adjlnop - Hilbert Space Explorer

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Theorem adjlnop 32176

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 31985	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ dom adj_ℎ)
2		dmadjop 31978	. . 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 1143	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑤 ∈ ℋ)
5		adjcl 32022	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑦 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
6		hvmulcl 31103	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
7	5, 6	sylan2 599	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇 ∈ dom adj_ℎ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
8	7	an12s 655	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
9	8	adantrr 723	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
10	9	3adant2 1137	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
11		adjcl 32022	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑧 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
12	11	adantrl 722	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
13	12	3adant2 1137	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
14		his7 31180	. . . . . . . . . . 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 1379	. . . . . . . . . 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 32024	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
17	16	3adant3l 1187	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
18	17	oveq2d 7373	. . . . . . . . . . . . 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 1208	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
20		dmadjop 31978	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ dom adj_ℎ → 𝑇: ℋ⟶ ℋ)
21	20	ffvelcdmda 7026	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑤) ∈ ℋ)
22	21	3adant3 1138	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
23		simp3r 1209	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
24		his5 31176	. . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) = ((∗‘𝑥) · ((𝑇‘𝑤) ·_ih 𝑦)))
26		simp2 1143	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑤 ∈ ℋ)
27	5	adantrl 722	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
28	27	3adant2 1137	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
29		his5 31176	. . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . 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 1188	. . . . . . . . . . 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 32024	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑧) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑧)))
34	33	3adant3l 1187	. . . . . . . . . . 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 7375	. . . . . . . . . 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 1138	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
37		hvmulcl 31103	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
38	37	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
39	38	3ad2ant3 1141	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
40		simp3r 1209	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑧 ∈ ℋ)
41		his7 31180	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 31102	. . . . . . . . . . . . 13 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
44	37, 43	sylan 586	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
45		adj2 32024	. . . . . . . . . . . 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 1171	. . . . . . . . . . 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 1132	. . . . . . . 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 1124	. . . . . . 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 3131	. . . . . 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 32022	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
53	44, 52	sylan2 599	. . . . . . 7 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
54		hvaddcl 31102	. . . . . . . . 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 602	. . . . . . . 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 684	. . . . . . 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 31197	. . . . . . 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 590	. . . . . 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 233	. . . . 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 421	. . . 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 3137	. . 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 3180	. 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 31948	. 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 589	1 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ LinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3053 dom cdm 5619 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ℂcc 11028 + caddc 11033 · cmul 11035 ∗ccj 15050 ℋchba 31009 +_ℎ cva 31010 ·_ℎ csm 31011 ·_ih csp 31012 LinOpclo 31037 adj_ℎcado 31045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-hilex 31089 ax-hfvadd 31090 ax-hvcom 31091 ax-hvass 31092 ax-hv0cl 31093 ax-hvaddid 31094 ax-hfvmul 31095 ax-hvmulid 31096 ax-hvdistr2 31099 ax-hvmul0 31100 ax-hfi 31169 ax-his1 31172 ax-his2 31173 ax-his3 31174 ax-his4 31175

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7808 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-er 8634 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-cj 15053 df-re 15054 df-im 15055 df-hvsub 31061 df-lnop 31931 df-adjh 31939

This theorem is referenced by: adjsslnop 32177

W3C validator