Theorem elunop2 32032

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 31939	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		elunop 31891	. . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))
3	2	simplbi 497	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
4		unopnorm 31936	. . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))
5	4	ralrimiva 3146	. . 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 2829	. . 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 2829	. . . . . . 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 6816	. . . . . . 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 6911	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘(𝑇‘𝑥)) = (normℎ‘(𝑇‘𝑦)))
11		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘𝑥) = (normℎ‘𝑦))
12	10, 11	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦)))
13	12	cbvralvw 3237	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
14		fveq1 6905	. . . . . . . . . 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 6914	. . . . . . . . 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 3178	. . . . . . . 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	bitrid 283	. . . . . . 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 1438	. . . . . 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 2829	. . . . . . 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 6816	. . . . . . 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 6905	. . . . . . . . 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 6914	. . . . . . . 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 3178	. . . . . . 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 1438	. . . . . 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 32011	. . . . . . 7 ⊢ ( I ↾ ℋ) ∈ LinOp
26		f1oi 6886	. . . . . . . 8 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
27		f1ofo 6855	. . . . . . . 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 7193	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
30	29	fveq2d 6910	. . . . . . . 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 4591	. . . . 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 32031	. . 3 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp
38	7, 37	dedth 4584	. 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)) → 𝑇 ∈ UniOp)
39	6, 38	impbii 209	1 ⊢ (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)))

Syntax hints:   ↔ wb 206   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ifcif 4525   I cid 5577   ↾ cres 5687  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   ℋchba 30938   ·_ih csp 30941  norm_ℎcno 30942  LinOpclo 30966  UniOpcuo 30968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-hilex 31018  ax-hfvadd 31019  ax-hvcom 31020  ax-hvass 31021  ax-hv0cl 31022  ax-hvaddid 31023  ax-hfvmul 31024  ax-hvmulid 31025  ax-hvdistr2 31028  ax-hvmul0 31029  ax-hfi 31098  ax-his1 31101  ax-his2 31102  ax-his3 31103  ax-his4 31104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-hnorm 30987  df-hvsub 30990  df-lnop 31860  df-unop 31862

This theorem is referenced by: (None)