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Mirrors > Home > HSE Home > Th. List > pjcmul1i | Structured version Visualization version GIF version |
Description: A necessary and sufficient condition for the product of two projectors to be a projector is that the projectors commute. Part 1 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjclem1.1 | ⊢ 𝐺 ∈ Cℋ |
pjclem1.2 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjcmul1i | ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) ∈ ran projℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjclem1.1 | . . . 4 ⊢ 𝐺 ∈ Cℋ | |
2 | pjclem1.2 | . . . 4 ⊢ 𝐻 ∈ Cℋ | |
3 | 1, 2 | pjclem4 32233 | . . 3 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = (projℎ‘(𝐺 ∩ 𝐻))) |
4 | pjmfn 31749 | . . . 4 ⊢ projℎ Fn Cℋ | |
5 | 1, 2 | chincli 31494 | . . . 4 ⊢ (𝐺 ∩ 𝐻) ∈ Cℋ |
6 | fnfvelrn 7116 | . . . 4 ⊢ ((projℎ Fn Cℋ ∧ (𝐺 ∩ 𝐻) ∈ Cℋ ) → (projℎ‘(𝐺 ∩ 𝐻)) ∈ ran projℎ) | |
7 | 4, 5, 6 | mp2an 691 | . . 3 ⊢ (projℎ‘(𝐺 ∩ 𝐻)) ∈ ran projℎ |
8 | 3, 7 | eqeltrdi 2852 | . 2 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) ∈ ran projℎ) |
9 | pjadj2 32221 | . . 3 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) ∈ ran projℎ → (adjℎ‘((projℎ‘𝐺) ∘ (projℎ‘𝐻))) = ((projℎ‘𝐺) ∘ (projℎ‘𝐻))) | |
10 | 1 | pjbdlni 32183 | . . . . 5 ⊢ (projℎ‘𝐺) ∈ BndLinOp |
11 | 2 | pjbdlni 32183 | . . . . 5 ⊢ (projℎ‘𝐻) ∈ BndLinOp |
12 | 10, 11 | adjcoi 32134 | . . . 4 ⊢ (adjℎ‘((projℎ‘𝐺) ∘ (projℎ‘𝐻))) = ((adjℎ‘(projℎ‘𝐻)) ∘ (adjℎ‘(projℎ‘𝐺))) |
13 | pjadj3 32222 | . . . . . 6 ⊢ (𝐻 ∈ Cℋ → (adjℎ‘(projℎ‘𝐻)) = (projℎ‘𝐻)) | |
14 | 2, 13 | ax-mp 5 | . . . . 5 ⊢ (adjℎ‘(projℎ‘𝐻)) = (projℎ‘𝐻) |
15 | pjadj3 32222 | . . . . . 6 ⊢ (𝐺 ∈ Cℋ → (adjℎ‘(projℎ‘𝐺)) = (projℎ‘𝐺)) | |
16 | 1, 15 | ax-mp 5 | . . . . 5 ⊢ (adjℎ‘(projℎ‘𝐺)) = (projℎ‘𝐺) |
17 | 14, 16 | coeq12i 5888 | . . . 4 ⊢ ((adjℎ‘(projℎ‘𝐻)) ∘ (adjℎ‘(projℎ‘𝐺))) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) |
18 | 12, 17 | eqtri 2768 | . . 3 ⊢ (adjℎ‘((projℎ‘𝐺) ∘ (projℎ‘𝐻))) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) |
19 | 9, 18 | eqtr3di 2795 | . 2 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) ∈ ran projℎ → ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺))) |
20 | 8, 19 | impbii 209 | 1 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ↔ ((projℎ‘𝐺) ∘ (projℎ‘𝐻)) ∈ ran projℎ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2108 ∩ cin 3975 ran crn 5701 ∘ ccom 5704 Fn wfn 6570 ‘cfv 6575 Cℋ cch 30963 projℎcpjh 30971 adjℎcado 30989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cc 10506 ax-dc 10517 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 ax-addf 11265 ax-mulf 11266 ax-hilex 31033 ax-hfvadd 31034 ax-hvcom 31035 ax-hvass 31036 ax-hv0cl 31037 ax-hvaddid 31038 ax-hfvmul 31039 ax-hvmulid 31040 ax-hvmulass 31041 ax-hvdistr1 31042 ax-hvdistr2 31043 ax-hvmul0 31044 ax-hfi 31113 ax-his1 31116 ax-his2 31117 ax-his3 31118 ax-his4 31119 ax-hcompl 31236 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-1st 8032 df-2nd 8033 df-supp 8204 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-oadd 8528 df-omul 8529 df-er 8765 df-map 8888 df-pm 8889 df-ixp 8958 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-fsupp 9434 df-fi 9482 df-sup 9513 df-inf 9514 df-oi 9581 df-card 10010 df-acn 10013 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-q 13016 df-rp 13060 df-xneg 13177 df-xadd 13178 df-xmul 13179 df-ioo 13413 df-ico 13415 df-icc 13416 df-fz 13570 df-fzo 13714 df-fl 13845 df-seq 14055 df-exp 14115 df-hash 14382 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15536 df-rlim 15537 df-sum 15737 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-rest 17484 df-topn 17485 df-0g 17503 df-gsum 17504 df-topgen 17505 df-pt 17506 df-prds 17509 df-xrs 17564 df-qtop 17569 df-imas 17570 df-xps 17572 df-mre 17646 df-mrc 17647 df-acs 17649 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-submnd 18821 df-mulg 19110 df-cntz 19359 df-cmn 19826 df-psmet 21381 df-xmet 21382 df-met 21383 df-bl 21384 df-mopn 21385 df-fbas 21386 df-fg 21387 df-cnfld 21390 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22976 df-cld 23050 df-ntr 23051 df-cls 23052 df-nei 23129 df-cn 23258 df-cnp 23259 df-lm 23260 df-t1 23345 df-haus 23346 df-cmp 23418 df-tx 23593 df-hmeo 23786 df-fil 23877 df-fm 23969 df-flim 23970 df-flf 23971 df-fcls 23972 df-xms 24353 df-ms 24354 df-tms 24355 df-cncf 24925 df-cfil 25310 df-cau 25311 df-cmet 25312 df-grpo 30527 df-gid 30528 df-ginv 30529 df-gdiv 30530 df-ablo 30579 df-vc 30593 df-nv 30626 df-va 30629 df-ba 30630 df-sm 30631 df-0v 30632 df-vs 30633 df-nmcv 30634 df-ims 30635 df-dip 30735 df-ssp 30756 df-lno 30778 df-nmoo 30779 df-blo 30780 df-0o 30781 df-ph 30847 df-cbn 30897 df-hlo 30920 df-hnorm 31002 df-hba 31003 df-hvsub 31005 df-hlim 31006 df-hcau 31007 df-sh 31241 df-ch 31255 df-oc 31286 df-ch0 31287 df-shs 31342 df-pjh 31429 df-h0op 31782 df-iop 31783 df-nmop 31873 df-cnop 31874 df-lnop 31875 df-bdop 31876 df-unop 31877 df-hmop 31878 df-nmfn 31879 df-nlfn 31880 df-cnfn 31881 df-lnfn 31882 df-adjh 31883 |
This theorem is referenced by: pjcmul2i 32236 |
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