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Mirrors > Home > HSE Home > Th. List > hilhhi | Structured version Visualization version GIF version |
Description: Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hilhh.1 | ⊢ ℋ = (BaseSet‘𝑈) |
hilhh.2 | ⊢ +ℎ = ( +𝑣 ‘𝑈) |
hilhh.3 | ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
hilhh.5 | ⊢ ·ih = (·𝑖OLD‘𝑈) |
hilhh.9 | ⊢ 𝑈 ∈ NrmCVec |
Ref | Expression |
---|---|
hilhhi | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hilhh.9 | . 2 ⊢ 𝑈 ∈ NrmCVec | |
2 | hilhh.2 | . . 3 ⊢ +ℎ = ( +𝑣 ‘𝑈) | |
3 | hilhh.3 | . . 3 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) | |
4 | hilhh.1 | . . . 4 ⊢ ℋ = (BaseSet‘𝑈) | |
5 | hilhh.5 | . . . 4 ⊢ ·ih = (·𝑖OLD‘𝑈) | |
6 | 4, 5, 1 | hilnormi 30105 | . . 3 ⊢ normℎ = (normCV‘𝑈) |
7 | 2, 3, 6 | nvop 29618 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
8 | 1, 7 | ax-mp 5 | 1 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 〈cop 4592 ‘cfv 6496 NrmCVeccnv 29526 +𝑣 cpv 29527 BaseSetcba 29528 ·𝑠OLD cns 29529 ·𝑖OLDcdip 29642 ℋchba 29861 +ℎ cva 29862 ·ℎ csm 29863 ·ih csp 29864 normℎcno 29865 |
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 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-hfi 30021 |
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-int 4908 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-se 5589 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-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-grpo 29435 df-gid 29436 df-ginv 29437 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-nmcv 29542 df-dip 29643 df-hnorm 29910 |
This theorem is referenced by: hilims 30139 hilcompl 30143 |
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