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Mirrors > Home > HSE Home > Th. List > hh0v | Structured version Visualization version GIF version |
Description: The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhnv.1 | โข ๐ = โจโจ +โ , ยทโ โฉ, normโโฉ |
Ref | Expression |
---|---|
hh0v | โข 0โ = (0vecโ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhnv.1 | . . . 4 โข ๐ = โจโจ +โ , ยทโ โฉ, normโโฉ | |
2 | 1 | hhnv 30110 | . . 3 โข ๐ โ NrmCVec |
3 | eqid 2737 | . . . 4 โข ( +๐ฃ โ๐) = ( +๐ฃ โ๐) | |
4 | eqid 2737 | . . . 4 โข (0vecโ๐) = (0vecโ๐) | |
5 | 3, 4 | 0vfval 29551 | . . 3 โข (๐ โ NrmCVec โ (0vecโ๐) = (GIdโ( +๐ฃ โ๐))) |
6 | 2, 5 | ax-mp 5 | . 2 โข (0vecโ๐) = (GIdโ( +๐ฃ โ๐)) |
7 | 1 | hhva 30111 | . . 3 โข +โ = ( +๐ฃ โ๐) |
8 | 7 | fveq2i 6846 | . 2 โข (GIdโ +โ ) = (GIdโ( +๐ฃ โ๐)) |
9 | hilid 30106 | . 2 โข (GIdโ +โ ) = 0โ | |
10 | 6, 8, 9 | 3eqtr2ri 2772 | 1 โข 0โ = (0vecโ๐) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 โจcop 4593 โcfv 6497 GIdcgi 29435 NrmCVeccnv 29529 +๐ฃ cpv 29530 0veccn0v 29533 +โ cva 29865 ยทโ csm 29866 normโcno 29868 0โc0v 29869 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 ax-hilex 29944 ax-hfvadd 29945 ax-hvcom 29946 ax-hvass 29947 ax-hv0cl 29948 ax-hvaddid 29949 ax-hfvmul 29950 ax-hvmulid 29951 ax-hvmulass 29952 ax-hvdistr1 29953 ax-hvdistr2 29954 ax-hvmul0 29955 ax-hfi 30024 ax-his1 30027 ax-his2 30028 ax-his3 30029 ax-his4 30030 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9379 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-grpo 29438 df-gid 29439 df-ablo 29490 df-vc 29504 df-nv 29537 df-va 29540 df-0v 29543 df-hnorm 29913 df-hvsub 29916 |
This theorem is referenced by: hhshsslem2 30213 hh0oi 30848 |
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