adjlnop - Hilbert Space Explorer

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Theorem adjlnop 32114

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 31923	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ dom adj_ℎ)
2		dmadjop 31916	. . 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 1136	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑤 ∈ ℋ)
5		adjcl 31960	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑦 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
6		hvmulcl 31041	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
7	5, 6	sylan2 593	. . . . . . . . . . . . . 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 1130	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
11		adjcl 31960	. . . . . . . . . . . . 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 1130	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
14		his7 31118	. . . . . . . . . . 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 1370	. . . . . . . . . 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 31962	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
17	16	3adant3l 1179	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
18	17	oveq2d 7446	. . . . . . . . . . . . 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 1200	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
20		dmadjop 31916	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ dom adj_ℎ → 𝑇: ℋ⟶ ℋ)
21	20	ffvelcdmda 7103	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑤) ∈ ℋ)
22	21	3adant3 1131	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
23		simp3r 1201	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
24		his5 31114	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) = ((∗‘𝑥) · ((𝑇‘𝑤) ·_ih 𝑦)))
26		simp2 1136	. . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
29		his5 31114	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 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 1180	. . . . . . . . . . 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 31962	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑧) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑧)))
34	33	3adant3l 1179	. . . . . . . . . . 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 7448	. . . . . . . . . 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 1131	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
37		hvmulcl 31041	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
38	37	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
39	38	3ad2ant3 1134	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
40		simp3r 1201	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑧 ∈ ℋ)
41		his7 31118	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 31040	. . . . . . . . . . . . 13 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
44	37, 43	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
45		adj2 31962	. . . . . . . . . . . 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 1164	. . . . . . . . . . 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 2776	. . . . . . . . . 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 2781	. . . . . . . . 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 1125	. . . . . . . 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 1117	. . . . . . 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 3143	. . . . . 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 31960	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
53	44, 52	sylan2 593	. . . . . . 7 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
54		hvaddcl 31040	. . . . . . . . 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 596	. . . . . . . 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 31135	. . . . . . 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 584	. . . . . 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 232	. . . . 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 420	. . . 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 3149	. . 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 3197	. 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 31886	. 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 583	1 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ LinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 dom cdm 5688 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 + caddc 11155 · cmul 11157 ∗ccj 15131 ℋchba 30947 +_ℎ cva 30948 ·_ℎ csm 30949 ·_ih csp 30950 LinOpclo 30975 adj_ℎcado 30983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-hilex 31027 ax-hfvadd 31028 ax-hvcom 31029 ax-hvass 31030 ax-hv0cl 31031 ax-hvaddid 31032 ax-hfvmul 31033 ax-hvmulid 31034 ax-hvdistr2 31037 ax-hvmul0 31038 ax-hfi 31107 ax-his1 31110 ax-his2 31111 ax-his3 31112 ax-his4 31113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-2 12326 df-cj 15134 df-re 15135 df-im 15136 df-hvsub 30999 df-lnop 31869 df-adjh 31877

This theorem is referenced by: adjsslnop 32115

W3C validator