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| Mirrors > Home > HSE Home > Th. List > nmopadjlem | Structured version Visualization version GIF version | ||
| Description: Lemma for nmopadji 31918. (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 31911 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → (adjℎ‘𝑇) ∈ BndLinOp) | |
| 3 | bdopf 31690 | . . . 4 ⊢ ((adjℎ‘𝑇) ∈ BndLinOp → (adjℎ‘𝑇): ℋ⟶ ℋ) | |
| 4 | 1, 2, 3 | mp2b 10 | . . 3 ⊢ (adjℎ‘𝑇): ℋ⟶ ℋ |
| 5 | bdopf 31690 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
| 6 | nmopxr 31694 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) ∈ ℝ*) | |
| 7 | 1, 5, 6 | mp2b 10 | . . 3 ⊢ (normop‘𝑇) ∈ ℝ* |
| 8 | nmopub 31736 | . . 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 690 | . 2 ⊢ ((normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) ↔ ∀𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇))) |
| 10 | 4 | ffvelcdmi 7096 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ) |
| 11 | normcl 30953 | . . . . . . 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 479 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ∈ ℝ) |
| 14 | nmopre 31698 | . . . . . . . 8 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | |
| 15 | 1, 14 | ax-mp 5 | . . . . . . 7 ⊢ (normop‘𝑇) ∈ ℝ |
| 16 | normcl 30953 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ) | |
| 17 | remulcl 11229 | . . . . . . 7 ⊢ (((normop‘𝑇) ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → ((normop‘𝑇) · (normℎ‘𝑦)) ∈ ℝ) | |
| 18 | 15, 16, 17 | sylancr 585 | . . . . . 6 ⊢ (𝑦 ∈ ℋ → ((normop‘𝑇) · (normℎ‘𝑦)) ∈ ℝ) |
| 19 | 18 | adantr 479 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ∈ ℝ) |
| 20 | 1re 11250 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
| 21 | 15, 20 | remulcli 11266 | . . . . . 6 ⊢ ((normop‘𝑇) · 1) ∈ ℝ |
| 22 | 21 | a1i 11 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · 1) ∈ ℝ) |
| 23 | 1 | nmopadjlei 31916 | . . . . . 6 ⊢ (𝑦 ∈ ℋ → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ ((normop‘𝑇) · (normℎ‘𝑦))) |
| 24 | 23 | adantr 479 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ ((normop‘𝑇) · (normℎ‘𝑦))) |
| 25 | nmopge0 31739 | . . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normop‘𝑇)) | |
| 26 | 1, 5, 25 | mp2b 10 | . . . . . . . . 9 ⊢ 0 ≤ (normop‘𝑇) |
| 27 | 15, 26 | pm3.2i 469 | . . . . . . . 8 ⊢ ((normop‘𝑇) ∈ ℝ ∧ 0 ≤ (normop‘𝑇)) |
| 28 | lemul2a 12105 | . . . . . . . 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 1453 | . . . . . . 7 ⊢ ((((normℎ‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 30 | 20, 29 | mpanl2 699 | . . . . . 6 ⊢ (((normℎ‘𝑦) ∈ ℝ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 31 | 16, 30 | sylan 578 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 32 | 13, 19, 22, 24, 31 | letrd 11407 | . . . 4 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 33 | 15 | recni 11264 | . . . . 5 ⊢ (normop‘𝑇) ∈ ℂ |
| 34 | 33 | mulridi 11254 | . . . 4 ⊢ ((normop‘𝑇) · 1) = (normop‘𝑇) |
| 35 | 32, 34 | breqtrdi 5191 | . . 3 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇)) |
| 36 | 35 | ex 411 | . 2 ⊢ (𝑦 ∈ ℋ → ((normℎ‘𝑦) ≤ 1 → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇))) |
| 37 | 9, 36 | mprgbir 3064 | 1 ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∀wral 3057 class class class wbr 5150 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 ℝcr 11143 0cc0 11144 1c1 11145 · cmul 11149 ℝ*cxr 11283 ≤ cle 11285 ℋchba 30747 normℎcno 30751 normopcnop 30773 BndLinOpcbo 30776 adjℎcado 30783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cc 10464 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223 ax-mulf 11224 ax-hilex 30827 ax-hfvadd 30828 ax-hvcom 30829 ax-hvass 30830 ax-hv0cl 30831 ax-hvaddid 30832 ax-hfvmul 30833 ax-hvmulid 30834 ax-hvmulass 30835 ax-hvdistr1 30836 ax-hvdistr2 30837 ax-hvmul0 30838 ax-hfi 30907 ax-his1 30910 ax-his2 30911 ax-his3 30912 ax-his4 30913 ax-hcompl 31030 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-oadd 8495 df-omul 8496 df-er 8729 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 df-acn 9971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-ioo 13366 df-ico 13368 df-icc 13369 df-fz 13523 df-fzo 13666 df-fl 13795 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-rlim 15471 df-sum 15671 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-mulg 19029 df-cntz 19273 df-cmn 19742 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-cn 23149 df-cnp 23150 df-lm 23151 df-t1 23236 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cfil 25201 df-cau 25202 df-cmet 25203 df-grpo 30321 df-gid 30322 df-ginv 30323 df-gdiv 30324 df-ablo 30373 df-vc 30387 df-nv 30420 df-va 30423 df-ba 30424 df-sm 30425 df-0v 30426 df-vs 30427 df-nmcv 30428 df-ims 30429 df-dip 30529 df-ssp 30550 df-ph 30641 df-cbn 30691 df-hnorm 30796 df-hba 30797 df-hvsub 30799 df-hlim 30800 df-hcau 30801 df-sh 31035 df-ch 31049 df-oc 31080 df-ch0 31081 df-shs 31136 df-pjh 31223 df-h0op 31576 df-nmop 31667 df-cnop 31668 df-lnop 31669 df-bdop 31670 df-unop 31671 df-hmop 31672 df-nmfn 31673 df-nlfn 31674 df-cnfn 31675 df-lnfn 31676 df-adjh 31677 |
| This theorem is referenced by: nmopadji 31918 |
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