Theorem climinf2lem 46823

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 46725	. 2 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
7	3	frnd 6721	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
8	3	ffnd 6713	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝑍)
9	2, 1	uzidd2 46533	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
10		fnfvelrn 7083	. . . . 5 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
11	8, 9, 10	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
12	11	ne0d 4287	. . 3 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
13		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
14		fvelrnb 6948	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝑍 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦))
15	8, 14	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦))
16	15	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦))
17	13, 16	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦)
18	17	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦)
19		nfv 1946	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝜑
20		nfra1 3286	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)
21	19, 20	nfan 1931	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))
22		nfv 1946	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑥 ≤ 𝑦
23		rspa 3251	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ∧ 𝑘 ∈ 𝑍) → 𝑥 ≤ (𝐹‘𝑘))
24		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → 𝑥 ≤ (𝐹‘𝑘))
25		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → (𝐹‘𝑘) = 𝑦)
26	24, 25	breqtrd 5130	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → 𝑥 ≤ 𝑦)
27	26	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≤ (𝐹‘𝑘) → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
28	23, 27	syl 18	. . . . . . . . . . . . 13 ⊢ ((∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
29	28	ex 418	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → (𝑘 ∈ 𝑍 → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦)))
30	29	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → (𝑘 ∈ 𝑍 → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦)))
31	21, 22, 30	rexlimd 3269	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → (∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
32	31	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ∧ 𝑦 ∈ ran 𝐹) → (∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
33	18, 32	mpd 16	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ∧ 𝑦 ∈ ran 𝐹) → 𝑥 ≤ 𝑦)
34	33	ralrimiva 3154	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
35	34	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
36	35	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
37	36	reximdva 3175	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
38	5, 37	mpd 16	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
39		infxrre 13472	. . 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 1398	. 2 ⊢ (𝜑 → inf(ran 𝐹, ℝ*, < ) = inf(ran 𝐹, ℝ, < ))
41	6, 40	breqtrrd 5132	1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3896  ∅c0 4277   class class class wbr 5102  ran crn 5651   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422  infcinf 9438  ℝcr 11204  1c1 11206   + caddc 11208  ℝ^*cxr 11347   < clt 11348   ≤ cle 11349  ℤcz 12698  ℤ_≥cuz 12970   ⇝ cli 15658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282  ax-pre-sup 11283

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-er 8722  df-en 8979  df-dom 8980  df-sdom 8981  df-sup 9439  df-inf 9440  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-div 11979  df-nn 12341  df-2 12410  df-3 12411  df-n0 12612  df-z 12699  df-uz 12971  df-rp 13126  df-fz 13645  df-seq 14150  df-exp 14210  df-cj 15273  df-re 15274  df-im 15275  df-sqrt 15409  df-abs 15410  df-clim 15662

This theorem is used by:  climinf2  46824