adjlnop - Hilbert Space Explorer

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Theorem adjlnop 31339

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 31148	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ dom adj_ℎ)
2		dmadjop 31141	. . 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 1138	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑤 ∈ ℋ)
5		adjcl 31185	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑦 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
6		hvmulcl 30266	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
7	5, 6	sylan2 594	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇 ∈ dom adj_ℎ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
8	7	an12s 648	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
9	8	adantrr 716	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
10	9	3adant2 1132	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
11		adjcl 31185	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑧 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
12	11	adantrl 715	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
13	12	3adant2 1132	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
14		his7 30343	. . . . . . . . . . 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 1372	. . . . . . . . . 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 31187	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
17	16	3adant3l 1181	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
18	17	oveq2d 7425	. . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
20		dmadjop 31141	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ dom adj_ℎ → 𝑇: ℋ⟶ ℋ)
21	20	ffvelcdmda 7087	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑤) ∈ ℋ)
22	21	3adant3 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
23		simp3r 1203	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
24		his5 30339	. . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) = ((∗‘𝑥) · ((𝑇‘𝑤) ·_ih 𝑦)))
26		simp2 1138	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑤 ∈ ℋ)
27	5	adantrl 715	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
28	27	3adant2 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
29		his5 30339	. . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . 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 2783	. . . . . . . . . . . 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 1182	. . . . . . . . . . 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 31187	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑧) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑧)))
34	33	3adant3l 1181	. . . . . . . . . . 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 7427	. . . . . . . . . 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 1133	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
37		hvmulcl 30266	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
38	37	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
39	38	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
40		simp3r 1203	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑧 ∈ ℋ)
41		his7 30343	. . . . . . . . . . . 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 1372	. . . . . . . . . . 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 30265	. . . . . . . . . . . . 13 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
44	37, 43	sylan 581	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
45		adj2 31187	. . . . . . . . . . . 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 1166	. . . . . . . . . . 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 2775	. . . . . . . . . 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 2780	. . . . . . . . 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 1127	. . . . . . . 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 1119	. . . . . . 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 3147	. . . . . 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 31185	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
53	44, 52	sylan2 594	. . . . . . 7 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
54		hvaddcl 30265	. . . . . . . . 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 597	. . . . . . . 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 677	. . . . . . 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 30360	. . . . . . 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 585	. . . . . 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 231	. . . . 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 422	. . . 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 3153	. . 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 3199	. 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 31111	. 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 584	1 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ LinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 + caddc 11113 · cmul 11115 ∗ccj 15043 ℋchba 30172 +_ℎ cva 30173 ·_ℎ csm 30174 ·_ih csp 30175 LinOpclo 30200 adj_ℎcado 30208

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-hilex 30252 ax-hfvadd 30253 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvdistr2 30262 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his2 30336 ax-his3 30337 ax-his4 30338

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-2 12275 df-cj 15046 df-re 15047 df-im 15048 df-hvsub 30224 df-lnop 31094 df-adjh 31102

This theorem is referenced by: adjsslnop 31340

W3C validator