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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 1135 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → 𝐴 ∈ ℝ) | |
2 | hmopd 32051 | . . . . . . 7 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → (𝑈 −op 𝑇) ∈ HrmOp) | |
3 | 2 | ancoms 458 | . . . . . 6 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑈 −op 𝑇) ∈ HrmOp) |
4 | 3 | 3adant1 1129 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑈 −op 𝑇) ∈ HrmOp) |
5 | leopmuli 32162 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑈 −op 𝑇) ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 0hop ≤op (𝑈 −op 𝑇))) → 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇))) | |
6 | 5 | exp32 420 | . . . . 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 584 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (0 ≤ 𝐴 → ( 0hop ≤op (𝑈 −op 𝑇) → 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇))))) |
8 | 7 | imp 406 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 0 ≤ 𝐴) → ( 0hop ≤op (𝑈 −op 𝑇) → 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇)))) |
9 | leop3 32154 | . . . . 5 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇))) | |
10 | 9 | 3adant1 1129 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇))) |
11 | 10 | adantr 480 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 0 ≤ 𝐴) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇))) |
12 | hmopm 32050 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → (𝐴 ·op 𝑇) ∈ HrmOp) | |
13 | hmopm 32050 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑈 ∈ HrmOp) → (𝐴 ·op 𝑈) ∈ HrmOp) | |
14 | leop3 32154 | . . . . . . 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 596 | . . . . . 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 1349 | . . . . 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 11243 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
18 | hmopf 31903 | . . . . . . . 8 ⊢ (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ) | |
19 | hmopf 31903 | . . . . . . . 8 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) | |
20 | hosubdi 31837 | . . . . . . . 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 1159 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑈 ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → (𝐴 ·op (𝑈 −op 𝑇)) = ((𝐴 ·op 𝑈) −op (𝐴 ·op 𝑇))) |
22 | 21 | 3com23 1125 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝐴 ·op (𝑈 −op 𝑇)) = ((𝐴 ·op 𝑈) −op (𝐴 ·op 𝑇))) |
23 | 22 | breq2d 5160 | . . . . 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 282 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈) ↔ 0hop ≤op (𝐴 ·op (𝑈 −op 𝑇)))) |
25 | 24 | adantr 480 | . . 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 294 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 0 ≤ 𝐴) → (𝑇 ≤op 𝑈 → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈))) |
27 | 26 | impr 454 | 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ⟶wf 6559 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 ≤ cle 11294 ℋchba 30948 ·op chot 30968 −op chod 30969 0hop ch0o 30972 HrmOpcho 30979 ≤op cleo 30987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 ax-hilex 31028 ax-hfvadd 31029 ax-hvcom 31030 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvmulass 31036 ax-hvdistr1 31037 ax-hvdistr2 31038 ax-hvmul0 31039 ax-hfi 31108 ax-his1 31111 ax-his2 31112 ax-his3 31113 ax-his4 31114 ax-hcompl 31231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-cn 23251 df-cnp 23252 df-lm 23253 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cfil 25303 df-cau 25304 df-cmet 25305 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-ims 30630 df-dip 30730 df-ssp 30751 df-ph 30842 df-cbn 30892 df-hnorm 30997 df-hba 30998 df-hvsub 31000 df-hlim 31001 df-hcau 31002 df-sh 31236 df-ch 31250 df-oc 31281 df-ch0 31282 df-shs 31337 df-pjh 31424 df-hosum 31759 df-homul 31760 df-hodif 31761 df-h0op 31777 df-hmop 31873 df-leop 31881 |
This theorem is referenced by: nmopleid 32168 |
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