adj1 - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > adj1		Structured version Visualization version GIF version

Theorem adj1 29712

Description: Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
adj1	⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))

Proof of Theorem adj1 Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funadj 29665	. . . . . . 7 ⊢ Fun adj_ℎ
2		funfvop 6822	. . . . . . 7 ⊢ ((Fun adj_ℎ ∧ 𝑇 ∈ dom adj_ℎ) → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ adj_ℎ)
3	1, 2	mpan 688	. . . . . 6 ⊢ (𝑇 ∈ dom adj_ℎ → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ adj_ℎ)
4		dfadj2 29664	. . . . . 6 ⊢ adj_ℎ = {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))}
5	3, 4	eleqtrdi 2925	. . . . 5 ⊢ (𝑇 ∈ dom adj_ℎ → ⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))})
6		fvex 6685	. . . . . 6 ⊢ (adj_ℎ‘𝑇) ∈ V
7		feq1 6497	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8		fveq1 6671	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → (𝑧‘𝑦) = (𝑇‘𝑦))
9	8	oveq2d 7174	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑥 ·_ih (𝑧‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
10	9	eqeq1d 2825	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)))
11	10	2ralbidv 3201	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)))
12	7, 11	3anbi13d 1434	. . . . . . 7 ⊢ (𝑧 = 𝑇 → ((𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))))
13		feq1 6497	. . . . . . . 8 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj_ℎ‘𝑇): ℋ⟶ ℋ))
14		fveq1 6671	. . . . . . . . . . 11 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (𝑤‘𝑥) = ((adj_ℎ‘𝑇)‘𝑥))
15	14	oveq1d 7173	. . . . . . . . . 10 ⊢ (𝑤 = (adj_ℎ‘𝑇) → ((𝑤‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))
16	15	eqeq2d 2834	. . . . . . . . 9 ⊢ (𝑤 = (adj_ℎ‘𝑇) → ((𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦)))
17	16	2ralbidv 3201	. . . . . . . 8 ⊢ (𝑤 = (adj_ℎ‘𝑇) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦)))
18	13, 17	3anbi23d 1435	. . . . . . 7 ⊢ (𝑤 = (adj_ℎ‘𝑇) → ((𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (adj_ℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))))
19	12, 18	opelopabg 5427	. . . . . 6 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (adj_ℎ‘𝑇) ∈ V) → (⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adj_ℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))))
20	6, 19	mpan2 689	. . . . 5 ⊢ (𝑇 ∈ dom adj_ℎ → (⟨𝑇, (adj_ℎ‘𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑧‘𝑦)) = ((𝑤‘𝑥) ·_ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adj_ℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))))
21	5, 20	mpbid 234	. . . 4 ⊢ (𝑇 ∈ dom adj_ℎ → (𝑇: ℋ⟶ ℋ ∧ (adj_ℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦)))
22	21	simp3d 1140	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦))
23		oveq1 7165	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝑦)))
24		fveq2 6672	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((adj_ℎ‘𝑇)‘𝑥) = ((adj_ℎ‘𝑇)‘𝐴))
25	24	oveq1d 7173	. . . . 5 ⊢ (𝑥 = 𝐴 → (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦))
26	23, 25	eqeq12d 2839	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦) ↔ (𝐴 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦)))
27		fveq2 6672	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
28	27	oveq2d 7174	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝐵)))
29		oveq2 7166	. . . . 5 ⊢ (𝑦 = 𝐵 → (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))
30	28, 29	eqeq12d 2839	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝑦) ↔ (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵)))
31	26, 30	rspc2v 3635	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = (((adj_ℎ‘𝑇)‘𝑥) ·_ih 𝑦) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵)))
32	22, 31	syl5com 31	. 2 ⊢ (𝑇 ∈ dom adj_ℎ → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵)))
33	32	3impib 1112	1 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = (((adj_ℎ‘𝑇)‘𝐴) ·_ih 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ⟨cop 4575 {copab 5130 dom cdm 5557 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℋchba 28698 ·_ih csp 28701 adj_ℎcado 28734

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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864

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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 df-hvsub 28750 df-adjh 29628

This theorem is referenced by: adj2 29713 adjadj 29715 hmopadj2 29720

W3C validator