Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > adjvalval | Structured version Visualization version GIF version |
Description: Value of the value of the adjoint function. (Contributed by NM, 22-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
adjvalval | ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) = (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adjcl 28175 | . . 3 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) ∈ ℋ) | |
2 | eqcom 2617 | . . . . . . 7 ⊢ (((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤) ↔ (𝑥 ·ih 𝑤) = ((𝑇‘𝑥) ·ih 𝐴)) | |
3 | adj2 28177 | . . . . . . . . . 10 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝐴))) | |
4 | 3 | 3com23 1263 | . . . . . . . . 9 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝐴))) |
5 | 4 | 3expa 1257 | . . . . . . . 8 ⊢ (((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝐴))) |
6 | 5 | eqeq2d 2620 | . . . . . . 7 ⊢ (((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝑤) = ((𝑇‘𝑥) ·ih 𝐴) ↔ (𝑥 ·ih 𝑤) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝐴)))) |
7 | 2, 6 | syl5bb 271 | . . . . . 6 ⊢ (((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤) ↔ (𝑥 ·ih 𝑤) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝐴)))) |
8 | 7 | ralbidva 2968 | . . . . 5 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤) ↔ ∀𝑥 ∈ ℋ (𝑥 ·ih 𝑤) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝐴)))) |
9 | 8 | adantr 480 | . . . 4 ⊢ (((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤) ↔ ∀𝑥 ∈ ℋ (𝑥 ·ih 𝑤) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝐴)))) |
10 | simpr 476 | . . . . 5 ⊢ (((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑤 ∈ ℋ) | |
11 | 1 | adantr 480 | . . . . 5 ⊢ (((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) ∈ ℋ) |
12 | hial2eq2 27348 | . . . . 5 ⊢ ((𝑤 ∈ ℋ ∧ ((adjℎ‘𝑇)‘𝐴) ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝑤) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝐴)) ↔ 𝑤 = ((adjℎ‘𝑇)‘𝐴))) | |
13 | 10, 11, 12 | syl2anc 691 | . . . 4 ⊢ (((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝑤) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝐴)) ↔ 𝑤 = ((adjℎ‘𝑇)‘𝐴))) |
14 | 9, 13 | bitrd 267 | . . 3 ⊢ (((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤) ↔ 𝑤 = ((adjℎ‘𝑇)‘𝐴))) |
15 | 1, 14 | riota5 6536 | . 2 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤)) = ((adjℎ‘𝑇)‘𝐴)) |
16 | 15 | eqcomd 2616 | 1 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) = (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 dom cdm 5038 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 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: nmopadjlei 28331 |
Copyright terms: Public domain | W3C validator |