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Mirrors > Home > HSE Home > Th. List > lnfnconi | Structured version Visualization version GIF version |
Description: A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnfncon.1 | ⊢ 𝑇 ∈ LinFn |
Ref | Expression |
---|---|
lnfnconi | ⊢ (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnfncon.1 | . . 3 ⊢ 𝑇 ∈ LinFn | |
2 | nmcfnex 30088 | . . 3 ⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn) → (normfn‘𝑇) ∈ ℝ) | |
3 | 1, 2 | mpan 690 | . 2 ⊢ (𝑇 ∈ ContFn → (normfn‘𝑇) ∈ ℝ) |
4 | nmcfnlb 30089 | . . 3 ⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn ∧ 𝑦 ∈ ℋ) → (abs‘(𝑇‘𝑦)) ≤ ((normfn‘𝑇) · (normℎ‘𝑦))) | |
5 | 1, 4 | mp3an1 1450 | . 2 ⊢ ((𝑇 ∈ ContFn ∧ 𝑦 ∈ ℋ) → (abs‘(𝑇‘𝑦)) ≤ ((normfn‘𝑇) · (normℎ‘𝑦))) |
6 | 1 | lnfnfi 30076 | . . 3 ⊢ 𝑇: ℋ⟶ℂ |
7 | elcnfn 29917 | . . 3 ⊢ (𝑇 ∈ ContFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (abs‘((𝑇‘𝑤) − (𝑇‘𝑥))) < 𝑧))) | |
8 | 6, 7 | mpbiran 709 | . 2 ⊢ (𝑇 ∈ ContFn ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (abs‘((𝑇‘𝑤) − (𝑇‘𝑥))) < 𝑧)) |
9 | 6 | ffvelrni 6881 | . . 3 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℂ) |
10 | 9 | abscld 14965 | . 2 ⊢ (𝑦 ∈ ℋ → (abs‘(𝑇‘𝑦)) ∈ ℝ) |
11 | 1 | lnfnsubi 30081 | . 2 ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤) − (𝑇‘𝑥))) |
12 | 3, 5, 8, 10, 11 | lnconi 30068 | 1 ⊢ (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 class class class wbr 5039 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 ℝcr 10693 · cmul 10699 < clt 10832 ≤ cle 10833 − cmin 11027 ℝ+crp 12551 abscabs 14762 ℋchba 28954 normℎcno 28958 −ℎ cmv 28960 normfncnmf 28986 ContFnccnfn 28988 LinFnclf 28989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-hilex 29034 ax-hfvadd 29035 ax-hv0cl 29038 ax-hvaddid 29039 ax-hfvmul 29040 ax-hvmulid 29041 ax-hvmulass 29042 ax-hvmul0 29045 ax-hfi 29114 ax-his1 29117 ax-his3 29119 ax-his4 29120 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-hnorm 29003 df-hvsub 29006 df-nmfn 29880 df-cnfn 29882 df-lnfn 29883 |
This theorem is referenced by: lnfncon 30091 |
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