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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climinf3 | Structured version Visualization version GIF version |
Description: A convergent, nonincreasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
climinf3.1 | ⊢ Ⅎ𝑘𝜑 |
climinf3.2 | ⊢ Ⅎ𝑘𝐹 |
climinf3.3 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climinf3.4 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climinf3.5 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
climinf3.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) |
climinf3.7 | ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
Ref | Expression |
---|---|
climinf3 | ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climinf3.1 | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | climinf3.2 | . 2 ⊢ Ⅎ𝑘𝐹 | |
3 | climinf3.4 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | climinf3.3 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | climinf3.5 | . 2 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
6 | climinf3.6 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) | |
7 | climinf3.7 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | |
8 | 5 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
9 | 8 | recnd 11287 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
10 | 1, 9 | ralrimia 3256 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) |
11 | 2, 3 | climbddf 45643 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
12 | 4, 7, 10, 11 | syl3anc 1370 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
13 | renegcl 11570 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) | |
14 | 13 | ad2antlr 727 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) → -𝑥 ∈ ℝ) |
15 | nfv 1912 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ ℝ | |
16 | 1, 15 | nfan 1897 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ ℝ) |
17 | nfra1 3282 | . . . . . . 7 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥 | |
18 | 16, 17 | nfan 1897 | . . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
19 | simpll 767 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) ∧ 𝑘 ∈ 𝑍) → (𝜑 ∧ 𝑥 ∈ ℝ)) | |
20 | simpr 484 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
21 | rspa 3246 | . . . . . . . . 9 ⊢ ((∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ≤ 𝑥) | |
22 | 21 | adantll 714 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
23 | simpr 484 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → (abs‘(𝐹‘𝑘)) ≤ 𝑥) | |
24 | 8 | ad4ant13 751 | . . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → (𝐹‘𝑘) ∈ ℝ) |
25 | simpllr 776 | . . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → 𝑥 ∈ ℝ) | |
26 | 24, 25 | absled 15466 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → ((abs‘(𝐹‘𝑘)) ≤ 𝑥 ↔ (-𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) ≤ 𝑥))) |
27 | 23, 26 | mpbid 232 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → (-𝑥 ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) ≤ 𝑥)) |
28 | 27 | simpld 494 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (abs‘(𝐹‘𝑘)) ≤ 𝑥) → -𝑥 ≤ (𝐹‘𝑘)) |
29 | 19, 20, 22, 28 | syl21anc 838 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) ∧ 𝑘 ∈ 𝑍) → -𝑥 ≤ (𝐹‘𝑘)) |
30 | 29 | ex 412 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) → (𝑘 ∈ 𝑍 → -𝑥 ≤ (𝐹‘𝑘))) |
31 | 18, 30 | ralrimi 3255 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) → ∀𝑘 ∈ 𝑍 -𝑥 ≤ (𝐹‘𝑘)) |
32 | breq1 5151 | . . . . . . 7 ⊢ (𝑦 = -𝑥 → (𝑦 ≤ (𝐹‘𝑘) ↔ -𝑥 ≤ (𝐹‘𝑘))) | |
33 | 32 | ralbidv 3176 | . . . . . 6 ⊢ (𝑦 = -𝑥 → (∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝑍 -𝑥 ≤ (𝐹‘𝑘))) |
34 | 33 | rspcev 3622 | . . . . 5 ⊢ ((-𝑥 ∈ ℝ ∧ ∀𝑘 ∈ 𝑍 -𝑥 ≤ (𝐹‘𝑘)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘)) |
35 | 14, 31, 34 | syl2anc 584 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘)) |
36 | 35 | rexlimdva2 3155 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘))) |
37 | 12, 36 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ (𝐹‘𝑘)) |
38 | 1, 2, 3, 4, 5, 6, 37 | climinf2 45663 | 1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 ∃wrex 3068 class class class wbr 5148 dom cdm 5689 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 infcinf 9479 ℂcc 11151 ℝcr 11152 1c1 11154 + caddc 11156 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 -cneg 11491 ℤcz 12611 ℤ≥cuz 12876 abscabs 15270 ⇝ cli 15517 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 |
This theorem is referenced by: (None) |
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