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Mirrors > Home > MPE Home > Th. List > dvfsumrlimge0 | Structured version Visualization version GIF version |
Description: Lemma for dvfsumrlim 25547. Satisfy the assumption of dvfsumlem4 25545. (Contributed by Mario Carneiro, 18-May-2016.) |
Ref | Expression |
---|---|
dvfsum.s | ⊢ 𝑆 = (𝑇(,)+∞) |
dvfsum.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
dvfsum.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
dvfsum.d | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
dvfsum.md | ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) |
dvfsum.t | ⊢ (𝜑 → 𝑇 ∈ ℝ) |
dvfsum.a | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) |
dvfsum.b1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) |
dvfsum.b2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) |
dvfsum.b3 | ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) |
dvfsum.c | ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) |
dvfsumrlim.l | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵) |
dvfsumrlim.g | ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) |
dvfsumrlim.k | ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) |
Ref | Expression |
---|---|
dvfsumrlimge0 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvfsum.s | . . . . . 6 ⊢ 𝑆 = (𝑇(,)+∞) | |
2 | ioossre 13384 | . . . . . 6 ⊢ (𝑇(,)+∞) ⊆ ℝ | |
3 | 1, 2 | eqsstri 4016 | . . . . 5 ⊢ 𝑆 ⊆ ℝ |
4 | simprl 769 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ 𝑆) | |
5 | 3, 4 | sselid 3980 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
6 | 5 | rexrd 11263 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ*) |
7 | 5 | renepnfd 11264 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ≠ +∞) |
8 | icopnfsup 13829 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑥 ≠ +∞) → sup((𝑥[,)+∞), ℝ*, < ) = +∞) | |
9 | 6, 7, 8 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → sup((𝑥[,)+∞), ℝ*, < ) = +∞) |
10 | dvfsum.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
11 | 10 | rexrd 11263 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ*) |
12 | 4, 1 | eleqtrdi 2843 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ (𝑇(,)+∞)) |
13 | 11 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑇 ∈ ℝ*) |
14 | elioopnf 13419 | . . . . . . . 8 ⊢ (𝑇 ∈ ℝ* → (𝑥 ∈ (𝑇(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑇 < 𝑥))) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ (𝑇(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑇 < 𝑥))) |
16 | 12, 15 | mpbid 231 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 𝑇 < 𝑥)) |
17 | 16 | simprd 496 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑇 < 𝑥) |
18 | df-ioo 13327 | . . . . . 6 ⊢ (,) = (𝑢 ∈ ℝ*, 𝑣 ∈ ℝ* ↦ {𝑤 ∈ ℝ* ∣ (𝑢 < 𝑤 ∧ 𝑤 < 𝑣)}) | |
19 | df-ico 13329 | . . . . . 6 ⊢ [,) = (𝑢 ∈ ℝ*, 𝑣 ∈ ℝ* ↦ {𝑤 ∈ ℝ* ∣ (𝑢 ≤ 𝑤 ∧ 𝑤 < 𝑣)}) | |
20 | xrltletr 13135 | . . . . . 6 ⊢ ((𝑇 ∈ ℝ* ∧ 𝑥 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑇 < 𝑥 ∧ 𝑥 ≤ 𝑧) → 𝑇 < 𝑧)) | |
21 | 18, 19, 20 | ixxss1 13341 | . . . . 5 ⊢ ((𝑇 ∈ ℝ* ∧ 𝑇 < 𝑥) → (𝑥[,)+∞) ⊆ (𝑇(,)+∞)) |
22 | 11, 17, 21 | syl2an2r 683 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥[,)+∞) ⊆ (𝑇(,)+∞)) |
23 | 22, 1 | sseqtrrdi 4033 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥[,)+∞) ⊆ 𝑆) |
24 | dvfsum.c | . . . . 5 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) | |
25 | 24 | cbvmptv 5261 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ 𝐵) = (𝑘 ∈ 𝑆 ↦ 𝐶) |
26 | dvfsumrlim.k | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) | |
27 | 26 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) |
28 | 25, 27 | eqbrtrrid 5184 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ 𝑆 ↦ 𝐶) ⇝𝑟 0) |
29 | 23, 28 | rlimres2 15504 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↦ 𝐶) ⇝𝑟 0) |
30 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ℝ) |
31 | dvfsum.a | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) | |
32 | dvfsum.b1 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) | |
33 | dvfsum.b3 | . . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) | |
34 | 30, 31, 32, 33 | dvmptrecl 25540 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ) |
35 | 34 | adantrr 715 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℝ) |
36 | 35 | recnd 11241 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℂ) |
37 | rlimconst 15487 | . . . 4 ⊢ ((𝑆 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑘 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 𝐵) | |
38 | 30, 36, 37 | syl2an2r 683 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 𝐵) |
39 | 23, 38 | rlimres2 15504 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↦ 𝐵) ⇝𝑟 𝐵) |
40 | 34 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ) |
41 | 40 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ) |
42 | 23 | sselda 3982 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑘 ∈ 𝑆) |
43 | 24 | eleq1d 2818 | . . . 4 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ)) |
44 | 43 | rspccva 3611 | . . 3 ⊢ ((∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ ∧ 𝑘 ∈ 𝑆) → 𝐶 ∈ ℝ) |
45 | 41, 42, 44 | syl2an2r 683 | . 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐶 ∈ ℝ) |
46 | 35 | adantr 481 | . 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐵 ∈ ℝ) |
47 | simpll 765 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝜑) | |
48 | simplrl 775 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑥 ∈ 𝑆) | |
49 | simplrr 776 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐷 ≤ 𝑥) | |
50 | elicopnf 13421 | . . . . 5 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ (𝑥[,)+∞) ↔ (𝑘 ∈ ℝ ∧ 𝑥 ≤ 𝑘))) | |
51 | 5, 50 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↔ (𝑘 ∈ ℝ ∧ 𝑥 ≤ 𝑘))) |
52 | 51 | simplbda 500 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑥 ≤ 𝑘) |
53 | dvfsumrlim.l | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵) | |
54 | 47, 48, 42, 49, 52, 53 | syl122anc 1379 | . 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐶 ≤ 𝐵) |
55 | 9, 29, 39, 45, 46, 54 | rlimle 15593 | 1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3948 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7408 supcsup 9434 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 +∞cpnf 11244 ℝ*cxr 11246 < clt 11247 ≤ cle 11248 − cmin 11443 ℤcz 12557 ℤ≥cuz 12821 (,)cioo 13323 [,)cico 13325 ...cfz 13483 ⌊cfl 13754 ⇝𝑟 crli 15428 Σcsu 15631 D cdv 25379 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ico 13329 df-icc 13330 df-fz 13484 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-rlim 15432 df-struct 17079 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17367 df-topn 17368 df-topgen 17388 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-cncf 24393 df-limc 25382 df-dv 25383 |
This theorem is referenced by: dvfsumrlim 25547 dvfsumrlim2 25548 |
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