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Theorem eigvecval 31658

Description: The set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
eigvecval	⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·_ℎ 𝑥)})

Distinct variable group: 𝑥,𝑦,𝑇

Proof of Theorem eigvecval Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 30761	. . . 4 ⊢ ℋ ∈ V
2		difexg 5320	. . . 4 ⊢ ( ℋ ∈ V → ( ℋ ∖ 0_ℋ) ∈ V)
3	1, 2	ax-mp 5	. . 3 ⊢ ( ℋ ∖ 0_ℋ) ∈ V
4	3	rabex 5325	. 2 ⊢ {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·_ℎ 𝑥)} ∈ V
5		fveq1 6884	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
6	5	eqeq1d 2728	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) = (𝑦 ·_ℎ 𝑥) ↔ (𝑇‘𝑥) = (𝑦 ·_ℎ 𝑥)))
7	6	rexbidv 3172	. . 3 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·_ℎ 𝑥) ↔ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·_ℎ 𝑥)))
8	7	rabbidv 3434	. 2 ⊢ (𝑡 = 𝑇 → {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·_ℎ 𝑥)} = {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·_ℎ 𝑥)})
9		df-eigvec 31615	. 2 ⊢ eigvec = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑦 ∈ ℂ (𝑡‘𝑥) = (𝑦 ·_ℎ 𝑥)})
10	4, 1, 1, 8, 9	fvmptmap 8877	1 ⊢ (𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·_ℎ 𝑥)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 {crab 3426 Vcvv 3468 ∖ cdif 3940 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 ℋchba 30681 ·_ℎ csm 30683 0_ℋc0h 30697 eigveccei 30721

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-hilex 30761

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8824 df-eigvec 31615

This theorem is referenced by: eleigvec 31719

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