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Mirrors > Home > HSE Home > Th. List > lnfncon | 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, 16-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnfncon | ⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2827 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) → (𝑇 ∈ ContFn ↔ if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) ∈ ContFn)) | |
2 | fveq1 6906 | . . . . . 6 ⊢ (𝑇 = if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) → (𝑇‘𝑦) = (if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0}))‘𝑦)) | |
3 | 2 | fveq2d 6911 | . . . . 5 ⊢ (𝑇 = if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) → (abs‘(𝑇‘𝑦)) = (abs‘(if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0}))‘𝑦))) |
4 | 3 | breq1d 5158 | . . . 4 ⊢ (𝑇 = if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) → ((abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ (abs‘(if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0}))‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) |
5 | 4 | rexralbidv 3221 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0}))‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) |
6 | 1, 5 | bibi12d 345 | . 2 ⊢ (𝑇 = if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) → ((𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) ↔ (if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0}))‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))))) |
7 | 0lnfn 32014 | . . . 4 ⊢ ( ℋ × {0}) ∈ LinFn | |
8 | 7 | elimel 4600 | . . 3 ⊢ if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) ∈ LinFn |
9 | 8 | lnfnconi 32084 | . 2 ⊢ (if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0})) ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(if(𝑇 ∈ LinFn, 𝑇, ( ℋ × {0}))‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) |
10 | 6, 9 | dedth 4589 | 1 ⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ifcif 4531 {csn 4631 class class class wbr 5148 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 · cmul 11158 ≤ cle 11294 abscabs 15270 ℋchba 30948 normℎcno 30952 ContFnccnfn 30982 LinFnclf 30983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-hilex 31028 ax-hfvadd 31029 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvmulass 31036 ax-hvmul0 31039 ax-hfi 31108 ax-his1 31111 ax-his3 31113 ax-his4 31114 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-hnorm 30997 df-hvsub 31000 df-nmfn 31874 df-cnfn 31876 df-lnfn 31877 |
This theorem is referenced by: lnfncnbd 32086 riesz1 32094 cnlnadjlem2 32097 |
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