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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 | β’ π = β¨β¨( +β βΎ (π» Γ π»)), ( Β·β βΎ (β Γ π»))β©, (normβ βΎ π»)β© |
hhssims2.3 | β’ π· = (IndMetβπ) |
hhssims2.2 | β’ π» β Sβ |
Ref | Expression |
---|---|
hhssims2 | β’ π· = ((normβ β ββ ) βΎ (π» Γ π»)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhssims2.3 | . 2 β’ π· = (IndMetβπ) | |
2 | hhssims2.1 | . . 3 β’ π = β¨β¨( +β βΎ (π» Γ π»)), ( Β·β βΎ (β Γ π»))β©, (normβ βΎ π»)β© | |
3 | hhssims2.2 | . . 3 β’ π» β Sβ | |
4 | eqid 2728 | . . 3 β’ ((normβ β ββ ) βΎ (π» Γ π»)) = ((normβ β ββ ) βΎ (π» Γ π»)) | |
5 | 2, 3, 4 | hhssims 31112 | . 2 β’ ((normβ β ββ ) βΎ (π» Γ π»)) = (IndMetβπ) |
6 | 1, 5 | eqtr4i 2759 | 1 β’ π· = ((normβ β ββ ) βΎ (π» Γ π»)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 β¨cop 4638 Γ cxp 5680 βΎ cres 5684 β ccom 5686 βcfv 6553 βcc 11146 IndMetcims 30429 +β cva 30758 Β·β csm 30759 normβcno 30761 ββ cmv 30763 Sβ csh 30766 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228 ax-hilex 30837 ax-hfvadd 30838 ax-hvcom 30839 ax-hvass 30840 ax-hv0cl 30841 ax-hvaddid 30842 ax-hfvmul 30843 ax-hvmulid 30844 ax-hvmulass 30845 ax-hvdistr1 30846 ax-hvdistr2 30847 ax-hvmul0 30848 ax-hfi 30917 ax-his1 30920 ax-his2 30921 ax-his3 30922 ax-his4 30923 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-map 8855 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-sup 9475 df-inf 9476 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-n0 12513 df-z 12599 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-icc 13373 df-seq 14009 df-exp 14069 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-topgen 17434 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-top 22824 df-topon 22841 df-bases 22877 df-lm 23161 df-haus 23247 df-grpo 30331 df-gid 30332 df-ginv 30333 df-gdiv 30334 df-ablo 30383 df-vc 30397 df-nv 30430 df-va 30433 df-ba 30434 df-sm 30435 df-0v 30436 df-vs 30437 df-nmcv 30438 df-ims 30439 df-ssp 30560 df-hnorm 30806 df-hba 30807 df-hvsub 30809 df-hlim 30810 df-sh 31045 df-ch 31059 df-ch0 31091 |
This theorem is referenced by: hhsscms 31116 |
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