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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 29200 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((projℎ‘𝐻)‘𝑥) ∈ 𝐻) |
3 | pjco.1 | . . . . . . . . 9 ⊢ 𝐺 ∈ Cℋ | |
4 | 3, 1 | chsscon2i 29246 | . . . . . . . 8 ⊢ (𝐺 ⊆ (⊥‘𝐻) ↔ 𝐻 ⊆ (⊥‘𝐺)) |
5 | ssel 3908 | . . . . . . . 8 ⊢ (𝐻 ⊆ (⊥‘𝐺) → (((projℎ‘𝐻)‘𝑥) ∈ 𝐻 → ((projℎ‘𝐻)‘𝑥) ∈ (⊥‘𝐺))) | |
6 | 4, 5 | sylbi 220 | . . . . . . 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 29201 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((projℎ‘𝐻)‘𝑥) ∈ ℋ) |
9 | pjoc2 29222 | . . . . . . 7 ⊢ ((𝐺 ∈ Cℋ ∧ ((projℎ‘𝐻)‘𝑥) ∈ ℋ) → (((projℎ‘𝐻)‘𝑥) ∈ (⊥‘𝐺) ↔ ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = 0ℎ)) | |
10 | 3, 8, 9 | sylancr 590 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐻)‘𝑥) ∈ (⊥‘𝐺) ↔ ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = 0ℎ)) |
11 | 7, 10 | sylibd 242 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ (⊥‘𝐻) → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = 0ℎ)) |
12 | 11 | impcom 411 | . . . 4 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ 𝑥 ∈ ℋ) → ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥)) = 0ℎ) |
13 | 3, 1 | pjcoi 29941 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥))) |
14 | 13 | adantl 485 | . . . 4 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ 𝑥 ∈ ℋ) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ((projℎ‘𝐺)‘((projℎ‘𝐻)‘𝑥))) |
15 | ho0val 29533 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ( 0hop ‘𝑥) = 0ℎ) | |
16 | 15 | adantl 485 | . . . 4 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ 𝑥 ∈ ℋ) → ( 0hop ‘𝑥) = 0ℎ) |
17 | 12, 14, 16 | 3eqtr4d 2843 | . . 3 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ 𝑥 ∈ ℋ) → (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ( 0hop ‘𝑥)) |
18 | 17 | ralrimiva 3149 | . 2 ⊢ (𝐺 ⊆ (⊥‘𝐻) → ∀𝑥 ∈ ℋ (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ( 0hop ‘𝑥)) |
19 | 3 | pjfi 29487 | . . . 4 ⊢ (projℎ‘𝐺): ℋ⟶ ℋ |
20 | 1 | pjfi 29487 | . . . 4 ⊢ (projℎ‘𝐻): ℋ⟶ ℋ |
21 | 19, 20 | hocofi 29549 | . . 3 ⊢ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)): ℋ⟶ ℋ |
22 | ho0f 29534 | . . 3 ⊢ 0hop : ℋ⟶ ℋ | |
23 | 21, 22 | hoeqi 29544 | . 2 ⊢ (∀𝑥 ∈ ℋ (((projℎ‘𝐺) ∘ (projℎ‘𝐻))‘𝑥) = ( 0hop ‘𝑥) ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = 0hop ) |
24 | 18, 23 | sylib 221 | 1 ⊢ (𝐺 ⊆ (⊥‘𝐻) → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = 0hop ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 ∘ ccom 5523 ‘cfv 6324 ℋchba 28702 0ℎc0v 28707 Cℋ cch 28712 ⊥cort 28713 projℎcpjh 28720 0hop ch0o 28726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 ax-hcompl 28985 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cn 21832 df-cnp 21833 df-lm 21834 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cfil 23859 df-cau 23860 df-cmet 23861 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-dip 28484 df-ssp 28505 df-ph 28596 df-cbn 28646 df-hnorm 28751 df-hba 28752 df-hvsub 28754 df-hlim 28755 df-hcau 28756 df-sh 28990 df-ch 29004 df-oc 29035 df-ch0 29036 df-shs 29091 df-pjh 29178 df-h0op 29531 |
This theorem is referenced by: pjoccoi 29961 pjclem1 29978 |
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