Theorem climinf2lem 46152

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 46054	. 2 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
7	3	frnd 6670	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
8	3	ffnd 6663	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝑍)
9	2, 1	uzidd2 45862	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
10		fnfvelrn 7026	. . . . 5 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
11	8, 9, 10	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
12	11	ne0d 4283	. . 3 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
13		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
14		fvelrnb 6894	. . . . . . . . . . . . 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 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑘 ∈ 𝑍 (𝐹‘𝑘) = 𝑦)
19		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝜑
20		nfra1 3262	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)
21	19, 20	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))
22		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑥 ≤ 𝑦
23		rspa 3227	. . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . 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 3245	. . . . . . . . . 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 3130	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) → ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
35	34	adantlr 716	. . . . . 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 3151	. . . 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 13280	. . 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 1374	. 2 ⊢ (𝜑 → inf(ran 𝐹, ℝ*, < ) = inf(ran 𝐹, ℝ, < ))
41	6, 40	breqtrrd 5114	1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ∅c0 4274   class class class wbr 5086  ran crn 5625   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  infcinf 9347  ℝcr 11028  1c1 11030   + caddc 11032  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171  ℤcz 12515  ℤ_≥cuz 12779   ⇝ cli 15437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441

This theorem is referenced by:  climinf2  46153