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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 31700. (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 31473 | . . . 4 β’ projβ Fn Cβ | |
| 2 | fvelrnb 6945 | . . . 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 31908 | . . . . . 6 β’ (π₯ β Cβ β (projββπ₯) β HrmOp) | |
| 5 | pjidmco 31939 | . . . . . 6 β’ (π₯ β Cβ β ((projββπ₯) β (projββπ₯)) = (projββπ₯)) | |
| 6 | 4, 5 | jca 511 | . . . . 5 β’ (π₯ β Cβ β ((projββπ₯) β HrmOp β§ ((projββπ₯) β (projββπ₯)) = (projββπ₯))) |
| 7 | eleq1 2815 | . . . . . 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 5857 | . . . . . . 7 β’ ((projββπ₯) = π β ((projββπ₯) β (projββπ₯)) = (π β π)) |
| 10 | 9, 8 | eqeq12d 2742 | . . . . . 6 β’ ((projββπ₯) = π β (((projββπ₯) β (projββπ₯)) = (projββπ₯) β (π β π) = π)) |
| 11 | 7, 10 | anbi12d 630 | . . . . 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 3142 | . . 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 31912 | . . 3 β’ ((π β HrmOp β§ (π β π) = π) β π = (projββran π)) | |
| 16 | hmopidmch 31911 | . . . 4 β’ ((π β HrmOp β§ (π β π) = π) β ran π β Cβ ) | |
| 17 | fnfvelrn 7075 | . . . 4 β’ ((projβ Fn Cβ β§ ran π β Cβ ) β (projββran π) β ran projβ) | |
| 18 | 1, 16, 17 | sylancr 586 | . . 3 β’ ((π β HrmOp β§ (π β π) = π) β (projββran π) β ran projβ) |
| 19 | 15, 18 | eqeltrd 2827 | . 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 395 = wceq 1533 β wcel 2098 βwrex 3064 ran crn 5670 β ccom 5673 Fn wfn 6531 βcfv 6536 Cβ cch 30687 projβcpjh 30695 HrmOpcho 30708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cc 10429 ax-dc 10440 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 ax-hilex 30757 ax-hfvadd 30758 ax-hvcom 30759 ax-hvass 30760 ax-hv0cl 30761 ax-hvaddid 30762 ax-hfvmul 30763 ax-hvmulid 30764 ax-hvmulass 30765 ax-hvdistr1 30766 ax-hvdistr2 30767 ax-hvmul0 30768 ax-hfi 30837 ax-his1 30840 ax-his2 30841 ax-his3 30842 ax-his4 30843 ax-hcompl 30960 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-cn 23082 df-cnp 23083 df-lm 23084 df-t1 23169 df-haus 23170 df-cmp 23242 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-fcls 23796 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-cfil 25134 df-cau 25135 df-cmet 25136 df-grpo 30251 df-gid 30252 df-ginv 30253 df-gdiv 30254 df-ablo 30303 df-vc 30317 df-nv 30350 df-va 30353 df-ba 30354 df-sm 30355 df-0v 30356 df-vs 30357 df-nmcv 30358 df-ims 30359 df-dip 30459 df-ssp 30480 df-lno 30502 df-nmoo 30503 df-blo 30504 df-0o 30505 df-ph 30571 df-cbn 30621 df-hlo 30644 df-hnorm 30726 df-hba 30727 df-hvsub 30729 df-hlim 30730 df-hcau 30731 df-sh 30965 df-ch 30979 df-oc 31010 df-ch0 31011 df-shs 31066 df-pjh 31153 df-h0op 31506 df-iop 31507 df-nmop 31597 df-cnop 31598 df-lnop 31599 df-bdop 31600 df-unop 31601 df-hmop 31602 |
| This theorem is referenced by: pjhmopidm 31941 elpjidm 31942 elpjhmop 31943 elpjch 31947 |
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