Theorem elunop2 32394

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 32301	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		elunop 32253	. . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))
3	2	simplbi 502	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
4		unopnorm 32298	. . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))
5	4	ralrimiva 3159	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))
6	1, 3, 5	3jca 1146	. 2 ⊢ (𝑇 ∈ UniOp → (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)))
7		eleq1 2853	. . 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 2853	. . . . . . 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 6792	. . . . . . 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 6890	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘(𝑇‘𝑥)) = (normℎ‘(𝑇‘𝑦)))
11		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘𝑥) = (normℎ‘𝑦))
12	10, 11	eqeq12d 2781	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦)))
13	12	cbvralvw 3245	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
14		fveq1 6884	. . . . . . . . . 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 6893	. . . . . . . . 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 3190	. . . . . . . 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 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 1464	. . . . . 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 2853	. . . . . . 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 6792	. . . . . . 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 6884	. . . . . . . . 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 6893	. . . . . . . 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 3190	. . . . . . 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 1464	. . . . . 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 32373	. . . . . . 7 ⊢ ( I ↾ ℋ) ∈ LinOp
26		f1oi 6863	. . . . . . . 8 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
27		f1ofo 6832	. . . . . . . 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 7175	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
30	29	fveq2d 6889	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦))
31	30	rgen 3083	. . . . . . 7 ⊢ ∀𝑦 ∈ ℋ (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦)
32	25, 28, 31	3pm3.2i 1358	. . . . . 6 ⊢ (( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦))
33	18, 24, 32	elimhyp 4555	. . . . 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 1157	. . . 4 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp
35	33	simp2i 1158	. . . 4 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ
36	33	simp3i 1159	. . . 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 32393	. . 3 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp
38	7, 37	dedth 4548	. 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)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ifcif 4489   I cid 5557   ↾ cres 5665  –onto→wfo 6538  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7416   ℋchba 31300   ·_ih csp 31303  norm_ℎcno 31304  LinOpclo 31328  UniOpcuo 31330

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188  ax-pre-sup 11189  ax-hilex 31380  ax-hfvadd 31381  ax-hvcom 31382  ax-hvass 31383  ax-hv0cl 31384  ax-hvaddid 31385  ax-hfvmul 31386  ax-hvmulid 31387  ax-hvdistr2 31390  ax-hvmul0 31391  ax-hfi 31460  ax-his1 31463  ax-his2 31464  ax-his3 31465  ax-his4 31466

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9405  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-div 11883  df-nn 12245  df-2 12314  df-3 12315  df-n0 12516  df-z 12603  df-uz 12874  df-rp 13028  df-seq 14051  df-exp 14111  df-cj 15169  df-re 15170  df-im 15171  df-sqrt 15305  df-hnorm 31349  df-hvsub 31352  df-lnop 32222  df-unop 32224

This theorem is used by: (None)