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Mirrors > Home > HSE Home > Th. List > ho0f | Structured version Visualization version GIF version |
Description: Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ho0f | ⊢ 0hop : ℋ⟶ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | h0elch 31142 | . . 3 ⊢ 0ℋ ∈ Cℋ | |
2 | 1 | pjfi 31591 | . 2 ⊢ (projℎ‘0ℋ): ℋ⟶ ℋ |
3 | df-h0op 31635 | . . 3 ⊢ 0hop = (projℎ‘0ℋ) | |
4 | 3 | feq1i 6714 | . 2 ⊢ ( 0hop : ℋ⟶ ℋ ↔ (projℎ‘0ℋ): ℋ⟶ ℋ) |
5 | 2, 4 | mpbir 230 | 1 ⊢ 0hop : ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ⟶wf 6545 ‘cfv 6549 ℋchba 30806 0ℋc0h 30822 projℎcpjh 30824 0hop ch0o 30830 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cc 10465 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 ax-mulf 11225 ax-hilex 30886 ax-hfvadd 30887 ax-hvcom 30888 ax-hvass 30889 ax-hv0cl 30890 ax-hvaddid 30891 ax-hfvmul 30892 ax-hvmulid 30893 ax-hvmulass 30894 ax-hvdistr1 30895 ax-hvdistr2 30896 ax-hvmul0 30897 ax-hfi 30966 ax-his1 30969 ax-his2 30970 ax-his3 30971 ax-his4 30972 ax-hcompl 31089 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-fi 9441 df-sup 9472 df-inf 9473 df-oi 9540 df-card 9969 df-acn 9972 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-seq 14008 df-exp 14068 df-hash 14331 df-cj 15087 df-re 15088 df-im 15089 df-sqrt 15223 df-abs 15224 df-clim 15473 df-rlim 15474 df-sum 15674 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-starv 17256 df-sca 17257 df-vsca 17258 df-ip 17259 df-tset 17260 df-ple 17261 df-ds 17263 df-unif 17264 df-hom 17265 df-cco 17266 df-rest 17412 df-topn 17413 df-0g 17431 df-gsum 17432 df-topgen 17433 df-pt 17434 df-prds 17437 df-xrs 17492 df-qtop 17497 df-imas 17498 df-xps 17500 df-mre 17574 df-mrc 17575 df-acs 17577 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18749 df-mulg 19037 df-cntz 19285 df-cmn 19754 df-psmet 21293 df-xmet 21294 df-met 21295 df-bl 21296 df-mopn 21297 df-fbas 21298 df-fg 21299 df-cnfld 21302 df-top 22845 df-topon 22862 df-topsp 22884 df-bases 22898 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-cn 23180 df-cnp 23181 df-lm 23182 df-haus 23268 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-xms 24275 df-ms 24276 df-tms 24277 df-cfil 25232 df-cau 25233 df-cmet 25234 df-grpo 30380 df-gid 30381 df-ginv 30382 df-gdiv 30383 df-ablo 30432 df-vc 30446 df-nv 30479 df-va 30482 df-ba 30483 df-sm 30484 df-0v 30485 df-vs 30486 df-nmcv 30487 df-ims 30488 df-dip 30588 df-ssp 30609 df-ph 30700 df-cbn 30750 df-hnorm 30855 df-hba 30856 df-hvsub 30858 df-hlim 30859 df-hcau 30860 df-sh 31094 df-ch 31108 df-oc 31139 df-ch0 31140 df-shs 31195 df-pjh 31282 df-h0op 31635 |
This theorem is referenced by: df0op2 31639 hosubcl 31660 hoaddcom 31661 hoaddass 31669 hocsubdir 31672 hoaddridi 31673 hodidi 31674 ho0coi 31675 hoaddrid 31678 hodid 31679 ho0subi 31682 ho0sub 31684 hosubid1 31685 honegsub 31686 hoaddsubass 31702 hosd1i 31709 hosubeq0i 31713 0cnop 31866 0hmop 31870 nmop0 31873 hoddi 31877 adj0 31881 nmlnop0iALT 31882 lnopco0i 31891 pjorthcoi 32056 |
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