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Mirrors > Home > HSE Home > Th. List > dmadjrnb | Structured version Visualization version GIF version |
Description: The adjoint of an operator belongs to the adjoint function's domain. (Note: the converse is dependent on our definition of function value, since it uses ndmfv 6700.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmadjrnb | ⊢ (𝑇 ∈ dom adjℎ ↔ (adjℎ‘𝑇) ∈ dom adjℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmadjrn 29672 | . 2 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ) | |
2 | ax-hv0cl 28780 | . . . . . . . . 9 ⊢ 0ℎ ∈ ℋ | |
3 | 2 | n0ii 4302 | . . . . . . . 8 ⊢ ¬ ℋ = ∅ |
4 | eqcom 2828 | . . . . . . . 8 ⊢ (∅ = ℋ ↔ ℋ = ∅) | |
5 | 3, 4 | mtbir 325 | . . . . . . 7 ⊢ ¬ ∅ = ℋ |
6 | dm0 5790 | . . . . . . . 8 ⊢ dom ∅ = ∅ | |
7 | 6 | eqeq1i 2826 | . . . . . . 7 ⊢ (dom ∅ = ℋ ↔ ∅ = ℋ) |
8 | 5, 7 | mtbir 325 | . . . . . 6 ⊢ ¬ dom ∅ = ℋ |
9 | fdm 6522 | . . . . . 6 ⊢ (∅: ℋ⟶ ℋ → dom ∅ = ℋ) | |
10 | 8, 9 | mto 199 | . . . . 5 ⊢ ¬ ∅: ℋ⟶ ℋ |
11 | dmadjop 29665 | . . . . 5 ⊢ (∅ ∈ dom adjℎ → ∅: ℋ⟶ ℋ) | |
12 | 10, 11 | mto 199 | . . . 4 ⊢ ¬ ∅ ∈ dom adjℎ |
13 | ndmfv 6700 | . . . . 5 ⊢ (¬ 𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) = ∅) | |
14 | 13 | eleq1d 2897 | . . . 4 ⊢ (¬ 𝑇 ∈ dom adjℎ → ((adjℎ‘𝑇) ∈ dom adjℎ ↔ ∅ ∈ dom adjℎ)) |
15 | 12, 14 | mtbiri 329 | . . 3 ⊢ (¬ 𝑇 ∈ dom adjℎ → ¬ (adjℎ‘𝑇) ∈ dom adjℎ) |
16 | 15 | con4i 114 | . 2 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → 𝑇 ∈ dom adjℎ) |
17 | 1, 16 | impbii 211 | 1 ⊢ (𝑇 ∈ dom adjℎ ↔ (adjℎ‘𝑇) ∈ dom adjℎ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∅c0 4291 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 ℋchba 28696 0ℎc0v 28701 adjℎcado 28732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvdistr2 28786 ax-hvmul0 28787 ax-hfi 28856 ax-his1 28859 ax-his2 28860 ax-his3 28861 ax-his4 28862 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-cj 14458 df-re 14459 df-im 14460 df-hvsub 28748 df-adjh 29626 |
This theorem is referenced by: adjbdlnb 29861 adjeq0 29868 |
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