adjlnop - Hilbert Space Explorer

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Theorem adjlnop 32688

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 32497	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ dom adj_ℎ)
2		dmadjop 32490	. . 3 ⊢ ((adj_ℎ‘𝑇) ∈ dom adj_ℎ → (adj_ℎ‘𝑇): ℋ⟶ ℋ)
3	1, 2	syl 18	. 2 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇): ℋ⟶ ℋ)
4		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑤 ∈ ℋ)
5		adjcl 32534	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑦 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
6		hvmulcl 31615	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
7	5, 6	sylan2 605	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇 ∈ dom adj_ℎ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
8	7	an12s 662	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
9	8	adantrr 730	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
10	9	3adant2 1149	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ ((adj_ℎ‘𝑇)‘𝑦)) ∈ ℋ)
11		adjcl 32534	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑧 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
12	11	adantrl 729	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
13	12	3adant2 1149	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑧) ∈ ℋ)
14		his7 31692	. . . . . . . . . . 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 1398	. . . . . . . . . 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 32536	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
17	16	3adant3l 1199	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih 𝑦) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑦)))
18	17	oveq2d 7436	. . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
20		dmadjop 32490	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ dom adj_ℎ → 𝑇: ℋ⟶ ℋ)
21	20	ffvelcdmda 7084	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑤) ∈ ℋ)
22	21	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
23		simp3r 1221	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
24		his5 31688	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑤) ·_ih (𝑥 ·_ℎ 𝑦)) = ((∗‘𝑥) · ((𝑇‘𝑤) ·_ih 𝑦)))
26		simp2 1155	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑤 ∈ ℋ)
27	5	adantrl 729	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
28	27	3adant2 1149	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘𝑦) ∈ ℋ)
29		his5 31688	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 2806	. . . . . . . . . . . 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 1200	. . . . . . . . . . 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 32536	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘𝑤) ·_ih 𝑧) = (𝑤 ·_ih ((adj_ℎ‘𝑇)‘𝑧)))
34	33	3adant3l 1199	. . . . . . . . . . 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 7438	. . . . . . . . . 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 1150	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘𝑤) ∈ ℋ)
37		hvmulcl 31615	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
38	37	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
39	38	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
40		simp3r 1221	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑤 ∈ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → 𝑧 ∈ ℋ)
41		his7 31692	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 31614	. . . . . . . . . . . . 13 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
44	37, 43	sylan 592	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
45		adj2 32536	. . . . . . . . . . . 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 1183	. . . . . . . . . . 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 2798	. . . . . . . . . 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 2803	. . . . . . . . 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 1144	. . . . . . . 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 1136	. . . . . . 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 3155	. . . . . 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 32534	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
53	44, 52	sylan2 605	. . . . . . 7 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((adj_ℎ‘𝑇)‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
54		hvaddcl 31614	. . . . . . . . 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 608	. . . . . . . 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 691	. . . . . . 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 31709	. . . . . . 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 596	. . . . . 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 426	. . . 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 3161	. . 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 3204	. 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 32460	. 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 595	1 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) ∈ LinOp)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 dom cdm 5651 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 + caddc 11203 · cmul 11205 ∗ccj 15263 ℋchba 31521 +_ℎ cva 31522 ·_ℎ csm 31523 ·_ih csp 31524 LinOpclo 31549 adj_ℎcado 31557

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-hilex 31601 ax-hfvadd 31602 ax-hvcom 31603 ax-hvass 31604 ax-hv0cl 31605 ax-hvaddid 31606 ax-hfvmul 31607 ax-hvmulid 31608 ax-hvdistr2 31611 ax-hvmul0 31612 ax-hfi 31681 ax-his1 31684 ax-his2 31685 ax-his3 31686 ax-his4 31687

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-cj 15266 df-re 15267 df-im 15268 df-hvsub 31573 df-lnop 32443 df-adjh 32451

This theorem is used by: adjsslnop 32689

W3C validator