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Mirrors > Home > HSE Home > Th. List > leopmul2i | Structured version Visualization version GIF version |
Description: Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
leopmul2i | ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 𝑇 ≤op 𝑈)) → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1167 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → 𝐴 ∈ ℝ) | |
2 | hmopd 29406 | . . . . . . 7 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → (𝑈 −op 𝑇) ∈ HrmOp) | |
3 | 2 | ancoms 451 | . . . . . 6 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑈 −op 𝑇) ∈ HrmOp) |
4 | 3 | 3adant1 1161 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑈 −op 𝑇) ∈ HrmOp) |
5 | leopmuli 29517 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑈 −op 𝑇) ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 0hop ≤op (𝑈 −op 𝑇))) → 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇))) | |
6 | 5 | exp32 412 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝑈 −op 𝑇) ∈ HrmOp) → (0 ≤ 𝐴 → ( 0hop ≤op (𝑈 −op 𝑇) → 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇))))) |
7 | 1, 4, 6 | syl2anc 580 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (0 ≤ 𝐴 → ( 0hop ≤op (𝑈 −op 𝑇) → 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇))))) |
8 | 7 | imp 396 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 0 ≤ 𝐴) → ( 0hop ≤op (𝑈 −op 𝑇) → 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇)))) |
9 | leop3 29509 | . . . . 5 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇))) | |
10 | 9 | 3adant1 1161 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇))) |
11 | 10 | adantr 473 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 0 ≤ 𝐴) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇))) |
12 | hmopm 29405 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → (𝐴 ·op 𝑇) ∈ HrmOp) | |
13 | hmopm 29405 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑈 ∈ HrmOp) → (𝐴 ·op 𝑈) ∈ HrmOp) | |
14 | leop3 29509 | . . . . . . 7 ⊢ (((𝐴 ·op 𝑇) ∈ HrmOp ∧ (𝐴 ·op 𝑈) ∈ HrmOp) → ((𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈) ↔ 0hop ≤op ((𝐴 ·op 𝑈) −op (𝐴 ·op 𝑇)))) | |
15 | 12, 13, 14 | syl2an 590 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ (𝐴 ∈ ℝ ∧ 𝑈 ∈ HrmOp)) → ((𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈) ↔ 0hop ≤op ((𝐴 ·op 𝑈) −op (𝐴 ·op 𝑇)))) |
16 | 15 | 3impdi 1460 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈) ↔ 0hop ≤op ((𝐴 ·op 𝑈) −op (𝐴 ·op 𝑇)))) |
17 | recn 10314 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
18 | hmopf 29258 | . . . . . . . 8 ⊢ (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ) | |
19 | hmopf 29258 | . . . . . . . 8 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) | |
20 | hosubdi 29192 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝑈 −op 𝑇)) = ((𝐴 ·op 𝑈) −op (𝐴 ·op 𝑇))) | |
21 | 17, 18, 19, 20 | syl3an 1200 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑈 ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → (𝐴 ·op (𝑈 −op 𝑇)) = ((𝐴 ·op 𝑈) −op (𝐴 ·op 𝑇))) |
22 | 21 | 3com23 1157 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝐴 ·op (𝑈 −op 𝑇)) = ((𝐴 ·op 𝑈) −op (𝐴 ·op 𝑇))) |
23 | 22 | breq2d 4855 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ( 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇)) ↔ 0hop ≤op ((𝐴 ·op 𝑈) −op (𝐴 ·op 𝑇)))) |
24 | 16, 23 | bitr4d 274 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈) ↔ 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇)))) |
25 | 24 | adantr 473 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 0 ≤ 𝐴) → ((𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈) ↔ 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇)))) |
26 | 8, 11, 25 | 3imtr4d 286 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 0 ≤ 𝐴) → (𝑇 ≤op 𝑈 → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈))) |
27 | 26 | impr 447 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 𝑇 ≤op 𝑈)) → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 class class class wbr 4843 ⟶wf 6097 (class class class)co 6878 ℂcc 10222 ℝcr 10223 0cc0 10224 ≤ cle 10364 ℋchba 28301 ·op chot 28321 −op chod 28322 0hop ch0o 28325 HrmOpcho 28332 ≤op cleo 28340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cc 9545 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304 ax-hilex 28381 ax-hfvadd 28382 ax-hvcom 28383 ax-hvass 28384 ax-hv0cl 28385 ax-hvaddid 28386 ax-hfvmul 28387 ax-hvmulid 28388 ax-hvmulass 28389 ax-hvdistr1 28390 ax-hvdistr2 28391 ax-hvmul0 28392 ax-hfi 28461 ax-his1 28464 ax-his2 28465 ax-his3 28466 ax-his4 28467 ax-hcompl 28584 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-omul 7804 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-acn 9054 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-clim 14560 df-rlim 14561 df-sum 14758 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-hom 16291 df-cco 16292 df-rest 16398 df-topn 16399 df-0g 16417 df-gsum 16418 df-topgen 16419 df-pt 16420 df-prds 16423 df-xrs 16477 df-qtop 16482 df-imas 16483 df-xps 16485 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-mulg 17857 df-cntz 18062 df-cmn 18510 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-fbas 20065 df-fg 20066 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-cld 21152 df-ntr 21153 df-cls 21154 df-nei 21231 df-cn 21360 df-cnp 21361 df-lm 21362 df-haus 21448 df-tx 21694 df-hmeo 21887 df-fil 21978 df-fm 22070 df-flim 22071 df-flf 22072 df-xms 22453 df-ms 22454 df-tms 22455 df-cfil 23381 df-cau 23382 df-cmet 23383 df-grpo 27873 df-gid 27874 df-ginv 27875 df-gdiv 27876 df-ablo 27925 df-vc 27939 df-nv 27972 df-va 27975 df-ba 27976 df-sm 27977 df-0v 27978 df-vs 27979 df-nmcv 27980 df-ims 27981 df-dip 28081 df-ssp 28102 df-ph 28193 df-cbn 28244 df-hnorm 28350 df-hba 28351 df-hvsub 28353 df-hlim 28354 df-hcau 28355 df-sh 28589 df-ch 28603 df-oc 28634 df-ch0 28635 df-shs 28692 df-pjh 28779 df-hosum 29114 df-homul 29115 df-hodif 29116 df-h0op 29132 df-hmop 29228 df-leop 29236 |
This theorem is referenced by: nmopleid 29523 |
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