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Mirrors > Home > MPE Home > Th. List > climub | Structured version Visualization version GIF version |
Description: The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.) |
Ref | Expression |
---|---|
clim2ser.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climub.2 | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
climub.3 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climub.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
climub.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) |
Ref | Expression |
---|---|
climub | ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . 2 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁) | |
2 | climub.2 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
3 | clim2ser.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrdi 2849 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | eluzelz 12886 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | fveq2 6907 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁)) | |
8 | 7 | eleq1d 2824 | . . . . 5 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑁) ∈ ℝ)) |
9 | 8 | imbi2d 340 | . . . 4 ⊢ (𝑘 = 𝑁 → ((𝜑 → (𝐹‘𝑘) ∈ ℝ) ↔ (𝜑 → (𝐹‘𝑁) ∈ ℝ))) |
10 | climub.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
11 | 10 | expcom 413 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → (𝐹‘𝑘) ∈ ℝ)) |
12 | 9, 11 | vtoclga 3577 | . . 3 ⊢ (𝑁 ∈ 𝑍 → (𝜑 → (𝐹‘𝑁) ∈ ℝ)) |
13 | 2, 12 | mpcom 38 | . 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ) |
14 | climub.3 | . 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
15 | 3 | uztrn2 12895 | . . . 4 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑁)) → 𝑗 ∈ 𝑍) |
16 | 2, 15 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑁)) → 𝑗 ∈ 𝑍) |
17 | fveq2 6907 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
18 | 17 | eleq1d 2824 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑗) ∈ ℝ)) |
19 | 18 | imbi2d 340 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 → (𝐹‘𝑘) ∈ ℝ) ↔ (𝜑 → (𝐹‘𝑗) ∈ ℝ))) |
20 | 19, 11 | vtoclga 3577 | . . . 4 ⊢ (𝑗 ∈ 𝑍 → (𝜑 → (𝐹‘𝑗) ∈ ℝ)) |
21 | 20 | impcom 407 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ) |
22 | 16, 21 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑗) ∈ ℝ) |
23 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑁)) → 𝑗 ∈ (ℤ≥‘𝑁)) | |
24 | elfzuz 13557 | . . . . 5 ⊢ (𝑘 ∈ (𝑁...𝑗) → 𝑘 ∈ (ℤ≥‘𝑁)) | |
25 | 3 | uztrn2 12895 | . . . . . . 7 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
26 | 2, 25 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
27 | 26, 10 | syldan 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ) |
28 | 24, 27 | sylan2 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝐹‘𝑘) ∈ ℝ) |
29 | 28 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑁)) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝐹‘𝑘) ∈ ℝ) |
30 | elfzuz 13557 | . . . . 5 ⊢ (𝑘 ∈ (𝑁...(𝑗 − 1)) → 𝑘 ∈ (ℤ≥‘𝑁)) | |
31 | climub.5 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) | |
32 | 26, 31 | syldan 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) |
33 | 30, 32 | sylan2 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑁...(𝑗 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) |
34 | 33 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑁)) ∧ 𝑘 ∈ (𝑁...(𝑗 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) |
35 | 23, 29, 34 | monoord 14070 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑁) ≤ (𝐹‘𝑗)) |
36 | 1, 6, 13, 14, 22, 35 | climlec2 15692 | 1 ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 1c1 11154 + caddc 11156 ≤ cle 11294 − cmin 11490 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 ⇝ 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-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 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-fl 13829 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 |
This theorem is referenced by: climserle 15696 itg2i1fseqle 25804 emcllem7 27060 |
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