unoplin - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > unoplin		Structured version Visualization version GIF version

Theorem unoplin 32006

Description: A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
unoplin	⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)

Proof of Theorem unoplin Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unopf1o 32002	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)
2		f1of 6774	. . 3 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3	1, 2	syl 17	. 2 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4		simplll 775	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑇 ∈ UniOp)
5		hvmulcl 31099	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
6		hvaddcl 31098	. . . . . . . . . . 11 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
7	5, 6	sylan 581	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
8	7	adantll 715	. . . . . . . . 9 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
9	8	adantr 480	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
10		simpr 484	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑤 ∈ ℋ)
11		unopadj 32005	. . . . . . . 8 ⊢ ((𝑇 ∈ UniOp ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (^◡𝑇‘𝑤)))
12	4, 9, 10, 11	syl3anc 1374	. . . . . . 7 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (^◡𝑇‘𝑤)))
13		simprl 771	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
14	13	ad2antrr 727	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑥 ∈ ℂ)
15		simprr 773	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
16	15	ad2antrr 727	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑦 ∈ ℋ)
17		simplr 769	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑧 ∈ ℋ)
18		cnvunop 32004	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ UniOp → ^◡𝑇 ∈ UniOp)
19		unopf1o 32002	. . . . . . . . . . . 12 ⊢ (^◡𝑇 ∈ UniOp → ^◡𝑇: ℋ–1-1-onto→ ℋ)
20		f1of 6774	. . . . . . . . . . . 12 ⊢ (^◡𝑇: ℋ–1-1-onto→ ℋ → ^◡𝑇: ℋ⟶ ℋ)
21	18, 19, 20	3syl 18	. . . . . . . . . . 11 ⊢ (𝑇 ∈ UniOp → ^◡𝑇: ℋ⟶ ℋ)
22	21	ffvelcdmda 7030	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
23	22	adantlr 716	. . . . . . . . 9 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
24	23	adantllr 720	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
25		hiassdi 31177	. . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ (^◡𝑇‘𝑤) ∈ ℋ)) → (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (^◡𝑇‘𝑤)) = ((𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))) + (𝑧 ·_ih (^◡𝑇‘𝑤))))
26	14, 16, 17, 24, 25	syl22anc 839	. . . . . . 7 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (^◡𝑇‘𝑤)) = ((𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))) + (𝑧 ·_ih (^◡𝑇‘𝑤))))
27	3	ffvelcdmda 7030	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
28	27	adantrl 717	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑦) ∈ ℋ)
29	28	ad2antrr 727	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
30	3	ffvelcdmda 7030	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
31	30	adantr 480	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
32	31	adantllr 720	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
33		hiassdi 31177	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) ∧ ((𝑇‘𝑧) ∈ ℋ ∧ 𝑤 ∈ ℋ)) → (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) = ((𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) + ((𝑇‘𝑧) ·_ih 𝑤)))
34	14, 29, 32, 10, 33	syl22anc 839	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) = ((𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) + ((𝑇‘𝑧) ·_ih 𝑤)))
35		unopadj 32005	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (^◡𝑇‘𝑤)))
36	35	3expa 1119	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (^◡𝑇‘𝑤)))
37	36	oveq2d 7376	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
38	37	adantlrl 721	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
39	38	adantlr 716	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
40		unopadj 32005	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
41	40	3expa 1119	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
42	41	adantllr 720	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
43	39, 42	oveq12d 7378	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) + ((𝑇‘𝑧) ·_ih 𝑤)) = ((𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))) + (𝑧 ·_ih (^◡𝑇‘𝑤))))
44	34, 43	eqtr2d 2773	. . . . . . 7 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))) + (𝑧 ·_ih (^◡𝑇‘𝑤))) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤))
45	12, 26, 44	3eqtrd 2776	. . . . . 6 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤))
46	45	ralrimiva 3130	. . . . 5 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ∀𝑤 ∈ ℋ ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤))
47		ffvelcdm 7027	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
48	7, 47	sylan2 594	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
49	48	anassrs 467	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
50		ffvelcdm 7027	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
51		hvmulcl 31099	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
52	50, 51	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
53	52	an12s 650	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
54	53	adantr 480	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
55		ffvelcdm 7027	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
56	55	adantlr 716	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
57		hvaddcl 31098	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
58	54, 56, 57	syl2anc 585	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
59		hial2eq 31192	. . . . . . 7 ⊢ (((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ ∧ ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ) → (∀𝑤 ∈ ℋ ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) ↔ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
60	49, 58, 59	syl2anc 585	. . . . . 6 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (∀𝑤 ∈ ℋ ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) ↔ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
61	3, 60	sylanl1 681	. . . . 5 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (∀𝑤 ∈ ℋ ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) ↔ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
62	46, 61	mpbid 232	. . . 4 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
63	62	ralrimiva 3130	. . 3 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
64	63	ralrimivva 3181	. 2 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
65		ellnop 31944	. 2 ⊢ (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
66	3, 64, 65	sylanbrc 584	1 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ^◡ccnv 5623 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 + caddc 11032 · cmul 11034 ℋchba 31005 +_ℎ cva 31006 ·_ℎ csm 31007 ·_ih csp 31008 LinOpclo 31033 UniOpcuo 31035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-hilex 31085 ax-hfvadd 31086 ax-hvcom 31087 ax-hvass 31088 ax-hv0cl 31089 ax-hvaddid 31090 ax-hfvmul 31091 ax-hvmulid 31092 ax-hvdistr2 31095 ax-hvmul0 31096 ax-hfi 31165 ax-his1 31168 ax-his2 31169 ax-his3 31170 ax-his4 31171

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-cj 15052 df-re 15053 df-im 15054 df-hvsub 31057 df-lnop 31927 df-unop 31929

This theorem is referenced by: unopadj2 32024 idlnop 32078 elunop2 32099 nmopun 32100 unopbd 32101

W3C validator