Theorem climinf2lem 43210

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 43110	. 2 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
7	3	frnd 6605	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
8	3	ffnd 6598	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝑍)
9	2, 1	uzidd2 42919	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
10		fnfvelrn 6953	. . . . 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 4275	. . 3 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
13		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
14		fvelrnb 6825	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦))
17	13, 16	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦)
18	17	adantlr 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦)
19		nfv 1921	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝜑
20		nfra1 3145	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)
21	19, 20	nfan 1906	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))
22		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑥 ≤ 𝑦
23		rspa 3133	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) ∧ 𝑘 ∈ 𝑍) → 𝑥 ≤ (𝐹‘𝑘))
24		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → 𝑥 ≤ (𝐹‘𝑘))
25		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → (𝐹‘𝑘) = 𝑦)
26	24, 25	breqtrd 5105	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) = 𝑦) → 𝑥 ≤ 𝑦)
27	26	ex 413	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → (𝑘 ∈ 𝑍 → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦)))
30	29	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → (𝑘 ∈ 𝑍 → ((𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦)))
31	21, 22, 30	rexlimd 3248	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → (∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦 → 𝑥 ≤ 𝑦))
32	31	adantr 481	. . . . . . . . 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 3110	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
35	34	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
36	35	ex 413	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
37	36	reximdva 3205	. . . 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 13061	. . 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 1370	. 2 ⊢ (𝜑 → inf(ran 𝐹, ℝ*, < ) = inf(ran 𝐹, ℝ, < ))
41	6, 40	breqtrrd 5107	1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ≠ wne 2945  ∀wral 3066  ∃wrex 3067   ⊆ wss 3892  ∅c0 4262   class class class wbr 5079  ran crn 5590   Fn wfn 6426  ⟶wf 6427  ‘cfv 6431  (class class class)co 7269  infcinf 9170  ℝcr 10863  1c1 10865   + caddc 10867  ℝ^*cxr 11001   < clt 11002   ≤ cle 11003  ℤcz 12311  ℤ_≥cuz 12573   ⇝ cli 15183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-cnex 10920  ax-resscn 10921  ax-1cn 10922  ax-icn 10923  ax-addcl 10924  ax-addrcl 10925  ax-mulcl 10926  ax-mulrcl 10927  ax-mulcom 10928  ax-addass 10929  ax-mulass 10930  ax-distr 10931  ax-i2m1 10932  ax-1ne0 10933  ax-1rid 10934  ax-rnegex 10935  ax-rrecex 10936  ax-cnre 10937  ax-pre-lttri 10938  ax-pre-lttrn 10939  ax-pre-ltadd 10940  ax-pre-mulgt0 10941  ax-pre-sup 10942

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7702  df-1st 7818  df-2nd 7819  df-frecs 8082  df-wrecs 8113  df-recs 8187  df-rdg 8226  df-er 8473  df-en 8709  df-dom 8710  df-sdom 8711  df-sup 9171  df-inf 9172  df-pnf 11004  df-mnf 11005  df-xr 11006  df-ltxr 11007  df-le 11008  df-sub 11199  df-neg 11200  df-div 11625  df-nn 11966  df-2 12028  df-3 12029  df-n0 12226  df-z 12312  df-uz 12574  df-rp 12722  df-fz 13231  df-seq 13712  df-exp 13773  df-cj 14800  df-re 14801  df-im 14802  df-sqrt 14936  df-abs 14937  df-clim 15187

This theorem is referenced by:  climinf2  43211