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Mirrors > Home > HSE Home > Th. List > adj2 | Structured version Visualization version GIF version |
Description: Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
adj2 | ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adj1 28176 | . . . 4 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·ih (𝑇‘𝐴)) = (((adjℎ‘𝑇)‘𝐵) ·ih 𝐴)) | |
2 | simp2 1055 | . . . . 5 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → 𝐵 ∈ ℋ) | |
3 | dmadjop 28131 | . . . . . . 7 ⊢ (𝑇 ∈ dom adjℎ → 𝑇: ℋ⟶ ℋ) | |
4 | 3 | ffvelrnda 6267 | . . . . . 6 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ) |
5 | 4 | 3adant2 1073 | . . . . 5 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ) |
6 | ax-his1 27323 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝑇‘𝐴) ∈ ℋ) → (𝐵 ·ih (𝑇‘𝐴)) = (∗‘((𝑇‘𝐴) ·ih 𝐵))) | |
7 | 2, 5, 6 | syl2anc 691 | . . . 4 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·ih (𝑇‘𝐴)) = (∗‘((𝑇‘𝐴) ·ih 𝐵))) |
8 | adjcl 28175 | . . . . . 6 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐵) ∈ ℋ) | |
9 | 8 | 3adant3 1074 | . . . . 5 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐵) ∈ ℋ) |
10 | simp3 1056 | . . . . 5 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → 𝐴 ∈ ℋ) | |
11 | ax-his1 27323 | . . . . 5 ⊢ ((((adjℎ‘𝑇)‘𝐵) ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((adjℎ‘𝑇)‘𝐵) ·ih 𝐴) = (∗‘(𝐴 ·ih ((adjℎ‘𝑇)‘𝐵)))) | |
12 | 9, 10, 11 | syl2anc 691 | . . . 4 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((adjℎ‘𝑇)‘𝐵) ·ih 𝐴) = (∗‘(𝐴 ·ih ((adjℎ‘𝑇)‘𝐵)))) |
13 | 1, 7, 12 | 3eqtr3d 2652 | . . 3 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝑇‘𝐴) ·ih 𝐵)) = (∗‘(𝐴 ·ih ((adjℎ‘𝑇)‘𝐵)))) |
14 | hicl 27321 | . . . . 5 ⊢ (((𝑇‘𝐴) ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) ∈ ℂ) | |
15 | 5, 2, 14 | syl2anc 691 | . . . 4 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) ∈ ℂ) |
16 | hicl 27321 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ((adjℎ‘𝑇)‘𝐵) ∈ ℋ) → (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵)) ∈ ℂ) | |
17 | 10, 9, 16 | syl2anc 691 | . . . 4 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵)) ∈ ℂ) |
18 | cj11 13750 | . . . 4 ⊢ ((((𝑇‘𝐴) ·ih 𝐵) ∈ ℂ ∧ (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵)) ∈ ℂ) → ((∗‘((𝑇‘𝐴) ·ih 𝐵)) = (∗‘(𝐴 ·ih ((adjℎ‘𝑇)‘𝐵))) ↔ ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵)))) | |
19 | 15, 17, 18 | syl2anc 691 | . . 3 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((∗‘((𝑇‘𝐴) ·ih 𝐵)) = (∗‘(𝐴 ·ih ((adjℎ‘𝑇)‘𝐵))) ↔ ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵)))) |
20 | 13, 19 | mpbid 221 | . 2 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵))) |
21 | 20 | 3com23 1263 | 1 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ∗ccj 13684 ℋchil 27160 ·ih csp 27163 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: adjadj 28179 adjvalval 28180 adjlnop 28329 adjmul 28335 adjadd 28336 adjcoi 28343 nmopcoadji 28344 |
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