Theorem elunop2 29948

Description: An operator is unitary iff it is linear, onto, and idempotent in the norm. Similar to theorem in [AkhiezerGlazman] p. 73, and its converse. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
elunop2	⊢ (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)))

Distinct variable group:   𝑥,𝑇

Proof of Theorem elunop2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unoplin 29855	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		elunop 29807	. . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))
3	2	simplbi 501	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
4		unopnorm 29852	. . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))
5	4	ralrimiva 3096	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))
6	1, 3, 5	3jca 1129	. 2 ⊢ (𝑇 ∈ UniOp → (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)))
7		eleq1 2820	. . 3 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (𝑇 ∈ UniOp ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp))
8		eleq1 2820	. . . . . . 7 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (𝑇 ∈ LinOp ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp))
9		foeq1 6588	. . . . . . 7 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (𝑇: ℋ–onto→ ℋ ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ))
10		2fveq3 6679	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘(𝑇‘𝑥)) = (normℎ‘(𝑇‘𝑦)))
11		fveq2 6674	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘𝑥) = (normℎ‘𝑦))
12	10, 11	eqeq12d 2754	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦)))
13	12	cbvralvw 3349	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
14		fveq1 6673	. . . . . . . . . 10 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (𝑇‘𝑦) = (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦))
15	14	fveqeq2d 6682	. . . . . . . . 9 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → ((normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦) ↔ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦)))
16	15	ralbidv 3109	. . . . . . . 8 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦) ↔ ∀𝑦 ∈ ℋ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦)))
17	13, 16	syl5bb 286	. . . . . . 7 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ ∀𝑦 ∈ ℋ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦)))
18	8, 9, 17	3anbi123d 1437	. . . . . 6 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → ((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)) ↔ (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦))))
19		eleq1 2820	. . . . . . 7 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (( I ↾ ℋ) ∈ LinOp ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp))
20		foeq1 6588	. . . . . . 7 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (( I ↾ ℋ): ℋ–onto→ ℋ ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ))
21		fveq1 6673	. . . . . . . . 9 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (( I ↾ ℋ)‘𝑦) = (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦))
22	21	fveqeq2d 6682	. . . . . . . 8 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → ((normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦) ↔ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦)))
23	22	ralbidv 3109	. . . . . . 7 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (∀𝑦 ∈ ℋ (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦) ↔ ∀𝑦 ∈ ℋ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦)))
24	19, 20, 23	3anbi123d 1437	. . . . . 6 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → ((( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦)) ↔ (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦))))
25		idlnop 29927	. . . . . . 7 ⊢ ( I ↾ ℋ) ∈ LinOp
26		f1oi 6655	. . . . . . . 8 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
27		f1ofo 6625	. . . . . . . 8 ⊢ (( I ↾ ℋ): ℋ–1-1-onto→ ℋ → ( I ↾ ℋ): ℋ–onto→ ℋ)
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ ℋ): ℋ–onto→ ℋ
29		fvresi 6945	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
30	29	fveq2d 6678	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦))
31	30	rgen 3063	. . . . . . 7 ⊢ ∀𝑦 ∈ ℋ (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦)
32	25, 28, 31	3pm3.2i 1340	. . . . . 6 ⊢ (( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦))
33	18, 24, 32	elimhyp 4479	. . . . 5 ⊢ (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦))
34	33	simp1i 1140	. . . 4 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp
35	33	simp2i 1141	. . . 4 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ
36	33	simp3i 1142	. . . 4 ⊢ ∀𝑦 ∈ ℋ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦)
37	34, 35, 36	lnopunii 29947	. . 3 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp
38	7, 37	dedth 4472	. 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)) → 𝑇 ∈ UniOp)
39	6, 38	impbii 212	1 ⊢ (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)))

Syntax hints:   ↔ wb 209   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3053  ifcif 4414   I cid 5428   ↾ cres 5527  –onto→wfo 6337  –1-1-onto→wf1o 6338  ‘cfv 6339  (class class class)co 7170   ℋchba 28854   ·_ih csp 28857  norm_ℎcno 28858  LinOpclo 28882  UniOpcuo 28884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693  ax-hilex 28934  ax-hfvadd 28935  ax-hvcom 28936  ax-hvass 28937  ax-hv0cl 28938  ax-hvaddid 28939  ax-hfvmul 28940  ax-hvmulid 28941  ax-hvdistr2 28944  ax-hvmul0 28945  ax-hfi 29014  ax-his1 29017  ax-his2 29018  ax-his3 29019  ax-his4 29020

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-sup 8979  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-rp 12473  df-seq 13461  df-exp 13522  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-hnorm 28903  df-hvsub 28906  df-lnop 29776  df-unop 29778

This theorem is referenced by: (None)