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Mirrors > Home > HSE Home > Th. List > lnopcon | 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.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopcon | ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2718 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) → (𝑇 ∈ ContOp ↔ if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) ∈ ContOp)) | |
2 | fveq1 6228 | . . . . . 6 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) → (𝑇‘𝑦) = (if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ))‘𝑦)) | |
3 | 2 | fveq2d 6233 | . . . . 5 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘(if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ))‘𝑦))) |
4 | 3 | breq1d 4695 | . . . 4 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) → ((normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ (normℎ‘(if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ))‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) |
5 | 4 | rexralbidv 3087 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ))‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) |
6 | 1, 5 | bibi12d 334 | . 2 ⊢ (𝑇 = if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) → ((𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) ↔ (if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ))‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))))) |
7 | idlnop 28979 | . . . 4 ⊢ ( I ↾ ℋ) ∈ LinOp | |
8 | 7 | elimel 4183 | . . 3 ⊢ if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) ∈ LinOp |
9 | 8 | lnopconi 29021 | . 2 ⊢ (if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ)) ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(if(𝑇 ∈ LinOp, 𝑇, ( I ↾ ℋ))‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) |
10 | 6, 9 | dedth 4172 | 1 ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ∃wrex 2942 ifcif 4119 class class class wbr 4685 I cid 5052 ↾ cres 5145 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 · cmul 9979 ≤ cle 10113 ℋchil 27904 normℎcno 27908 ContOpccop 27931 LinOpclo 27932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-hilex 27984 ax-hfvadd 27985 ax-hvcom 27986 ax-hvass 27987 ax-hv0cl 27988 ax-hvaddid 27989 ax-hfvmul 27990 ax-hvmulid 27991 ax-hvmulass 27992 ax-hvdistr1 27993 ax-hvdistr2 27994 ax-hvmul0 27995 ax-hfi 28064 ax-his1 28067 ax-his2 28068 ax-his3 28069 ax-his4 28070 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-grpo 27475 df-gid 27476 df-ablo 27527 df-vc 27542 df-nv 27575 df-va 27578 df-ba 27579 df-sm 27580 df-0v 27581 df-nmcv 27583 df-hnorm 27953 df-hba 27954 df-hvsub 27956 df-nmop 28826 df-cnop 28827 df-lnop 28828 df-unop 28830 |
This theorem is referenced by: lnopcnbd 29023 |
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