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| Mirrors > Home > HSE Home > Th. List > nmopadjlem | Structured version Visualization version GIF version | ||
| Description: Lemma for nmopadji 32109. (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 32102 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → (adjℎ‘𝑇) ∈ BndLinOp) | |
| 3 | bdopf 31881 | . . . 4 ⊢ ((adjℎ‘𝑇) ∈ BndLinOp → (adjℎ‘𝑇): ℋ⟶ ℋ) | |
| 4 | 1, 2, 3 | mp2b 10 | . . 3 ⊢ (adjℎ‘𝑇): ℋ⟶ ℋ |
| 5 | bdopf 31881 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
| 6 | nmopxr 31885 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) ∈ ℝ*) | |
| 7 | 1, 5, 6 | mp2b 10 | . . 3 ⊢ (normop‘𝑇) ∈ ℝ* |
| 8 | nmopub 31927 | . . 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 692 | . 2 ⊢ ((normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) ↔ ∀𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇))) |
| 10 | 4 | ffvelcdmi 7103 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ) |
| 11 | normcl 31144 | . . . . . . 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 480 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ∈ ℝ) |
| 14 | nmopre 31889 | . . . . . . . 8 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | |
| 15 | 1, 14 | ax-mp 5 | . . . . . . 7 ⊢ (normop‘𝑇) ∈ ℝ |
| 16 | normcl 31144 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ) | |
| 17 | remulcl 11240 | . . . . . . 7 ⊢ (((normop‘𝑇) ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → ((normop‘𝑇) · (normℎ‘𝑦)) ∈ ℝ) | |
| 18 | 15, 16, 17 | sylancr 587 | . . . . . 6 ⊢ (𝑦 ∈ ℋ → ((normop‘𝑇) · (normℎ‘𝑦)) ∈ ℝ) |
| 19 | 18 | adantr 480 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ∈ ℝ) |
| 20 | 1re 11261 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
| 21 | 15, 20 | remulcli 11277 | . . . . . 6 ⊢ ((normop‘𝑇) · 1) ∈ ℝ |
| 22 | 21 | a1i 11 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · 1) ∈ ℝ) |
| 23 | 1 | nmopadjlei 32107 | . . . . . 6 ⊢ (𝑦 ∈ ℋ → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ ((normop‘𝑇) · (normℎ‘𝑦))) |
| 24 | 23 | adantr 480 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ ((normop‘𝑇) · (normℎ‘𝑦))) |
| 25 | nmopge0 31930 | . . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normop‘𝑇)) | |
| 26 | 1, 5, 25 | mp2b 10 | . . . . . . . . 9 ⊢ 0 ≤ (normop‘𝑇) |
| 27 | 15, 26 | pm3.2i 470 | . . . . . . . 8 ⊢ ((normop‘𝑇) ∈ ℝ ∧ 0 ≤ (normop‘𝑇)) |
| 28 | lemul2a 12122 | . . . . . . . 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 1459 | . . . . . . 7 ⊢ ((((normℎ‘𝑦) ∈ ℝ ∧ 1 ∈ ℝ) ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 30 | 20, 29 | mpanl2 701 | . . . . . 6 ⊢ (((normℎ‘𝑦) ∈ ℝ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 31 | 16, 30 | sylan 580 | . . . . 5 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → ((normop‘𝑇) · (normℎ‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 32 | 13, 19, 22, 24, 31 | letrd 11418 | . . . 4 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ ((normop‘𝑇) · 1)) |
| 33 | 15 | recni 11275 | . . . . 5 ⊢ (normop‘𝑇) ∈ ℂ |
| 34 | 33 | mulridi 11265 | . . . 4 ⊢ ((normop‘𝑇) · 1) = (normop‘𝑇) |
| 35 | 32, 34 | breqtrdi 5184 | . . 3 ⊢ ((𝑦 ∈ ℋ ∧ (normℎ‘𝑦) ≤ 1) → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇)) |
| 36 | 35 | ex 412 | . 2 ⊢ (𝑦 ∈ ℋ → ((normℎ‘𝑦) ≤ 1 → (normℎ‘((adjℎ‘𝑇)‘𝑦)) ≤ (normop‘𝑇))) |
| 37 | 9, 36 | mprgbir 3068 | 1 ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 ℝ*cxr 11294 ≤ cle 11296 ℋchba 30938 normℎcno 30942 normopcnop 30964 BndLinOpcbo 30967 adjℎcado 30974 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvmulass 31026 ax-hvdistr1 31027 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his1 31101 ax-his2 31102 ax-his3 31103 ax-his4 31104 ax-hcompl 31221 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-cnp 23236 df-lm 23237 df-t1 23322 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cfil 25289 df-cau 25290 df-cmet 25291 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620 df-dip 30720 df-ssp 30741 df-ph 30832 df-cbn 30882 df-hnorm 30987 df-hba 30988 df-hvsub 30990 df-hlim 30991 df-hcau 30992 df-sh 31226 df-ch 31240 df-oc 31271 df-ch0 31272 df-shs 31327 df-pjh 31414 df-h0op 31767 df-nmop 31858 df-cnop 31859 df-lnop 31860 df-bdop 31861 df-unop 31862 df-hmop 31863 df-nmfn 31864 df-nlfn 31865 df-cnfn 31866 df-lnfn 31867 df-adjh 31868 |
| This theorem is referenced by: nmopadji 32109 |
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