unoplin - Hilbert Space Explorer

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Theorem unoplin 30570

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 30566	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)
2		f1of 6767	. . 3 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3	1, 2	syl 17	. 2 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4		simplll 772	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑇 ∈ UniOp)
5		hvmulcl 29663	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
6		hvaddcl 29662	. . . . . . . . . . 11 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
7	5, 6	sylan 580	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
8	7	adantll 711	. . . . . . . . 9 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
9	8	adantr 481	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
10		simpr 485	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑤 ∈ ℋ)
11		unopadj 30569	. . . . . . . 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 1370	. . . . . . 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 768	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
14	13	ad2antrr 723	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑥 ∈ ℂ)
15		simprr 770	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
16	15	ad2antrr 723	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑦 ∈ ℋ)
17		simplr 766	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑧 ∈ ℋ)
18		cnvunop 30568	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ UniOp → ^◡𝑇 ∈ UniOp)
19		unopf1o 30566	. . . . . . . . . . . 12 ⊢ (^◡𝑇 ∈ UniOp → ^◡𝑇: ℋ–1-1-onto→ ℋ)
20		f1of 6767	. . . . . . . . . . . 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 7017	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
23	22	adantlr 712	. . . . . . . . 9 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
24	23	adantllr 716	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
25		hiassdi 29741	. . . . . . . 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 836	. . . . . . 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 7017	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
28	27	adantrl 713	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑦) ∈ ℋ)
29	28	ad2antrr 723	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
30	3	ffvelcdmda 7017	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
31	30	adantr 481	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
32	31	adantllr 716	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
33		hiassdi 29741	. . . . . . . . 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 836	. . . . . . . 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 30569	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (^◡𝑇‘𝑤)))
36	35	3expa 1117	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (^◡𝑇‘𝑤)))
37	36	oveq2d 7353	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
38	37	adantlrl 717	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
39	38	adantlr 712	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
40		unopadj 30569	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
41	40	3expa 1117	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
42	41	adantllr 716	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
43	39, 42	oveq12d 7355	. . . . . . . 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 2777	. . . . . . 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 2780	. . . . . 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 3139	. . . . 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 7015	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
48	7, 47	sylan2 593	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
49	48	anassrs 468	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
50		ffvelcdm 7015	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
51		hvmulcl 29663	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
52	50, 51	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
53	52	an12s 646	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
54	53	adantr 481	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
55		ffvelcdm 7015	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
56	55	adantlr 712	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
57		hvaddcl 29662	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
58	54, 56, 57	syl2anc 584	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
59		hial2eq 29756	. . . . . . 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 584	. . . . . 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 677	. . . . 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 231	. . . 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 3139	. . 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 3193	. 2 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
65		ellnop 30508	. 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 583	1 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 ^◡ccnv 5619 ⟶wf 6475 –1-1-onto→wf1o 6478 ‘cfv 6479 (class class class)co 7337 ℂcc 10970 + caddc 10975 · cmul 10977 ℋchba 29569 +_ℎ cva 29570 ·_ℎ csm 29571 ·_ih csp 29572 LinOpclo 29597 UniOpcuo 29599

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-hilex 29649 ax-hfvadd 29650 ax-hvcom 29651 ax-hvass 29652 ax-hv0cl 29653 ax-hvaddid 29654 ax-hfvmul 29655 ax-hvmulid 29656 ax-hvdistr2 29659 ax-hvmul0 29660 ax-hfi 29729 ax-his1 29732 ax-his2 29733 ax-his3 29734 ax-his4 29735

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-po 5532 df-so 5533 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-2 12137 df-cj 14909 df-re 14910 df-im 14911 df-hvsub 29621 df-lnop 30491 df-unop 30493

This theorem is referenced by: unopadj2 30588 idlnop 30642 elunop2 30663 nmopun 30664 unopbd 30665

W3C validator