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Mirrors > Home > MPE Home > Th. List > iseraltlem1 | Structured version Visualization version GIF version |
Description: Lemma for iseralt 15035. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.) |
Ref | Expression |
---|---|
iseralt.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
iseralt.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
iseralt.3 | ⊢ (𝜑 → 𝐺:𝑍⟶ℝ) |
iseralt.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘)) |
iseralt.5 | ⊢ (𝜑 → 𝐺 ⇝ 0) |
Ref | Expression |
---|---|
iseraltlem1 | ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → 0 ≤ (𝐺‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . 2 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
2 | eluzelz 12247 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
3 | iseralt.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleq2s 2931 | . . 3 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
5 | 4 | adantl 484 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → 𝑁 ∈ ℤ) |
6 | iseralt.5 | . . 3 ⊢ (𝜑 → 𝐺 ⇝ 0) | |
7 | 6 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → 𝐺 ⇝ 0) |
8 | iseralt.3 | . . . . 5 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ) | |
9 | 8 | ffvelrnda 6845 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → (𝐺‘𝑁) ∈ ℝ) |
10 | 9 | recnd 10663 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → (𝐺‘𝑁) ∈ ℂ) |
11 | 1z 12006 | . . 3 ⊢ 1 ∈ ℤ | |
12 | uzssz 12258 | . . . 4 ⊢ (ℤ≥‘1) ⊆ ℤ | |
13 | zex 11984 | . . . 4 ⊢ ℤ ∈ V | |
14 | 12, 13 | climconst2 14899 | . . 3 ⊢ (((𝐺‘𝑁) ∈ ℂ ∧ 1 ∈ ℤ) → (ℤ × {(𝐺‘𝑁)}) ⇝ (𝐺‘𝑁)) |
15 | 10, 11, 14 | sylancl 588 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → (ℤ × {(𝐺‘𝑁)}) ⇝ (𝐺‘𝑁)) |
16 | 8 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝐺:𝑍⟶ℝ) |
17 | 3 | uztrn2 12256 | . . . 4 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍) |
18 | 17 | adantll 712 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍) |
19 | 16, 18 | ffvelrnd 6846 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑛) ∈ ℝ) |
20 | eluzelz 12247 | . . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘𝑁) → 𝑛 ∈ ℤ) | |
21 | 20 | adantl 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ ℤ) |
22 | fvex 6677 | . . . . 5 ⊢ (𝐺‘𝑁) ∈ V | |
23 | 22 | fvconst2 6960 | . . . 4 ⊢ (𝑛 ∈ ℤ → ((ℤ × {(𝐺‘𝑁)})‘𝑛) = (𝐺‘𝑁)) |
24 | 21, 23 | syl 17 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → ((ℤ × {(𝐺‘𝑁)})‘𝑛) = (𝐺‘𝑁)) |
25 | 9 | adantr 483 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑁) ∈ ℝ) |
26 | 24, 25 | eqeltrd 2913 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → ((ℤ × {(𝐺‘𝑁)})‘𝑛) ∈ ℝ) |
27 | simpr 487 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ (ℤ≥‘𝑁)) | |
28 | 16 | adantr 483 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝐺:𝑍⟶ℝ) |
29 | simplr 767 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ 𝑍) | |
30 | elfzuz 12898 | . . . . . 6 ⊢ (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ≥‘𝑁)) | |
31 | 3 | uztrn2 12256 | . . . . . 6 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
32 | 29, 30, 31 | syl2an 597 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝑘 ∈ 𝑍) |
33 | 28, 32 | ffvelrnd 6846 | . . . 4 ⊢ ((((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝐺‘𝑘) ∈ ℝ) |
34 | simpl 485 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝜑 ∧ 𝑁 ∈ 𝑍)) | |
35 | elfzuz 12898 | . . . . 5 ⊢ (𝑘 ∈ (𝑁...(𝑛 − 1)) → 𝑘 ∈ (ℤ≥‘𝑁)) | |
36 | 31 | adantll 712 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
37 | iseralt.4 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘)) | |
38 | 37 | adantlr 713 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘)) |
39 | 36, 38 | syldan 593 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘)) |
40 | 34, 35, 39 | syl2an 597 | . . . 4 ⊢ ((((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) ∧ 𝑘 ∈ (𝑁...(𝑛 − 1))) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘)) |
41 | 27, 33, 40 | monoord2 13395 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑛) ≤ (𝐺‘𝑁)) |
42 | 41, 24 | breqtrrd 5086 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑛) ≤ ((ℤ × {(𝐺‘𝑁)})‘𝑛)) |
43 | 1, 5, 7, 15, 19, 26, 42 | climle 14990 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → 0 ≤ (𝐺‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {csn 4560 class class class wbr 5058 × cxp 5547 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 ≤ cle 10670 − cmin 10864 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 ⇝ cli 14835 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fl 13156 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 |
This theorem is referenced by: iseraltlem3 15034 iseralt 15035 |
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