![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > hhssims2 | Structured version Visualization version GIF version |
Description: Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhssims2.1 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
hhssims2.3 | ⊢ 𝐷 = (IndMet‘𝑊) |
hhssims2.2 | ⊢ 𝐻 ∈ Sℋ |
Ref | Expression |
---|---|
hhssims2 | ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhssims2.3 | . 2 ⊢ 𝐷 = (IndMet‘𝑊) | |
2 | hhssims2.1 | . . 3 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
3 | hhssims2.2 | . . 3 ⊢ 𝐻 ∈ Sℋ | |
4 | eqid 2736 | . . 3 ⊢ ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) | |
5 | 2, 3, 4 | hhssims 30163 | . 2 ⊢ ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) = (IndMet‘𝑊) |
6 | 1, 5 | eqtr4i 2767 | 1 ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 〈cop 4592 × cxp 5631 ↾ cres 5635 ∘ ccom 5637 ‘cfv 6496 ℂcc 11048 IndMetcims 29480 +ℎ cva 29809 ·ℎ csm 29810 normℎcno 29812 −ℎ cmv 29814 Sℋ csh 29817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 ax-addf 11129 ax-mulf 11130 ax-hilex 29888 ax-hfvadd 29889 ax-hvcom 29890 ax-hvass 29891 ax-hv0cl 29892 ax-hvaddid 29893 ax-hfvmul 29894 ax-hvmulid 29895 ax-hvmulass 29896 ax-hvdistr1 29897 ax-hvdistr2 29898 ax-hvmul0 29899 ax-hfi 29968 ax-his1 29971 ax-his2 29972 ax-his3 29973 ax-his4 29974 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-map 8766 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-sup 9377 df-inf 9378 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-n0 12413 df-z 12499 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-icc 13270 df-seq 13906 df-exp 13967 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-topgen 17324 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-top 22241 df-topon 22258 df-bases 22294 df-lm 22578 df-haus 22664 df-grpo 29382 df-gid 29383 df-ginv 29384 df-gdiv 29385 df-ablo 29434 df-vc 29448 df-nv 29481 df-va 29484 df-ba 29485 df-sm 29486 df-0v 29487 df-vs 29488 df-nmcv 29489 df-ims 29490 df-ssp 29611 df-hnorm 29857 df-hba 29858 df-hvsub 29860 df-hlim 29861 df-sh 30096 df-ch 30110 df-ch0 30142 |
This theorem is referenced by: hhsscms 30167 |
Copyright terms: Public domain | W3C validator |