lnophmi - Hilbert Space Explorer

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Theorem lnophmi 31271

Description: A linear operator is Hermitian if 𝑥 ·_ih (𝑇‘𝑥) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
lnophm.1	⊢ 𝑇 ∈ LinOp
lnophm.2	⊢ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑥)) ∈ ℝ

**Assertion**
Ref	Expression
lnophmi	⊢ 𝑇 ∈ HrmOp

Distinct variable group: 𝑥,𝑇

Proof of Theorem lnophmi Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnophm.1	. . 3 ⊢ 𝑇 ∈ LinOp
2	1	lnopfi 31222	. 2 ⊢ 𝑇: ℋ⟶ ℋ
3		oveq1 7416	. . . . 5 ⊢ (𝑦 = if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) → (𝑦 ·_ih (𝑇‘𝑧)) = (if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) ·_ih (𝑇‘𝑧)))
4		fveq2 6892	. . . . . 6 ⊢ (𝑦 = if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) → (𝑇‘𝑦) = (𝑇‘if(𝑦 ∈ ℋ, 𝑦, 0_ℎ)))
5	4	oveq1d 7424	. . . . 5 ⊢ (𝑦 = if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) → ((𝑇‘𝑦) ·_ih 𝑧) = ((𝑇‘if(𝑦 ∈ ℋ, 𝑦, 0_ℎ)) ·_ih 𝑧))
6	3, 5	eqeq12d 2749	. . . 4 ⊢ (𝑦 = if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) → ((𝑦 ·_ih (𝑇‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧) ↔ (if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) ·_ih (𝑇‘𝑧)) = ((𝑇‘if(𝑦 ∈ ℋ, 𝑦, 0_ℎ)) ·_ih 𝑧)))
7		fveq2 6892	. . . . . 6 ⊢ (𝑧 = if(𝑧 ∈ ℋ, 𝑧, 0_ℎ) → (𝑇‘𝑧) = (𝑇‘if(𝑧 ∈ ℋ, 𝑧, 0_ℎ)))
8	7	oveq2d 7425	. . . . 5 ⊢ (𝑧 = if(𝑧 ∈ ℋ, 𝑧, 0_ℎ) → (if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) ·_ih (𝑇‘𝑧)) = (if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) ·_ih (𝑇‘if(𝑧 ∈ ℋ, 𝑧, 0_ℎ))))
9		oveq2 7417	. . . . 5 ⊢ (𝑧 = if(𝑧 ∈ ℋ, 𝑧, 0_ℎ) → ((𝑇‘if(𝑦 ∈ ℋ, 𝑦, 0_ℎ)) ·_ih 𝑧) = ((𝑇‘if(𝑦 ∈ ℋ, 𝑦, 0_ℎ)) ·_ih if(𝑧 ∈ ℋ, 𝑧, 0_ℎ)))
10	8, 9	eqeq12d 2749	. . . 4 ⊢ (𝑧 = if(𝑧 ∈ ℋ, 𝑧, 0_ℎ) → ((if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) ·_ih (𝑇‘𝑧)) = ((𝑇‘if(𝑦 ∈ ℋ, 𝑦, 0_ℎ)) ·_ih 𝑧) ↔ (if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) ·_ih (𝑇‘if(𝑧 ∈ ℋ, 𝑧, 0_ℎ))) = ((𝑇‘if(𝑦 ∈ ℋ, 𝑦, 0_ℎ)) ·_ih if(𝑧 ∈ ℋ, 𝑧, 0_ℎ))))
11		ifhvhv0 30275	. . . . 5 ⊢ if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) ∈ ℋ
12		ifhvhv0 30275	. . . . 5 ⊢ if(𝑧 ∈ ℋ, 𝑧, 0_ℎ) ∈ ℋ
13		lnophm.2	. . . . 5 ⊢ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑥)) ∈ ℝ
14	11, 12, 1, 13	lnophmlem2 31270	. . . 4 ⊢ (if(𝑦 ∈ ℋ, 𝑦, 0_ℎ) ·_ih (𝑇‘if(𝑧 ∈ ℋ, 𝑧, 0_ℎ))) = ((𝑇‘if(𝑦 ∈ ℋ, 𝑦, 0_ℎ)) ·_ih if(𝑧 ∈ ℋ, 𝑧, 0_ℎ))
15	6, 10, 14	dedth2h 4588	. . 3 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 ·_ih (𝑇‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧))
16	15	rgen2 3198	. 2 ⊢ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑦 ·_ih (𝑇‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧)
17		elhmop 31126	. 2 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑦 ·_ih (𝑇‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧)))
18	2, 16, 17	mpbir2an 710	1 ⊢ 𝑇 ∈ HrmOp

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2107 ∀wral 3062 ifcif 4529 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 ℋchba 30172 ·_ih csp 30175 0_ℎc0v 30177 LinOpclo 30200 HrmOpcho 30203

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-hilex 30252 ax-hfvadd 30253 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvmulass 30260 ax-hvdistr1 30261 ax-hvdistr2 30262 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his2 30336 ax-his3 30337

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-2 12275 df-3 12276 df-4 12277 df-cj 15046 df-re 15047 df-im 15048 df-hvsub 30224 df-lnop 31094 df-hmop 31097

This theorem is referenced by: lnophm 31272

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