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Mirrors > Home > HSE Home > Th. List > pjorthcoi | Structured version Visualization version GIF version |
Description: Composition of projections of orthogonal subspaces. Part (i)->(iia) of Theorem 27.4 of [Halmos] p. 45. (Contributed by NM, 6-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjco.1 | ⊢ 𝐺 ∈ Cℋ |
pjco.2 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjorthcoi | ⊢ (𝐺 ⊆ (⊥‘𝐻) → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = 0hop ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjco.2 | . . . . . . . 8 ⊢ 𝐻 ∈ Cℋ | |
2 | 1 | pjcli 29775 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((projℎ‘𝐻)‘𝑥) ∈ 𝐻) |
3 | pjco.1 | . . . . . . . . 9 ⊢ 𝐺 ∈ Cℋ | |
4 | 3, 1 | chsscon2i 29821 | . . . . . . . 8 ⊢ (𝐺 ⊆ (⊥‘𝐻) ↔ 𝐻 ⊆ (⊥‘𝐺)) |
5 | ssel 3919 | . . . . . . . 8 ⊢ (𝐻 ⊆ (⊥‘𝐺) → (((projℎ‘𝐻)‘𝑥) ∈ 𝐻 → ((projℎ‘𝐻)‘𝑥) ∈ (⊥‘𝐺))) | |
6 | 4, 5 | sylbi 216 | . . . . . . 7 ⊢ (𝐺 ⊆ (⊥‘𝐻) → (((projℎ‘𝐻)‘𝑥) ∈ 𝐻 → ((projℎ‘𝐻)‘𝑥) ∈ (⊥‘𝐺))) |
7 | 2, 6 | syl5com 31 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ (⊥‘𝐻) → ((projℎ‘𝐻)‘𝑥) ∈ (⊥‘𝐺))) |
8 | 1 | pjhcli 29776 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((projℎ‘𝐻)‘𝑥) ∈ ℋ) |
9 | pjoc2 29797 | . . . . . . 7 ⊢ ((𝐺 ∈ Cℋ ∧ ((projℎ‘𝐻)‘𝑥) ∈ ℋ) → (((projℎ‘𝐻)‘𝑥) ∈ (⊥‘𝐺) ↔ ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = 0ℎ)) | |
10 | 3, 8, 9 | sylancr 587 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐻)‘𝑥) ∈ (⊥‘𝐺) ↔ ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = 0ℎ)) |
11 | 7, 10 | sylibd 238 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ (⊥‘𝐻) → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = 0ℎ)) |
12 | 11 | impcom 408 | . . . 4 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ 𝑥 ∈ ℋ) → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = 0ℎ) |
13 | 3, 1 | pjcoi 30516 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥))) |
14 | 13 | adantl 482 | . . . 4 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ 𝑥 ∈ ℋ) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥))) |
15 | ho0val 30108 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ( 0hop ‘𝑥) = 0ℎ) | |
16 | 15 | adantl 482 | . . . 4 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ 𝑥 ∈ ℋ) → ( 0hop ‘𝑥) = 0ℎ) |
17 | 12, 14, 16 | 3eqtr4d 2790 | . . 3 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ 𝑥 ∈ ℋ) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ( 0hop ‘𝑥)) |
18 | 17 | ralrimiva 3110 | . 2 ⊢ (𝐺 ⊆ (⊥‘𝐻) → ∀𝑥 ∈ ℋ (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ( 0hop ‘𝑥)) |
19 | 3 | pjfi 30062 | . . . 4 ⊢ (projℎ‘𝐺): ℋ⟶ ℋ |
20 | 1 | pjfi 30062 | . . . 4 ⊢ (projℎ‘𝐻): ℋ⟶ ℋ |
21 | 19, 20 | hocofi 30124 | . . 3 ⊢ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)): ℋ⟶ ℋ |
22 | ho0f 30109 | . . 3 ⊢ 0hop : ℋ⟶ ℋ | |
23 | 21, 22 | hoeqi 30119 | . 2 ⊢ (∀𝑥 ∈ ℋ (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ( 0hop ‘𝑥) ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = 0hop ) |
24 | 18, 23 | sylib 217 | 1 ⊢ (𝐺 ⊆ (⊥‘𝐻) → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = 0hop ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ⊆ wss 3892 ∘ ccom 5594 ‘cfv 6432 ℋchba 29277 0ℎc0v 29282 Cℋ cch 29287 ⊥cort 29288 projℎcpjh 29295 0hop ch0o 29301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cc 10192 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 ax-hilex 29357 ax-hfvadd 29358 ax-hvcom 29359 ax-hvass 29360 ax-hv0cl 29361 ax-hvaddid 29362 ax-hfvmul 29363 ax-hvmulid 29364 ax-hvmulass 29365 ax-hvdistr1 29366 ax-hvdistr2 29367 ax-hvmul0 29368 ax-hfi 29437 ax-his1 29440 ax-his2 29441 ax-his3 29442 ax-his4 29443 ax-hcompl 29560 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-oadd 8292 df-omul 8293 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-acn 9701 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-rlim 15196 df-sum 15396 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-cn 22376 df-cnp 22377 df-lm 22378 df-haus 22464 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cfil 24417 df-cau 24418 df-cmet 24419 df-grpo 28851 df-gid 28852 df-ginv 28853 df-gdiv 28854 df-ablo 28903 df-vc 28917 df-nv 28950 df-va 28953 df-ba 28954 df-sm 28955 df-0v 28956 df-vs 28957 df-nmcv 28958 df-ims 28959 df-dip 29059 df-ssp 29080 df-ph 29171 df-cbn 29221 df-hnorm 29326 df-hba 29327 df-hvsub 29329 df-hlim 29330 df-hcau 29331 df-sh 29565 df-ch 29579 df-oc 29610 df-ch0 29611 df-shs 29666 df-pjh 29753 df-h0op 30106 |
This theorem is referenced by: pjoccoi 30536 pjclem1 30553 |
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