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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 29832 | . . 3 ⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn) → (normfn‘𝑇) ∈ ℝ) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝑇 ∈ ContFn → (normfn‘𝑇) ∈ ℝ) |
4 | nmcfnlb 29833 | . . 3 ⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn ∧ 𝑦 ∈ ℋ) → (abs‘(𝑇‘𝑦)) ≤ ((normfn‘𝑇) · (normℎ‘𝑦))) | |
5 | 1, 4 | mp3an1 1444 | . 2 ⊢ ((𝑇 ∈ ContFn ∧ 𝑦 ∈ ℋ) → (abs‘(𝑇‘𝑦)) ≤ ((normfn‘𝑇) · (normℎ‘𝑦))) |
6 | 1 | lnfnfi 29820 | . . 3 ⊢ 𝑇: ℋ⟶ℂ |
7 | elcnfn 29661 | . . 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 707 | . 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 6852 | . . 3 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℂ) |
10 | 9 | abscld 14798 | . 2 ⊢ (𝑦 ∈ ℋ → (abs‘(𝑇‘𝑦)) ∈ ℝ) |
11 | 1 | lnfnsubi 29825 | . 2 ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤) − (𝑇‘𝑥))) |
12 | 3, 5, 8, 10, 11 | lnconi 29812 | 1 ⊢ (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 · cmul 10544 < clt 10677 ≤ cle 10678 − cmin 10872 ℝ+crp 12392 abscabs 14595 ℋchba 28698 normℎcno 28702 −ℎ cmv 28704 normfncnmf 28730 ContFnccnfn 28732 LinFnclf 28733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-hilex 28778 ax-hfvadd 28779 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his3 28863 ax-his4 28864 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-hnorm 28747 df-hvsub 28750 df-nmfn 29624 df-cnfn 29626 df-lnfn 29627 |
This theorem is referenced by: lnfncon 29835 |
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