘 ∈ 𝑍) → 𝑥 ≤ (𝐹‘𝑘))
24		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → 𝑥 ≤ (𝐹‘𝑘))
25		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → (𝐹‘𝑘) = 𝑦)
26	24, 25	breqtrd 5145	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → 𝑥 ≤ 𝑦)
27	26	ex 412	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → (𝑘 ∈ 𝑍 → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦)))
30	29	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → (𝑘 ∈ 𝑍 → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦)))
31	21, 22, 30	rexlimd 3249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → (∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
32	31	adantr 480	. . . . . . . . 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 3132	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
35	34	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
36	35	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
37	36	reximdva 3153	. . . 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 13353	. . 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 1373	. 2 ⊢ (𝜑 → inf(ran 𝐹, ℝ*, < ) = inf(ran 𝐹, ℝ, < ))
41	6, 40	breqtrrd 5147	1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  ∅c0 4308   class class class wbr 5119  ran crn 5655   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  infcinf 9453  ℝcr 11128  1c1 11130   + caddc 11132  ℝ^*cxr 11268   < clt 11269   ≤ cle 11270  ℤcz 12588  ℤ_≥cuz 12852   ⇝ cli 15500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-n0 12502  df-z 12589  df-uz 12853  df-rp 13009  df-fz 13525  df-seq 14020  df-exp 14080  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-clim 15504

This theorem is referenced by:  climinf2  45736