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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 30411 | . . 3 ⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn) → (normfn‘𝑇) ∈ ℝ) | |
3 | 1, 2 | mpan 687 | . 2 ⊢ (𝑇 ∈ ContFn → (normfn‘𝑇) ∈ ℝ) |
4 | nmcfnlb 30412 | . . 3 ⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn ∧ 𝑦 ∈ ℋ) → (abs‘(𝑇‘𝑦)) ≤ ((normfn‘𝑇) · (normℎ‘𝑦))) | |
5 | 1, 4 | mp3an1 1447 | . 2 ⊢ ((𝑇 ∈ ContFn ∧ 𝑦 ∈ ℋ) → (abs‘(𝑇‘𝑦)) ≤ ((normfn‘𝑇) · (normℎ‘𝑦))) |
6 | 1 | lnfnfi 30399 | . . 3 ⊢ 𝑇: ℋ⟶ℂ |
7 | elcnfn 30240 | . . 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 706 | . 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 6957 | . . 3 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℂ) |
10 | 9 | abscld 15146 | . 2 ⊢ (𝑦 ∈ ℋ → (abs‘(𝑇‘𝑦)) ∈ ℝ) |
11 | 1 | lnfnsubi 30404 | . 2 ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤) − (𝑇‘𝑥))) |
12 | 3, 5, 8, 10, 11 | lnconi 30391 | 1 ⊢ (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 class class class wbr 5079 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 ℂcc 10870 ℝcr 10871 · cmul 10877 < clt 11010 ≤ cle 11011 − cmin 11205 ℝ+crp 12729 abscabs 14943 ℋchba 29277 normℎcno 29281 −ℎ cmv 29283 normfncnmf 29309 ContFnccnfn 29311 LinFnclf 29312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-hilex 29357 ax-hfvadd 29358 ax-hv0cl 29361 ax-hvaddid 29362 ax-hfvmul 29363 ax-hvmulid 29364 ax-hvmulass 29365 ax-hvmul0 29368 ax-hfi 29437 ax-his1 29440 ax-his3 29442 ax-his4 29443 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-sup 9179 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12582 df-rp 12730 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-hnorm 29326 df-hvsub 29329 df-nmfn 30203 df-cnfn 30205 df-lnfn 30206 |
This theorem is referenced by: lnfncon 30414 |
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