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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 6128.) (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 28138 | . 2 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ) | |
2 | ax-hv0cl 27244 | . . . . . . . . 9 ⊢ 0ℎ ∈ ℋ | |
3 | 2 | n0ii 3881 | . . . . . . . 8 ⊢ ¬ ℋ = ∅ |
4 | eqcom 2617 | . . . . . . . 8 ⊢ (∅ = ℋ ↔ ℋ = ∅) | |
5 | 3, 4 | mtbir 312 | . . . . . . 7 ⊢ ¬ ∅ = ℋ |
6 | dm0 5260 | . . . . . . . 8 ⊢ dom ∅ = ∅ | |
7 | 6 | eqeq1i 2615 | . . . . . . 7 ⊢ (dom ∅ = ℋ ↔ ∅ = ℋ) |
8 | 5, 7 | mtbir 312 | . . . . . 6 ⊢ ¬ dom ∅ = ℋ |
9 | fdm 5964 | . . . . . 6 ⊢ (∅: ℋ⟶ ℋ → dom ∅ = ℋ) | |
10 | 8, 9 | mto 187 | . . . . 5 ⊢ ¬ ∅: ℋ⟶ ℋ |
11 | dmadjop 28131 | . . . . 5 ⊢ (∅ ∈ dom adjℎ → ∅: ℋ⟶ ℋ) | |
12 | 10, 11 | mto 187 | . . . 4 ⊢ ¬ ∅ ∈ dom adjℎ |
13 | ndmfv 6128 | . . . . 5 ⊢ (¬ 𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) = ∅) | |
14 | 13 | eleq1d 2672 | . . . 4 ⊢ (¬ 𝑇 ∈ dom adjℎ → ((adjℎ‘𝑇) ∈ dom adjℎ ↔ ∅ ∈ dom adjℎ)) |
15 | 12, 14 | mtbiri 316 | . . 3 ⊢ (¬ 𝑇 ∈ dom adjℎ → ¬ (adjℎ‘𝑇) ∈ dom adjℎ) |
16 | 15 | con4i 112 | . 2 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → 𝑇 ∈ dom adjℎ) |
17 | 1, 16 | impbii 198 | 1 ⊢ (𝑇 ∈ dom adjℎ ↔ (adjℎ‘𝑇) ∈ dom adjℎ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∅c0 3874 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 ℋchil 27160 0ℎc0v 27165 adjℎcado 27196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 df-im 13689 df-hvsub 27212 df-adjh 28092 |
This theorem is referenced by: adjbdlnb 28327 adjeq0 28334 |
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