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Mirrors > Home > HSE Home > Th. List > pjssposi | Structured version Visualization version GIF version |
Description: Projector ordering can be expressed by the subset relationship between their projection subspaces. (i)<->(iii) of Theorem 29.2 of [Halmos] p. 48. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjco.1 | ⊢ 𝐺 ∈ Cℋ |
pjco.2 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjssposi | ⊢ (∀𝑥 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) ↔ 𝐺 ⊆ 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjco.2 | . . . . . . . 8 ⊢ 𝐻 ∈ Cℋ | |
2 | 1 | pjhcli 28617 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((projℎ‘𝐻)‘𝑥) ∈ ℋ) |
3 | normcl 28322 | . . . . . . 7 ⊢ (((projℎ‘𝐻)‘𝑥) ∈ ℋ → (normℎ‘((projℎ‘𝐻)‘𝑥)) ∈ ℝ) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((projℎ‘𝐻)‘𝑥)) ∈ ℝ) |
5 | 4 | resqcld 13242 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((normℎ‘((projℎ‘𝐻)‘𝑥))↑2) ∈ ℝ) |
6 | pjco.1 | . . . . . . . 8 ⊢ 𝐺 ∈ Cℋ | |
7 | 6 | pjhcli 28617 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((projℎ‘𝐺)‘𝑥) ∈ ℋ) |
8 | normcl 28322 | . . . . . . 7 ⊢ (((projℎ‘𝐺)‘𝑥) ∈ ℋ → (normℎ‘((projℎ‘𝐺)‘𝑥)) ∈ ℝ) | |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((projℎ‘𝐺)‘𝑥)) ∈ ℝ) |
10 | 9 | resqcld 13242 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((normℎ‘((projℎ‘𝐺)‘𝑥))↑2) ∈ ℝ) |
11 | 5, 10 | subge0d 10823 | . . . 4 ⊢ (𝑥 ∈ ℋ → (0 ≤ (((normℎ‘((projℎ‘𝐻)‘𝑥))↑2) − ((normℎ‘((projℎ‘𝐺)‘𝑥))↑2)) ↔ ((normℎ‘((projℎ‘𝐺)‘𝑥))↑2) ≤ ((normℎ‘((projℎ‘𝐻)‘𝑥))↑2))) |
12 | 1 | pjfi 28903 | . . . . . . . 8 ⊢ (projℎ‘𝐻): ℋ⟶ ℋ |
13 | 6 | pjfi 28903 | . . . . . . . 8 ⊢ (projℎ‘𝐺): ℋ⟶ ℋ |
14 | hodval 28941 | . . . . . . . 8 ⊢ (((projℎ‘𝐻): ℋ⟶ ℋ ∧ (projℎ‘𝐺): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) = (((projℎ‘𝐻)‘𝑥) −ℎ ((projℎ‘𝐺)‘𝑥))) | |
15 | 12, 13, 14 | mp3an12 1562 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) = (((projℎ‘𝐻)‘𝑥) −ℎ ((projℎ‘𝐺)‘𝑥))) |
16 | 15 | oveq1d 6811 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) = ((((projℎ‘𝐻)‘𝑥) −ℎ ((projℎ‘𝐺)‘𝑥)) ·ih 𝑥)) |
17 | id 22 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → 𝑥 ∈ ℋ) | |
18 | his2sub 28289 | . . . . . . 7 ⊢ ((((projℎ‘𝐻)‘𝑥) ∈ ℋ ∧ ((projℎ‘𝐺)‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((((projℎ‘𝐻)‘𝑥) −ℎ ((projℎ‘𝐺)‘𝑥)) ·ih 𝑥) = ((((projℎ‘𝐻)‘𝑥) ·ih 𝑥) − (((projℎ‘𝐺)‘𝑥) ·ih 𝑥))) | |
19 | 2, 7, 17, 18 | syl3anc 1476 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((((projℎ‘𝐻)‘𝑥) −ℎ ((projℎ‘𝐺)‘𝑥)) ·ih 𝑥) = ((((projℎ‘𝐻)‘𝑥) ·ih 𝑥) − (((projℎ‘𝐺)‘𝑥) ·ih 𝑥))) |
20 | 1 | pjinormi 28886 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐻)‘𝑥) ·ih 𝑥) = ((normℎ‘((projℎ‘𝐻)‘𝑥))↑2)) |
21 | 6 | pjinormi 28886 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐺)‘𝑥) ·ih 𝑥) = ((normℎ‘((projℎ‘𝐺)‘𝑥))↑2)) |
22 | 20, 21 | oveq12d 6814 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((((projℎ‘𝐻)‘𝑥) ·ih 𝑥) − (((projℎ‘𝐺)‘𝑥) ·ih 𝑥)) = (((normℎ‘((projℎ‘𝐻)‘𝑥))↑2) − ((normℎ‘((projℎ‘𝐺)‘𝑥))↑2))) |
23 | 16, 19, 22 | 3eqtrd 2809 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) = (((normℎ‘((projℎ‘𝐻)‘𝑥))↑2) − ((normℎ‘((projℎ‘𝐺)‘𝑥))↑2))) |
24 | 23 | breq2d 4799 | . . . 4 ⊢ (𝑥 ∈ ℋ → (0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((normℎ‘((projℎ‘𝐻)‘𝑥))↑2) − ((normℎ‘((projℎ‘𝐺)‘𝑥))↑2)))) |
