Theorem elunop2 28872

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 28779	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2		elunop 28731	. . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))
3	2	simplbi 476	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
4		unopnorm 28776	. . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))
5	4	ralrimiva 2966	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))
6	1, 3, 5	3jca 1242	. 2 ⊢ (𝑇 ∈ UniOp → (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)))
7		eleq1 2689	. . 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 2689	. . . . . . 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 6111	. . . . . . 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		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑇‘𝑥) = (𝑇‘𝑦))
11	10	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘(𝑇‘𝑥)) = (normℎ‘(𝑇‘𝑦)))
12		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (normℎ‘𝑥) = (normℎ‘𝑦))
13	11, 12	eqeq12d 2637	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦)))
14	13	cbvralv 3171	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ↔ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) = (normℎ‘𝑦))
15		fveq1 6190	. . . . . . . . . . 11 ⊢ (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))
16	15	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (normℎ‘(𝑇‘𝑦)) = (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)))
17	16	eqeq1d 2624	. . . . . . . . 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)))
18	17	ralbidv 2986	. . . . . . . 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)))
19	14, 18	syl5bb 272	. . . . . . 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)))
20	8, 9, 19	3anbi123d 1399	. . . . . 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))))
21		eleq1 2689	. . . . . . 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))
22		foeq1 6111	. . . . . . 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))
23		fveq1 6190	. . . . . . . . . 10 ⊢ (( 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))
24	23	fveq2d 6195	. . . . . . . . 9 ⊢ (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) → (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)))
25	24	eqeq1d 2624	. . . . . . . 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)))
26	25	ralbidv 2986	. . . . . . 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)))
27	21, 22, 26	3anbi123d 1399	. . . . . 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))))
28		idlnop 28851	. . . . . . 7 ⊢ ( I ↾ ℋ) ∈ LinOp
29		f1oi 6174	. . . . . . . 8 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
30		f1ofo 6144	. . . . . . . 8 ⊢ (( I ↾ ℋ): ℋ–1-1-onto→ ℋ → ( I ↾ ℋ): ℋ–onto→ ℋ)
31	29, 30	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ ℋ): ℋ–onto→ ℋ
32		fvresi 6439	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
33	32	fveq2d 6195	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦))
34	33	rgen 2922	. . . . . . 7 ⊢ ∀𝑦 ∈ ℋ (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦)
35	28, 31, 34	3pm3.2i 1239	. . . . . 6 ⊢ (( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (normℎ‘(( I ↾ ℋ)‘𝑦)) = (normℎ‘𝑦))
36	20, 27, 35	elimhyp 4146	. . . . 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))
37	36	simp1i 1070	. . . 4 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp
38	36	simp2i 1071	. . . 4 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ
39	36	simp3i 1072	. . . 4 ⊢ ∀𝑦 ∈ ℋ (normℎ‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (normℎ‘𝑦)
40	37, 38, 39	lnopunii 28871	. . 3 ⊢ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp
41	7, 40	dedth 4139	. 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)) → 𝑇 ∈ UniOp)
42	6, 41	impbii 199	1 ⊢ (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥)))

Syntax hints:   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ifcif 4086   I cid 5023   ↾ cres 5116  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ℋchil 27776   ·_ih csp 27779  norm_ℎcno 27780  LinOpclo 27804  UniOpcuo 27806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-hnorm 27825  df-hvsub 27828  df-lnop 28700  df-unop 28702

This theorem is referenced by: (None)