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Mirrors > Home > MPE Home > Th. List > dvfsumrlim3 | Structured version Visualization version GIF version |
Description: Conjoin the statements of dvfsumrlim 23598 and dvfsumrlim2 23599. (This is useful as a target for lemmas, because the hypotheses to this theorem are complex, and we don't want to repeat ourselves.) (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) |
dvfsumrlim3.1 | ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐸) |
Ref | Expression |
---|---|
dvfsumrlim3 | ⊢ (𝜑 → (𝐺:𝑆⟶ℝ ∧ 𝐺 ∈ dom ⇝𝑟 ∧ ((𝐺 ⇝𝑟 𝐿 ∧ 𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvfsum.s | . . 3 ⊢ 𝑆 = (𝑇(,)+∞) | |
2 | dvfsum.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | dvfsum.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | dvfsum.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
5 | dvfsum.md | . . 3 ⊢ (𝜑 → 𝑀 ≤ (𝐷 + 1)) | |
6 | dvfsum.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
7 | dvfsum.a | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) | |
8 | dvfsum.b1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) | |
9 | dvfsum.b2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) | |
10 | dvfsum.b3 | . . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) | |
11 | dvfsum.c | . . 3 ⊢ (𝑥 = 𝑘 → 𝐵 = 𝐶) | |
12 | dvfsumrlim.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝑆 ↦ (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | dvfsumrlimf 23592 | . 2 ⊢ (𝜑 → 𝐺:𝑆⟶ℝ) |
14 | dvfsumrlim.l | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵) | |
15 | dvfsumrlim.k | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 12, 15 | dvfsumrlim 23598 | . 2 ⊢ (𝜑 → 𝐺 ∈ dom ⇝𝑟 ) |
17 | 3 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝑀 ∈ ℤ) |
18 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝐷 ∈ ℝ) |
19 | 5 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝑀 ≤ (𝐷 + 1)) |
20 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝑇 ∈ ℝ) |
21 | 7 | adantlr 747 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) |
22 | 8 | adantlr 747 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉) |
23 | 9 | adantlr 747 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ) |
24 | 10 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵)) |
25 | 14 | 3adant1r 1311 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘)) → 𝐶 ≤ 𝐵) |
26 | 15 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → (𝑥 ∈ 𝑆 ↦ 𝐵) ⇝𝑟 0) |
27 | simprr 792 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝑋 ∈ 𝑆) | |
28 | simprl 790 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) → 𝐷 ≤ 𝑋) | |
29 | 1, 2, 17, 18, 19, 20, 21, 22, 23, 24, 11, 25, 12, 26, 27, 28 | dvfsumrlim2 23599 | . . . . . 6 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝐺 ⇝𝑟 𝐿) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ ⦋𝑋 / 𝑥⦌𝐵) |
30 | 27 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝐺 ⇝𝑟 𝐿) → 𝑋 ∈ 𝑆) |
31 | nfcvd 2752 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → Ⅎ𝑥𝐸) | |
32 | dvfsumrlim3.1 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐸) | |
33 | 31, 32 | csbiegf 3523 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → ⦋𝑋 / 𝑥⦌𝐵 = 𝐸) |
34 | 30, 33 | syl 17 | . . . . . 6 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝐺 ⇝𝑟 𝐿) → ⦋𝑋 / 𝑥⦌𝐵 = 𝐸) |
35 | 29, 34 | breqtrd 4609 | . . . . 5 ⊢ (((𝜑 ∧ (𝐷 ≤ 𝑋 ∧ 𝑋 ∈ 𝑆)) ∧ 𝐺 ⇝𝑟 𝐿) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸) |
36 | 35 | exp42 637 | . . . 4 ⊢ (𝜑 → (𝐷 ≤ 𝑋 → (𝑋 ∈ 𝑆 → (𝐺 ⇝𝑟 𝐿 → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸)))) |
37 | 36 | com24 93 | . . 3 ⊢ (𝜑 → (𝐺 ⇝𝑟 𝐿 → (𝑋 ∈ 𝑆 → (𝐷 ≤ 𝑋 → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸)))) |
38 | 37 | 3impd 1273 | . 2 ⊢ (𝜑 → ((𝐺 ⇝𝑟 𝐿 ∧ 𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸)) |
39 | 13, 16, 38 | 3jca 1235 | 1 ⊢ (𝜑 → (𝐺:𝑆⟶ℝ ∧ 𝐺 ∈ dom ⇝𝑟 ∧ ((𝐺 ⇝𝑟 𝐿 ∧ 𝑋 ∈ 𝑆 ∧ 𝐷 ≤ 𝑋) → (abs‘((𝐺‘𝑋) − 𝐿)) ≤ 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⦋csb 3499 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 +∞cpnf 9950 ≤ cle 9954 − cmin 10145 ℤcz 11254 ℤ≥cuz 11563 (,)cioo 12046 ...cfz 12197 ⌊cfl 12453 abscabs 13822 ⇝𝑟 crli 14064 Σcsu 14264 D cdv 23433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 |
This theorem is referenced by: divsqrtsumlem 24506 logdivsum 25022 |
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