25 | normge0 28323 | . . . . . 6 ⊢ (((projℎ‘𝐺)‘𝑥) ∈ ℋ → 0 ≤ (normℎ‘((projℎ‘𝐺)‘𝑥))) | |
26 | 7, 25 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ ℋ → 0 ≤ (normℎ‘((projℎ‘𝐺)‘𝑥))) |
27 | normge0 28323 | . . . . . 6 ⊢ (((projℎ‘𝐻)‘𝑥) ∈ ℋ → 0 ≤ (normℎ‘((projℎ‘𝐻)‘𝑥))) | |
28 | 2, 27 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ ℋ → 0 ≤ (normℎ‘((projℎ‘𝐻)‘𝑥))) |
29 | 9, 4, 26, 28 | le2sqd 13251 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((normℎ‘((projℎ‘𝐺)‘𝑥)) ≤ (normℎ‘((projℎ‘𝐻)‘𝑥)) ↔ ((normℎ‘((projℎ‘𝐺)‘𝑥))↑2) ≤ ((normℎ‘((projℎ‘𝐻)‘𝑥))↑2))) |
30 | 11, 24, 29 | 3bitr4d 300 | . . 3 ⊢ (𝑥 ∈ ℋ → (0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) ↔ (normℎ‘((projℎ‘𝐺)‘𝑥)) ≤ (normℎ‘((projℎ‘𝐻)‘𝑥)))) |
31 | 30 | ralbiia 3128 | . 2 ⊢ (∀𝑥 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ (normℎ‘((projℎ‘𝐺)‘𝑥)) ≤ (normℎ‘((projℎ‘𝐻)‘𝑥))) |
32 | 6, 1 | pjnormssi 29367 | . 2 ⊢ (𝐺 ⊆ 𝐻 ↔ ∀𝑥 ∈ ℋ (normℎ‘((projℎ‘𝐺)‘𝑥)) ≤ (normℎ‘((projℎ‘𝐻)‘𝑥))) |
33 | 31, 32 | bitr4i 267 | 1 ⊢ (∀𝑥 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑥) ·ih 𝑥) ↔ 𝐺 ⊆ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ⊆ wss 3723 class class class wbr 4787 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 ℝcr 10141 0cc0 10142 ≤ cle 10281 − cmin 10472 2c2 11276 ↑cexp 13067 ℋchil 28116 ·ih csp 28119 normℎcno 28120 −ℎ cmv 28122 Cℋ cch 28126 projℎcpjh 28134 −op chod 28137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cc 9463 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 ax-addf 10221 ax-mulf 10222 ax-hilex 28196 ax-hfvadd 28197 ax-hvcom 28198 ax-hvass 28199 ax-hv0cl 28200 ax-hvaddid 28201 ax-hfvmul 28202 ax-hvmulid 28203 ax-hvmulass 28204 ax-hvdistr1 28205 ax-hvdistr2 28206 ax-hvmul0 28207 ax-hfi 28276 ax-his1 28279 ax-his2 28280 ax-his3 28281 ax-his4 28282 ax-hcompl 28399 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-of 7048 df-om 7217 df-1st 7319 df-2nd 7320 df-supp 7451 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-omul 7722 df-er 7900 df-map 8015 df-pm 8016 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8436 df-fi 8477 df-sup 8508 df-inf 8509 df-oi 8575 df-card 8969 df-acn 8972 df-cda 9196 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-z 11585 df-dec 11701 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-rlim 14428 df-sum 14625 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-mulg 17749 df-cntz 17957 df-cmn 18402 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-cn 21252 df-cnp 21253 df-lm 21254 df-haus 21340 df-tx 21586 df-hmeo 21779 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-xms 22345 df-ms 22346 df-tms 22347 df-cfil 23272 df-cau 23273 df-cmet 23274 df-grpo 27687 df-gid 27688 df-ginv 27689 df-gdiv 27690 df-ablo 27739 df-vc 27754 df-nv 27787 df-va 27790 df-ba 27791 df-sm 27792 df-0v 27793 df-vs 27794 df-nmcv 27795 df-ims 27796 df-dip 27896 df-ssp 27917 df-ph 28008 df-cbn 28059 df-hnorm 28165 df-hba 28166 df-hvsub 28168 df-hlim 28169 df-hcau 28170 df-sh 28404 df-ch 28418 df-oc 28449 df-ch0 28450 df-shs 28507 df-pjh 28594 df-hodif 28931 |
This theorem is referenced by: pjordi 29372 pjssdif2i 29373 pjssdif1i 29374 |
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