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Mirrors > Home > HSE Home > Th. List > hmdmadj | Structured version Visualization version GIF version |
Description: Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hmdmadj | ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ dom adjℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmopf 31761 | . . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) | |
2 | hon0 31680 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑇 ∈ HrmOp → ¬ 𝑇 = ∅) |
4 | hmopadj 31826 | . . . 4 ⊢ (𝑇 ∈ HrmOp → (adjℎ‘𝑇) = 𝑇) | |
5 | 4 | eqeq1d 2727 | . . 3 ⊢ (𝑇 ∈ HrmOp → ((adjℎ‘𝑇) = ∅ ↔ 𝑇 = ∅)) |
6 | 3, 5 | mtbird 324 | . 2 ⊢ (𝑇 ∈ HrmOp → ¬ (adjℎ‘𝑇) = ∅) |
7 | ndmfv 6931 | . 2 ⊢ (¬ 𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) = ∅) | |
8 | 6, 7 | nsyl2 141 | 1 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ dom adjℎ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ∅c0 4322 dom cdm 5678 ⟶wf 6545 ‘cfv 6549 ℋchba 30806 HrmOpcho 30837 adjℎcado 30842 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-hilex 30886 ax-hfvadd 30887 ax-hvcom 30888 ax-hvass 30889 ax-hv0cl 30890 ax-hvaddid 30891 ax-hfvmul 30892 ax-hvmulid 30893 ax-hvdistr2 30896 ax-hvmul0 30897 ax-hfi 30966 ax-his1 30969 ax-his2 30970 ax-his3 30971 ax-his4 30972 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-2 12313 df-cj 15087 df-re 15088 df-im 15089 df-hvsub 30858 df-hmop 31731 df-adjh 31736 |
This theorem is referenced by: (None) |
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