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| Mirrors > Home > HSE Home > Th. List > nmopadjlem | Structured version Visualization version GIF version | ||
| Description: Lemma for nmopadji 32180. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nmopadjle.1 | ⊢ 𝑇 ∈ BndLinOp |
| Ref | Expression |
|---|---|
| nmopadjlem | ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nmopadjle.1 | . . . 4 ⊢ 𝑇 ∈ BndLinOp | |
| 2 | adjbdln 32173 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → (adjℎ‘𝑇) ∈ BndLinOp) | |
| 3 | bdopf 31952 | . . . 4 ⊢ ((adjℎ‘𝑇) ∈ BndLinOp → (adjℎ‘𝑇): ℋ⟶ ℋ) | |
| 4 | 1, 2, 3 | mp2b 10 | . . 3 ⊢ (adjℎ‘𝑇): ℋ⟶ ℋ |
| 5 | bdopf 31952 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
| 6 | nmopxr 31956 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) ∈ ℝ*) | |
| 7 | 1, 5, 6 | mp2b 10 | . . 3 ⊢ (normop‘𝑇) ∈ ℝ* |
| 8 | nmopub 31998 | . . 3 ⊢ (((adjℎ‘𝑇): ℋ⟶ ℋ ∧ (normop‘𝑇) ∈ ℝ*) → ((normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) ↔ ∀𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇)))) | |
| 9 | 4, 7, 8 | mp2an 698 | . 2 ⊢ ((normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) ↔ ∀𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇))) |
| 10 | 4 | ffvelcdmi 7025 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ) |
| 11 | normcl 31215 | . . . . . . 7 ⊢ (((adjℎ‘𝑇)‘𝑦) ∈ ℋ → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ∈ ℝ) | |
| 12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ ℋ → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ∈ ℝ) |
| 13 | 12 | adantr 481 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ∈ ℝ) |
| 14 | nmopre 31960 | . . . . . . . 8 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | |
| 15 | 1, 14 | ax-mp 5 | . . . . . . 7 ⊢ (normop‘𝑇) ∈ ℝ |
| 16 | normcl 31215 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ) | |
| 17 | remulcl 11115 | . . . . . . 7 ⊢ (((normop‘𝑇) ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → ((normop‘𝑇) · (normℎ‘𝑦)) ∈ ℝ) | |
| 18 | 15, 16, 17 | sylancr 593 | . . . . . 6 ⊢ (𝑦 ∈ ℋ → ((normop‘𝑇) · (normℎ‘𝑦)) ∈ ℝ) |
| 19 | 18 | adantr 481 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ∈ ℝ) |
| 20 | 1re 11136 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
| 21 | 15, 20 | remulcli 11153 | . . . . . 6 ⊢ ((normop‘𝑇) · 1) ∈ ℝ |
| 22 | 21 | a1i 11 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · 1) ∈ ℝ) |
| 23 | 1 | nmopadjlei 32178 | . . . . . 6 ⊢ (𝑦 ∈ ℋ → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ ((normop‘𝑇) · (normℎ‘𝑦))) |
| 24 | 23 | adantr 481 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ ((normop‘𝑇) · (normℎ‘𝑦))) |
| 25 | nmopge0 32001 | . . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normop‘𝑇)) | |
| 26 | 1, 5, 25 | mp2b 10 | . . . . . . . . 9 ⊢ 0 ≤ (normop‘𝑇) |
| 27 | 15, 26 | pm3.2i 471 | . . . . . . . 8 ⊢ ((normop‘𝑇) ∈ ℝ ∧ 0 ≤ (normop‘𝑇)) |
| 28 | lemul2a 12002 | . . . . . . . 8 ⊢ ((((normℎ‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((normop‘𝑇) ∈ ℝ ∧ 0 ≤ (normop‘𝑇))) ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) | |
| 29 | 27, 28 | mp3anl3 1465 | . . . . . . 7 ⊢ ((((normℎ‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 30 | 20, 29 | mpanl2 707 | . . . . . 6 ⊢ (((normℎ‘𝑦) ∈ ℝ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 31 | 16, 30 | sylan 586 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 32 | 13, 19, 22, 24, 31 | letrd 11295 | . . . 4 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 33 | 15 | recni 11151 | . . . . 5 ⊢ (normop‘𝑇) ∈ ℂ |
| 34 | 33 | mulridi 11141 | . . . 4 ⊢ ((normop‘𝑇) · 1) = (normop‘𝑇) |
| 35 | 32, 34 | breqtrdi 5114 | . . 3 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇)) |
| 36 | 35 | ex 413 | . 2 ⊢ (𝑦 ∈ ℋ → ((normℎ‘𝑦) ≤ 1 → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇))) |
| 37 | 9, 36 | mprgbir 3060 | 1 ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2119 ∀wral 3053 class class class wbr 5073 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ℝcr 11029 0cc0 11030 1c1 11031 · cmul 11035 ℝ*cxr 11170 ≤ cle 11172 ℋchba 31009 normℎcno 31013 normopcnop 31035 BndLinOpcbo 31038 adjℎcado 31045 |
| 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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110 ax-hilex 31089 ax-hfvadd 31090 ax-hvcom 31091 ax-hvass 31092 ax-hv0cl 31093 ax-hvaddid 31094 ax-hfvmul 31095 ax-hvmulid 31096 ax-hvmulass 31097 ax-hvdistr1 31098 ax-hvdistr2 31099 ax-hvmul0 31100 ax-hfi 31169 ax-his1 31172 ax-his2 31173 ax-his3 31174 ax-his4 31175 ax-hcompl 31292 |
| This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9855 df-acn 9858 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ioo 13294 df-ico 13296 df-icc 13297 df-fz 13454 df-fzo 13601 df-fl 13743 df-seq 13956 df-exp 14016 df-hash 14285 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15442 df-rlim 15443 df-sum 15641 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-ntr 23004 df-cls 23005 df-nei 23082 df-cn 23211 df-cnp 23212 df-lm 23213 df-t1 23298 df-haus 23299 df-tx 23546 df-hmeo 23739 df-fil 23830 df-fm 23922 df-flim 23923 df-flf 23924 df-xms 24304 df-ms 24305 df-tms 24306 df-cfil 25241 df-cau 25242 df-cmet 25243 df-grpo 30583 df-gid 30584 df-ginv 30585 df-gdiv 30586 df-ablo 30635 df-vc 30649 df-nv 30682 df-va 30685 df-ba 30686 df-sm 30687 df-0v 30688 df-vs 30689 df-nmcv 30690 df-ims 30691 df-dip 30791 df-ssp 30812 df-ph 30903 df-cbn 30953 df-hnorm 31058 df-hba 31059 df-hvsub 31061 df-hlim 31062 df-hcau 31063 df-sh 31297 df-ch 31311 df-oc 31342 df-ch0 31343 df-shs 31398 df-pjh 31485 df-h0op 31838 df-nmop 31929 df-cnop 31930 df-lnop 31931 df-bdop 31932 df-unop 31933 df-hmop 31934 df-nmfn 31935 df-nlfn 31936 df-cnfn 31937 df-lnfn 31938 df-adjh 31939 |
| This theorem is referenced by: nmopadji 32180 |
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