hmoplin - Hilbert Space Explorer

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Theorem hmoplin 31933

Description: A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hmoplin	⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)

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

Step	Hyp	Ref	Expression
1		hmopf 31865	. 2 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2		simplll 774	. . . . . . . 8 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑇 ∈ HrmOp)
3		hvmulcl 31004	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
4		hvaddcl 31003	. . . . . . . . . . 11 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
5	3, 4	sylan 580	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
6	5	adantll 714	. . . . . . . . 9 ⊢ (((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
7	6	adantr 480	. . . . . . . 8 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
8		simpr 484	. . . . . . . 8 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑤 ∈ ℋ)
9		hmop 31913	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ ∧ 𝑤 ∈ ℋ) → (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (𝑇‘𝑤)) = ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤))
10	9	eqcomd 2739	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (𝑇‘𝑤)))
11	2, 7, 8, 10	syl3anc 1373	. . . . . . 7 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (𝑇‘𝑤)))
12		simprl 770	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
13	12	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑥 ∈ ℂ)
14		simprr 772	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
15	14	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑦 ∈ ℋ)
16		simplr 768	. . . . . . . 8 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑧 ∈ ℋ)
17	1	ffvelcdmda 7026	. . . . . . . . . 10 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑤) ∈ ℋ)
18	17	adantlr 715	. . . . . . . . 9 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑤) ∈ ℋ)
19	18	adantllr 719	. . . . . . . 8 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑤) ∈ ℋ)
20		hiassdi 31082	. . . . . . . 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))))
21	13, 15, 16, 19, 20	syl22anc 838	. . . . . . 7 ⊢ ((((𝑇 ∈ HrmOp ∧ (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))))
22	1	ffvelcdmda 7026	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
23	22	adantrl 716	. . . . . . . . . 10 ⊢ ((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑦) ∈ ℋ)
24	23	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
25	1	ffvelcdmda 7026	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
26	25	adantr 480	. . . . . . . . . 10 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
27	26	adantllr 719	. . . . . . . . 9 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
28		hiassdi 31082	. . . . . . . . 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)))
29	13, 24, 27, 8, 28	syl22anc 838	. . . . . . . 8 ⊢ ((((𝑇 ∈ HrmOp ∧ (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)))
30		hmop 31913	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑦 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑦 ·_ih (𝑇‘𝑤)) = ((𝑇‘𝑦) ·_ih 𝑤))
31	30	eqcomd 2739	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑦 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (𝑇‘𝑤)))
32	31	3expa 1118	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (𝑇‘𝑤)))
33	32	oveq2d 7371	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (𝑇‘𝑤))))
34	33	adantlrl 720	. . . . . . . . . 10 ⊢ (((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (𝑇‘𝑤))))
35	34	adantlr 715	. . . . . . . . 9 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (𝑇‘𝑤))))
36		hmop 31913	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑧 ·_ih (𝑇‘𝑤)) = ((𝑇‘𝑧) ·_ih 𝑤))
37	36	eqcomd 2739	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (𝑇‘𝑤)))
38	37	3expa 1118	. . . . . . . . . 10 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (𝑇‘𝑤)))
39	38	adantllr 719	. . . . . . . . 9 ⊢ ((((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (𝑇‘𝑤)))
40	35, 39	oveq12d 7373	. . . . . . . 8 ⊢ ((((𝑇 ∈ HrmOp ∧ (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))))
41	29, 40	eqtr2d 2769	. . . . . . 7 ⊢ ((((𝑇 ∈ HrmOp ∧ (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))
42	11, 21, 41	3eqtrd 2772	. . . . . 6 ⊢ ((((𝑇 ∈ HrmOp ∧ (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))
43	42	ralrimiva 3126	. . . . 5 ⊢ (((𝑇 ∈ HrmOp ∧ (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))
44		ffvelcdm 7023	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
45	5, 44	sylan2 593	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
46	45	anassrs 467	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
47		ffvelcdm 7023	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
48		hvmulcl 31004	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
49	47, 48	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
50	49	an12s 649	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
51	50	adantr 480	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
52		ffvelcdm 7023	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
53	52	adantlr 715	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
54		hvaddcl 31003	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
55	51, 53, 54	syl2anc 584	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
56		hial2eq 31097	. . . . . . 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))))
57	46, 55, 56	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))))
58	1, 57	sylanl1 680	. . . . 5 ⊢ (((𝑇 ∈ HrmOp ∧ (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))))
59	43, 58	mpbid 232	. . . 4 ⊢ (((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
60	59	ralrimiva 3126	. . 3 ⊢ ((𝑇 ∈ HrmOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
61	60	ralrimivva 3177	. 2 ⊢ (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
62		ellnop 31849	. 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))))
63	1, 61, 62	sylanbrc 583	1 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3049 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ℂcc 11014 + caddc 11019 · cmul 11021 ℋchba 30910 +_ℎ cva 30911 ·_ℎ csm 30912 ·_ih csp 30913 LinOpclo 30938 HrmOpcho 30941

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-hilex 30990 ax-hfvadd 30991 ax-hvcom 30992 ax-hvass 30993 ax-hv0cl 30994 ax-hvaddid 30995 ax-hfvmul 30996 ax-hvmulid 30997 ax-hvdistr2 31000 ax-hvmul0 31001 ax-hfi 31070 ax-his2 31074 ax-his3 31075 ax-his4 31076

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-po 5529 df-so 5530 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-er 8631 df-map 8761 df-en 8879 df-dom 8880 df-sdom 8881 df-pnf 11158 df-mnf 11159 df-ltxr 11161 df-sub 11356 df-neg 11357 df-hvsub 30962 df-lnop 31832 df-hmop 31835

This theorem is referenced by: 0lnop 31975 hmopbdoptHIL 31979 leoptri 32127 leopnmid 32129 nmopleid 32130 opsqrlem1 32131 opsqrlem6 32136 pjlnopi 32138 hmopidmchi 32142 hmopidmpji 32143

W3C validator