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Mirrors > Home > HSE Home > Th. List > leopmuli | Structured version Visualization version GIF version |
Description: The scalar product of a nonnegative real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
leopmuli | ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 0hop ≤op 𝑇)) → 0hop ≤op (𝐴 ·op 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmopre 29700 | . . . . . . . . . 10 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑥) ∈ ℝ) | |
2 | mulge0 11158 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (((𝑇‘𝑥) ·ih 𝑥) ∈ ℝ ∧ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥))) → 0 ≤ (𝐴 · ((𝑇‘𝑥) ·ih 𝑥))) | |
3 | 1, 2 | sylanr1 680 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) ∧ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥))) → 0 ≤ (𝐴 · ((𝑇‘𝑥) ·ih 𝑥))) |
4 | 3 | expr 459 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ)) → (0 ≤ ((𝑇‘𝑥) ·ih 𝑥) → 0 ≤ (𝐴 · ((𝑇‘𝑥) ·ih 𝑥)))) |
5 | 4 | an4s 658 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 𝑥 ∈ ℋ)) → (0 ≤ ((𝑇‘𝑥) ·ih 𝑥) → 0 ≤ (𝐴 · ((𝑇‘𝑥) ·ih 𝑥)))) |
6 | 5 | anassrs 470 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℋ) → (0 ≤ ((𝑇‘𝑥) ·ih 𝑥) → 0 ≤ (𝐴 · ((𝑇‘𝑥) ·ih 𝑥)))) |
7 | recn 10627 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
8 | hmopf 29651 | . . . . . . . . . 10 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) | |
9 | 7, 8 | anim12i 614 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → (𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) |
10 | homval 29518 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥))) | |
11 | 10 | 3expa 1114 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥))) |
12 | 11 | oveq1d 7171 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥) = ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑥)) |
13 | simpll 765 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℂ) | |
14 | ffvelrn 6849 | . . . . . . . . . . . 12 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ) | |
15 | 14 | adantll 712 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ) |
16 | simpr 487 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → 𝑥 ∈ ℋ) | |
17 | ax-his3 28861 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑥) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑥))) | |
18 | 13, 15, 16, 17 | syl3anc 1367 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝑥)) ·ih 𝑥) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑥))) |
19 | 12, 18 | eqtrd 2856 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑥))) |
20 | 9, 19 | sylan 582 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥) = (𝐴 · ((𝑇‘𝑥) ·ih 𝑥))) |
21 | 20 | breq2d 5078 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ 𝑥 ∈ ℋ) → (0 ≤ (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (𝐴 · ((𝑇‘𝑥) ·ih 𝑥)))) |
22 | 21 | adantlr 713 | . . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℋ) → (0 ≤ (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (𝐴 · ((𝑇‘𝑥) ·ih 𝑥)))) |
23 | 6, 22 | sylibrd 261 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℋ) → (0 ≤ ((𝑇‘𝑥) ·ih 𝑥) → 0 ≤ (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥))) |
24 | 23 | ralimdva 3177 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ 0 ≤ 𝐴) → (∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥) → ∀𝑥 ∈ ℋ 0 ≤ (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥))) |
25 | 24 | expimpd 456 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → ((0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥)) → ∀𝑥 ∈ ℋ 0 ≤ (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥))) |
26 | leoppos 29903 | . . . . 5 ⊢ (𝑇 ∈ HrmOp → ( 0hop ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥))) | |
27 | 26 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → ( 0hop ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥))) |
28 | 27 | anbi2d 630 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → ((0 ≤ 𝐴 ∧ 0hop ≤op 𝑇) ↔ (0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥)))) |
29 | hmopm 29798 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → (𝐴 ·op 𝑇) ∈ HrmOp) | |
30 | leoppos 29903 | . . . 4 ⊢ ((𝐴 ·op 𝑇) ∈ HrmOp → ( 0hop ≤op (𝐴 ·op 𝑇) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥))) | |
31 | 29, 30 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → ( 0hop ≤op (𝐴 ·op 𝑇) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝐴 ·op 𝑇)‘𝑥) ·ih 𝑥))) |
32 | 25, 28, 31 | 3imtr4d 296 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → ((0 ≤ 𝐴 ∧ 0hop ≤op 𝑇) → 0hop ≤op (𝐴 ·op 𝑇))) |
33 | 32 | imp 409 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 0hop ≤op 𝑇)) → 0hop ≤op (𝐴 ·op 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 · cmul 10542 ≤ cle 10676 ℋchba 28696 ·ℎ csm 28698 ·ih csp 28699 ·op chot 28716 0hop ch0o 28720 HrmOpcho 28727 ≤op cleo 28735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvmulass 28784 ax-hvdistr1 28785 ax-hvdistr2 28786 ax-hvmul0 28787 ax-hfi 28856 ax-his1 28859 ax-his2 28860 ax-his3 28861 ax-his4 28862 ax-hcompl 28979 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-cn 21835 df-cnp 21836 df-lm 21837 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cfil 23858 df-cau 23859 df-cmet 23860 df-grpo 28270 df-gid 28271 df-ginv 28272 df-gdiv 28273 df-ablo 28322 df-vc 28336 df-nv 28369 df-va 28372 df-ba 28373 df-sm 28374 df-0v 28375 df-vs 28376 df-nmcv 28377 df-ims 28378 df-dip 28478 df-ssp 28499 df-ph 28590 df-cbn 28640 df-hnorm 28745 df-hba 28746 df-hvsub 28748 df-hlim 28749 df-hcau 28750 df-sh 28984 df-ch 28998 df-oc 29029 df-ch0 29030 df-shs 29085 df-pjh 29172 df-hosum 29507 df-homul 29508 df-hodif 29509 df-h0op 29525 df-hmop 29621 df-leop 29629 |
This theorem is referenced by: leopmul 29911 leopmul2i 29912 opsqrlem1 29917 |
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