Theorem elunop2 32672

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 32579	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		elunop 32531	. . . 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 32576	. . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))
5	4	ralrimiva 3158	. . 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 2852	. . 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 2852	. . . . . . 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 6822	. . . . . . 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 6921	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘(𝑇‘𝑥)) = (normℎ‘(𝑇‘𝑦)))
11		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘𝑥) = (normℎ‘𝑦))
12	10, 11	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦)))
13	12	cbvralvw 3244	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
14		fveq1 6915	. . . . . . . . . 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 6924	. . . . . . . . 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 3189	. . . . . . . 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 1465	. . . . . 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 2852	. . . . . . 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 6822	. . . . . . 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 6915	. . . . . . . . 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 6924	. . . . . . . 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 3189	. . . . . . 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 1465	. . . . . 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 32651	. . . . . . 7 ⊢ ( I ↾ ℋ) ∈ LinOp
26		f1oi 6894	. . . . . . . 8 ⊢ ( I ↾ ℋ): ℋ⟶1-1onto ℋ
27		f1ofo 6862	. . . . . . . 8 ⊢ (( I ↾ ℋ): ℋ⟶1-1onto ℋ → ( I ↾ ℋ): ℋ⟶onto ℋ)
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ ℋ): ℋ⟶onto ℋ
29		fvresi 7209	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
30	29	fveq2d 6920	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦))
31	30	rgen 3082	. . . . . . 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 4551	. . . . 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 32671	. . 3 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ⟶onto ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp
38	7, 37	dedth 4544	. 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 2147  ∀wral 3080  ifcif 4485   I cid 5552   ↾ cres 5660  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451   ℋchba 31578   ·_ih csp 31581  norm_ℎcno 31582  LinOpclo 31606  UniOpcuo 31608

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317  ax-hilex 31658  ax-hfvadd 31659  ax-hvcom 31660  ax-hvass 31661  ax-hv0cl 31662  ax-hvaddid 31663  ax-hfvmul 31664  ax-hvmulid 31665  ax-hvdistr2 31668  ax-hvmul0 31669  ax-hfi 31738  ax-his1 31741  ax-his2 31742  ax-his3 31743  ax-his4 31744

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-sup 9471  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-seq 14185  df-exp 14245  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-hnorm 31627  df-hvsub 31630  df-lnop 32500  df-unop 32502

This theorem is used by: (None)