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Mirrors > Home > MPE Home > Th. List > dvfsumrlimge0 | Structured version Visualization version GIF version |
Description: Lemma for dvfsumrlim 25548. Satisfy the assumption of dvfsumlem4 25546. (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 13385 | . . . . . 6 ⊢ (𝑇(,)+∞) ⊆ ℝ | |
3 | 1, 2 | eqsstri 4017 | . . . . 5 ⊢ 𝑆 ⊆ ℝ |
4 | simprl 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ 𝑆) | |
5 | 3, 4 | sselid 3981 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
6 | 5 | rexrd 11264 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ*) |
7 | 5 | renepnfd 11265 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ≠ +∞) |
8 | icopnfsup 13830 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑥 ≠ +∞) → sup((𝑥[,)+∞), ℝ*, < ) = +∞) | |
9 | 6, 7, 8 | syl2anc 585 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → sup((𝑥[,)+∞), ℝ*, < ) = +∞) |
10 | dvfsum.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
11 | 10 | rexrd 11264 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ*) |
12 | 4, 1 | eleqtrdi 2844 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ (𝑇(,)+∞)) |
13 | 11 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑇 ∈ ℝ*) |
14 | elioopnf 13420 | . . . . . . . 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 497 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝑇 < 𝑥) |
18 | df-ioo 13328 | . . . . . 6 ⊢ (,) = (𝑢 ∈ ℝ*, 𝑣 ∈ ℝ* ↦ {𝑤 ∈ ℝ* ∣ (𝑢 < 𝑤 ∧ 𝑤 < 𝑣)}) | |
19 | df-ico 13330 | . . . . . 6 ⊢ [,) = (𝑢 ∈ ℝ*, 𝑣 ∈ ℝ* ↦ {𝑤 ∈ ℝ* ∣ (𝑢 ≤ 𝑤 ∧ 𝑤 < 𝑣)}) | |
20 | xrltletr 13136 | . . . . . 6 ⊢ ((𝑇 ∈ ℝ* ∧ 𝑥 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → ((𝑇 < 𝑥 ∧ 𝑥 ≤ 𝑧) → 𝑇 < 𝑧)) | |
21 | 18, 19, 20 | ixxss1 13342 | . . . . 5 ⊢ ((𝑇 ∈ ℝ* ∧ 𝑇 < 𝑥) → (𝑥[,)+∞) ⊆ (𝑇(,)+∞)) |
22 | 11, 17, 21 | syl2an2r 684 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥[,)+∞) ⊆ (𝑇(,)+∞)) |
23 | 22, 1 | sseqtrrdi 4034 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥[,)+∞) ⊆ 𝑆) |
24 | dvfsum.c | . . . . 5 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) | |
25 | 24 | cbvmptv 5262 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ 𝐵) = (𝑘 ∈ 𝑆 ↦ 𝐶) |
26 | dvfsumrlim.k | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) | |
27 | 26 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) |
28 | 25, 27 | eqbrtrrid 5185 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ 𝑆 ↦ 𝐶) ⇝𝑟 0) |
29 | 23, 28 | rlimres2 15505 | . 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 25541 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ) |
35 | 34 | adantrr 716 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℝ) |
36 | 35 | recnd 11242 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 𝐵 ∈ ℂ) |
37 | rlimconst 15488 | . . . 4 ⊢ ((𝑆 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑘 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 𝐵) | |
38 | 30, 36, 37 | syl2an2r 684 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 𝐵) |
39 | 23, 38 | rlimres2 15505 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → (𝑘 ∈ (𝑥[,)+∞) ↦ 𝐵) ⇝𝑟 𝐵) |
40 | 34 | ralrimiva 3147 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ) |
41 | 40 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → ∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ) |
42 | 23 | sselda 3983 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑘 ∈ 𝑆) |
43 | 24 | eleq1d 2819 | . . . 4 ⊢ (𝑥 = 𝑘 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ)) |
44 | 43 | rspccva 3612 | . . 3 ⊢ ((∀𝑥 ∈ 𝑆 𝐵 ∈ ℝ ∧ 𝑘 ∈ 𝑆) → 𝐶 ∈ ℝ) |
45 | 41, 42, 44 | syl2an2r 684 | . 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐶 ∈ ℝ) |
46 | 35 | adantr 482 | . 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐵 ∈ ℝ) |
47 | simpll 766 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝜑) | |
48 | simplrl 776 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝑥 ∈ 𝑆) | |
49 | simplrr 777 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐷 ≤ 𝑥) | |
50 | elicopnf 13422 | . . . . 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 501 | . . 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 1380 | . 2 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑘 ∈ (𝑥[,)+∞)) → 𝐶 ≤ 𝐵) |
55 | 9, 29, 39, 45, 46, 54 | rlimle 15594 | 1 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 supcsup 9435 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 +∞cpnf 11245 ℝ*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 ℤcz 12558 ℤ≥cuz 12822 (,)cioo 13324 [,)cico 13326 ...cfz 13484 ⌊cfl 13755 ⇝𝑟 crli 15429 Σcsu 15632 D cdv 25380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-rlim 15433 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-topn 17369 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-cncf 24394 df-limc 25383 df-dv 25384 |
This theorem is referenced by: dvfsumrlim 25548 dvfsumrlim2 25549 |
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