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Mirrors > Home > HSE Home > Th. List > adjcl | Structured version Visualization version GIF version |
Description: Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
adjcl | ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmadjrn 30251 | . . 3 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ) | |
2 | dmadjop 30244 | . . 3 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ) |
4 | 3 | ffvelrnda 6956 | 1 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2110 dom cdm 5589 ⟶wf 6427 ‘cfv 6431 ℋchba 29275 adjℎcado 29311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-hilex 29355 ax-hfvadd 29356 ax-hvcom 29357 ax-hvass 29358 ax-hv0cl 29359 ax-hvaddid 29360 ax-hfvmul 29361 ax-hvmulid 29362 ax-hvdistr2 29365 ax-hvmul0 29366 ax-hfi 29435 ax-his1 29438 ax-his2 29439 ax-his3 29440 ax-his4 29441 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-er 8479 df-map 8598 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-2 12034 df-cj 14806 df-re 14807 df-im 14808 df-hvsub 29327 df-adjh 30205 |
This theorem is referenced by: adj2 30290 adjvalval 30293 adjlnop 30442 adjmul 30448 adjadd 30449 |
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