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| Mirrors > Home > HSE Home > Th. List > dfpjop | Structured version Visualization version GIF version | ||
| Description: Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 31195. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dfpjop | β’ (π β ran projβ β (π β HrmOp β§ (π β π) = π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pjmfn 30968 | . . . 4 β’ projβ Fn Cβ | |
| 2 | fvelrnb 6953 | . . . 4 β’ (projβ Fn Cβ β (π β ran projβ β βπ₯ β Cβ (projββπ₯) = π)) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 β’ (π β ran projβ β βπ₯ β Cβ (projββπ₯) = π) |
| 4 | pjhmop 31403 | . . . . . 6 β’ (π₯ β Cβ β (projββπ₯) β HrmOp) | |
| 5 | pjidmco 31434 | . . . . . 6 β’ (π₯ β Cβ β ((projββπ₯) β (projββπ₯)) = (projββπ₯)) | |
| 6 | 4, 5 | jca 513 | . . . . 5 β’ (π₯ β Cβ β ((projββπ₯) β HrmOp β§ ((projββπ₯) β (projββπ₯)) = (projββπ₯))) |
| 7 | eleq1 2822 | . . . . . 6 β’ ((projββπ₯) = π β ((projββπ₯) β HrmOp β π β HrmOp)) | |
| 8 | id 22 | . . . . . . . 8 β’ ((projββπ₯) = π β (projββπ₯) = π) | |
| 9 | 8, 8 | coeq12d 5865 | . . . . . . 7 β’ ((projββπ₯) = π β ((projββπ₯) β (projββπ₯)) = (π β π)) |
| 10 | 9, 8 | eqeq12d 2749 | . . . . . 6 β’ ((projββπ₯) = π β (((projββπ₯) β (projββπ₯)) = (projββπ₯) β (π β π) = π)) |
| 11 | 7, 10 | anbi12d 632 | . . . . 5 β’ ((projββπ₯) = π β (((projββπ₯) β HrmOp β§ ((projββπ₯) β (projββπ₯)) = (projββπ₯)) β (π β HrmOp β§ (π β π) = π))) |
| 12 | 6, 11 | syl5ibcom 244 | . . . 4 β’ (π₯ β Cβ β ((projββπ₯) = π β (π β HrmOp β§ (π β π) = π))) |
| 13 | 12 | rexlimiv 3149 | . . 3 β’ (βπ₯ β Cβ (projββπ₯) = π β (π β HrmOp β§ (π β π) = π)) |
| 14 | 3, 13 | sylbi 216 | . 2 β’ (π β ran projβ β (π β HrmOp β§ (π β π) = π)) |
| 15 | hmopidmpj 31407 | . . 3 β’ ((π β HrmOp β§ (π β π) = π) β π = (projββran π)) | |
| 16 | hmopidmch 31406 | . . . 4 β’ ((π β HrmOp β§ (π β π) = π) β ran π β Cβ ) | |
| 17 | fnfvelrn 7083 | . . . 4 β’ ((projβ Fn Cβ β§ ran π β Cβ ) β (projββran π) β ran projβ) | |
| 18 | 1, 16, 17 | sylancr 588 | . . 3 β’ ((π β HrmOp β§ (π β π) = π) β (projββran π) β ran projβ) |
| 19 | 15, 18 | eqeltrd 2834 | . 2 β’ ((π β HrmOp β§ (π β π) = π) β π β ran projβ) |
| 20 | 14, 19 | impbii 208 | 1 β’ (π β ran projβ β (π β HrmOp β§ (π β π) = π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3071 ran crn 5678 β ccom 5681 Fn wfn 6539 βcfv 6544 Cβ cch 30182 projβcpjh 30190 HrmOpcho 30203 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-dc 10441 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 ax-hilex 30252 ax-hfvadd 30253 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvmulass 30260 ax-hvdistr1 30261 ax-hvdistr2 30262 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his2 30336 ax-his3 30337 ax-his4 30338 ax-hcompl 30455 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-cn 22731 df-cnp 22732 df-lm 22733 df-t1 22818 df-haus 22819 df-cmp 22891 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-fcls 23445 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-cfil 24772 df-cau 24773 df-cmet 24774 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-vs 29852 df-nmcv 29853 df-ims 29854 df-dip 29954 df-ssp 29975 df-lno 29997 df-nmoo 29998 df-blo 29999 df-0o 30000 df-ph 30066 df-cbn 30116 df-hlo 30139 df-hnorm 30221 df-hba 30222 df-hvsub 30224 df-hlim 30225 df-hcau 30226 df-sh 30460 df-ch 30474 df-oc 30505 df-ch0 30506 df-shs 30561 df-pjh 30648 df-h0op 31001 df-iop 31002 df-nmop 31092 df-cnop 31093 df-lnop 31094 df-bdop 31095 df-unop 31096 df-hmop 31097 |
| This theorem is referenced by: pjhmopidm 31436 elpjidm 31437 elpjhmop 31438 elpjch 31442 |
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