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Mirrors > Home > HSE Home > Th. List > nmopre | Structured version Visualization version GIF version |
Description: The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmopre | ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdopf 30702 | . . 3 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
2 | nmopgtmnf 30708 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → -∞ < (normop‘𝑇)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑇 ∈ BndLinOp → -∞ < (normop‘𝑇)) |
4 | elbdop 30700 | . . 3 ⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) < +∞)) | |
5 | 4 | simprbi 497 | . 2 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) < +∞) |
6 | nmopxr 30706 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) ∈ ℝ*) | |
7 | xrrebnd 13084 | . . 3 ⊢ ((normop‘𝑇) ∈ ℝ* → ((normop‘𝑇) ∈ ℝ ↔ (-∞ < (normop‘𝑇) ∧ (normop‘𝑇) < +∞))) | |
8 | 1, 6, 7 | 3syl 18 | . 2 ⊢ (𝑇 ∈ BndLinOp → ((normop‘𝑇) ∈ ℝ ↔ (-∞ < (normop‘𝑇) ∧ (normop‘𝑇) < +∞))) |
9 | 3, 5, 8 | mpbir2and 711 | 1 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5104 ⟶wf 6490 ‘cfv 6494 ℝcr 11047 +∞cpnf 11183 -∞cmnf 11184 ℝ*cxr 11185 < clt 11186 ℋchba 29759 normopcnop 29785 LinOpclo 29787 BndLinOpcbo 29788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 ax-hilex 29839 ax-hfvadd 29840 ax-hvcom 29841 ax-hvass 29842 ax-hv0cl 29843 ax-hvaddid 29844 ax-hfvmul 29845 ax-hvmulid 29846 ax-hvmulass 29847 ax-hvdistr1 29848 ax-hvdistr2 29849 ax-hvmul0 29850 ax-hfi 29919 ax-his1 29922 ax-his2 29923 ax-his3 29924 ax-his4 29925 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-map 8764 df-en 8881 df-dom 8882 df-sdom 8883 df-sup 9375 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-n0 12411 df-z 12497 df-uz 12761 df-rp 12913 df-seq 13904 df-exp 13965 df-cj 14981 df-re 14982 df-im 14983 df-sqrt 15117 df-abs 15118 df-grpo 29333 df-gid 29334 df-ablo 29385 df-vc 29399 df-nv 29432 df-va 29435 df-ba 29436 df-sm 29437 df-0v 29438 df-nmcv 29440 df-hnorm 29808 df-hba 29809 df-hvsub 29811 df-nmop 30679 df-lnop 30681 df-bdop 30682 |
This theorem is referenced by: nmbdoplbi 30864 nmophmi 30871 bdophmi 30872 lnopcnbd 30876 nmopadjlem 30929 nmopadji 30930 nmoptrii 30934 nmopcoi 30935 bdophsi 30936 bdopcoi 30938 nmoptri2i 30939 nmopcoadji 30941 nmopcoadj0i 30943 unierri 30944 |
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