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Mirrors > Home > HSE Home > Th. List > adjsslnop | Structured version Visualization version GIF version |
Description: Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
adjsslnop | ⊢ dom adjℎ ⊆ LinOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adjadj 31864 | . . 3 ⊢ (𝑡 ∈ dom adjℎ → (adjℎ‘(adjℎ‘𝑡)) = 𝑡) | |
2 | dmadjrn 31823 | . . . 4 ⊢ (𝑡 ∈ dom adjℎ → (adjℎ‘𝑡) ∈ dom adjℎ) | |
3 | adjlnop 32014 | . . . 4 ⊢ ((adjℎ‘𝑡) ∈ dom adjℎ → (adjℎ‘(adjℎ‘𝑡)) ∈ LinOp) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑡 ∈ dom adjℎ → (adjℎ‘(adjℎ‘𝑡)) ∈ LinOp) |
5 | 1, 4 | eqeltrrd 2827 | . 2 ⊢ (𝑡 ∈ dom adjℎ → 𝑡 ∈ LinOp) |
6 | 5 | ssriv 3983 | 1 ⊢ dom adjℎ ⊆ LinOp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ⊆ wss 3947 dom cdm 5673 ‘cfv 6544 LinOpclo 30875 adjℎcado 30883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-hilex 30927 ax-hfvadd 30928 ax-hvcom 30929 ax-hvass 30930 ax-hv0cl 30931 ax-hvaddid 30932 ax-hfvmul 30933 ax-hvmulid 30934 ax-hvdistr2 30937 ax-hvmul0 30938 ax-hfi 31007 ax-his1 31010 ax-his2 31011 ax-his3 31012 ax-his4 31013 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-2 12319 df-cj 15097 df-re 15098 df-im 15099 df-hvsub 30899 df-lnop 31769 df-adjh 31777 |
This theorem is referenced by: (None) |
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