adjlnop - Hilbert Space Explorer

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Theorem adjlnop 30167

Description: The adjoint of an operator is linear. Proposition 1 of [AkhiezerGlazman] p. 80. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
adjlnop	⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ LinOp)

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

Step	Hyp	Ref	Expression
1		dmadjrn 29976	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ dom adj_ℎ)
2		dmadjop 29969	. . 3 ⊢ ((adj_ℎ‘𝑇) ∈ dom adj_ℎ → (adj_ℎ‘𝑇): ℋ⟶ ℋ)
3	1, 2	syl 17	. 2 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇): ℋ⟶ ℋ)
4		simp2 1139	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑤 ∈ ℋ)
5		adjcl 30013	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑦 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
6		hvmulcl 29094	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
7	5, 6	sylan2 596	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇 ∈ dom adj_ℎ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
8	7	an12s 649	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
9	8	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
10	9	3adant2 1133	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
11		adjcl 30013	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑧 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
12	11	adantrl 716	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
13	12	3adant2 1133	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
14		his7 29171	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℋ ∧ (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ ∧ ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ) → (𝑤 ·_ih ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))) = ((𝑤 ·_ih (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦))) + (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑧))))
15	4, 10, 13, 14	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑤 ·_ih ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))) = ((𝑤 ·_ih (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦))) + (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑧))))
16		adj2 30015	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
17	16	3adant3l 1182	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
18	17	oveq2d 7229	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((∗‘𝑥) · ((𝑇‘𝑤) ·_ih 𝑦)) = ((∗‘𝑥) · (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦))))
19		simp3l 1203	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
20		dmadjop 29969	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ dom adj_ℎ → 𝑇: ℋ⟶ ℋ)
21	20	ffvelrnda 6904	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑤) ∈ ℋ)
22	21	3adant3 1134	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
23		simp3r 1204	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
24		his5 29167	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑤) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) = ((∗‘𝑥) · ((𝑇‘𝑤) ·_ih 𝑦)))
25	19, 22, 23, 24	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) = ((∗‘𝑥) · ((𝑇‘𝑤) ·_ih 𝑦)))
26		simp2 1139	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑤 ∈ ℋ)
27	5	adantrl 716	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
28	27	3adant2 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
29		his5 29167	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ ∧ ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑤 ·_ih (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦))) = ((∗‘𝑥) · (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦))))
30	19, 26, 28, 29	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑤 ·_ih (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦))) = ((∗‘𝑥) · (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦))))
31	18, 25, 30	3eqtr4d 2787	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) = (𝑤 ·_ih (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦))))
32	31	3adant3r 1183	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) = (𝑤 ·_ih (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦))))
33		adj2 30015	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑧) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑧)))
34	33	3adant3l 1182	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih 𝑧) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑧)))
35	32, 34	oveq12d 7231	. . . . . . . . . 10 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) + ((𝑇‘𝑤) ·_ih 𝑧)) = ((𝑤 ·_ih (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦))) + (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑧))))
36	21	3adant3 1134	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
37		hvmulcl 29094	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
38	37	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
39	38	3ad2ant3 1137	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
40		simp3r 1204	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑧 ∈ ℋ)
41		his7 29171	. . . . . . . . . . . 12 ⊢ (((𝑇‘𝑤) ∈ ℋ ∧ (𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑤) ·_ih ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = (((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) + ((𝑇‘𝑤) ·_ih 𝑧)))
42	36, 39, 40, 41	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = (((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) + ((𝑇‘𝑤) ·_ih 𝑧)))
43		hvaddcl 29093	. . . . . . . . . . . . 13 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
44	37, 43	sylan 583	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
45		adj2 30015	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → ((𝑇‘𝑤) ·_ih ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))))
46	44, 45	syl3an3 1167	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))))
47	42, 46	eqtr3d 2779	. . . . . . . . . 10 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) + ((𝑇‘𝑤) ·_ih 𝑧)) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))))
48	15, 35, 47	3eqtr2rd 2784	. . . . . . . . 9 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑤 ·_ih ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))) = (𝑤 ·_ih ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))))
49	48	3com23 1128	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑤 ·_ih ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))) = (𝑤 ·_ih ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))))
50	49	3expa 1120	. . . . . . 7 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) ∧ 𝑤 ∈ ℋ) → (𝑤 ·_ih ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))) = (𝑤 ·_ih ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))))
51	50	ralrimiva 3105	. . . . . 6 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ∀𝑤 ∈ ℋ (𝑤 ·_ih ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))) = (𝑤 ·_ih ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))))
52		adjcl 30013	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
53	44, 52	sylan2 596	. . . . . . 7 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
54		hvaddcl 29093	. . . . . . . . 9 ⊢ (((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ ∧ ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ) → ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧)) ∈ ℋ)
55	8, 11, 54	syl2an 599	. . . . . . . 8 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ (𝑇 ∈ dom adj_ℎ ∧ 𝑧 ∈ ℋ)) → ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧)) ∈ ℋ)
56	55	anandis 678	. . . . . . 7 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧)) ∈ ℋ)
57		hial2eq2 29188	. . . . . . 7 ⊢ ((((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ ∧ ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧)) ∈ ℋ) → (∀𝑤 ∈ ℋ (𝑤 ·_ih ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))) = (𝑤 ·_ih ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))) ↔ ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))))
58	53, 56, 57	syl2anc 587	. . . . . 6 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (∀𝑤 ∈ ℋ (𝑤 ·_ih ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧))) = (𝑤 ·_ih ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))) ↔ ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))))
59	51, 58	mpbid 235	. . . . 5 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧)))
60	59	exp32 424	. . . 4 ⊢ (𝑇 ∈ dom adj_ℎ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑧 ∈ ℋ → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧)))))
61	60	ralrimdv 3109	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ∀𝑧 ∈ ℋ ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))))
62	61	ralrimivv 3111	. 2 ⊢ (𝑇 ∈ dom adj_ℎ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧)))
63		ellnop 29939	. 2 ⊢ ((adj_ℎ‘𝑇) ∈ LinOp ↔ ((adj_ℎ‘𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) +_ℎ ((adj_ℎ‘𝑇)‘𝑧))))
64	3, 62, 63	sylanbrc 586	1 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ LinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 dom cdm 5551 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 + caddc 10732 · cmul 10734 ∗ccj 14659 ℋchba 29000 +_ℎ cva 29001 ·_ℎ csm 29002 ·_ih csp 29003 LinOpclo 29028 adj_ℎcado 29036

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-hilex 29080 ax-hfvadd 29081 ax-hvcom 29082 ax-hvass 29083 ax-hv0cl 29084 ax-hvaddid 29085 ax-hfvmul 29086 ax-hvmulid 29087 ax-hvdistr2 29090 ax-hvmul0 29091 ax-hfi 29160 ax-his1 29163 ax-his2 29164 ax-his3 29165 ax-his4 29166

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-2 11893 df-cj 14662 df-re 14663 df-im 14664 df-hvsub 29052 df-lnop 29922 df-adjh 29930

This theorem is referenced by: adjsslnop 30168

W3C validator