unoplin - Hilbert Space Explorer

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Theorem unoplin 29699

Description: A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
unoplin	⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)

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

Step	Hyp	Ref	Expression
1		unopf1o 29695	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)
2		f1of 6617	. . 3 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3	1, 2	syl 17	. 2 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4		simplll 773	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑇 ∈ UniOp)
5		hvmulcl 28792	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
6		hvaddcl 28791	. . . . . . . . . . 11 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
7	5, 6	sylan 582	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
8	7	adantll 712	. . . . . . . . 9 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
9	8	adantr 483	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
10		simpr 487	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑤 ∈ ℋ)
11		unopadj 29698	. . . . . . . 8 ⊢ ((𝑇 ∈ UniOp ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (^◡𝑇‘𝑤)))
12	4, 9, 10, 11	syl3anc 1367	. . . . . . 7 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (^◡𝑇‘𝑤)))
13		simprl 769	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
14	13	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑥 ∈ ℂ)
15		simprr 771	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
16	15	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑦 ∈ ℋ)
17		simplr 767	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑧 ∈ ℋ)
18		cnvunop 29697	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ UniOp → ^◡𝑇 ∈ UniOp)
19		unopf1o 29695	. . . . . . . . . . . 12 ⊢ (^◡𝑇 ∈ UniOp → ^◡𝑇: ℋ–1-1-onto→ ℋ)
20		f1of 6617	. . . . . . . . . . . 12 ⊢ (^◡𝑇: ℋ–1-1-onto→ ℋ → ^◡𝑇: ℋ⟶ ℋ)
21	18, 19, 20	3syl 18	. . . . . . . . . . 11 ⊢ (𝑇 ∈ UniOp → ^◡𝑇: ℋ⟶ ℋ)
22	21	ffvelrnda 6853	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
23	22	adantlr 713	. . . . . . . . 9 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
24	23	adantllr 717	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
25		hiassdi 28870	. . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ (^◡𝑇‘𝑤) ∈ ℋ)) → (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (^◡𝑇‘𝑤)) = ((𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))) + (𝑧 ·_ih (^◡𝑇‘𝑤))))
26	14, 16, 17, 24, 25	syl22anc 836	. . . . . . 7 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ·_ih (^◡𝑇‘𝑤)) = ((𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))) + (𝑧 ·_ih (^◡𝑇‘𝑤))))
27	3	ffvelrnda 6853	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
28	27	adantrl 714	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑦) ∈ ℋ)
29	28	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
30	3	ffvelrnda 6853	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
31	30	adantr 483	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
32	31	adantllr 717	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
33		hiassdi 28870	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) ∧ ((𝑇‘𝑧) ∈ ℋ ∧ 𝑤 ∈ ℋ)) → (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) = ((𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) + ((𝑇‘𝑧) ·_ih 𝑤)))
34	14, 29, 32, 10, 33	syl22anc 836	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) = ((𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) + ((𝑇‘𝑧) ·_ih 𝑤)))
35		unopadj 29698	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (^◡𝑇‘𝑤)))
36	35	3expa 1114	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (^◡𝑇‘𝑤)))
37	36	oveq2d 7174	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
38	37	adantlrl 718	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
39	38	adantlr 713	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
40		unopadj 29698	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
41	40	3expa 1114	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
42	41	adantllr 717	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
43	39, 42	oveq12d 7176	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) + ((𝑇‘𝑧) ·_ih 𝑤)) = ((𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))) + (𝑧 ·_ih (^◡𝑇‘𝑤))))
44	34, 43	eqtr2d 2859	. . . . . . 7 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))) + (𝑧 ·_ih (^◡𝑇‘𝑤))) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤))
45	12, 26, 44	3eqtrd 2862	. . . . . 6 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤))
46	45	ralrimiva 3184	. . . . 5 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ∀𝑤 ∈ ℋ ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤))
47		ffvelrn 6851	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
48	7, 47	sylan2 594	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
49	48	anassrs 470	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
50		ffvelrn 6851	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
51		hvmulcl 28792	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
52	50, 51	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
53	52	an12s 647	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
54	53	adantr 483	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
55		ffvelrn 6851	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
56	55	adantlr 713	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
57		hvaddcl 28791	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
58	54, 56, 57	syl2anc 586	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
59		hial2eq 28885	. . . . . . 7 ⊢ (((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ ∧ ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ) → (∀𝑤 ∈ ℋ ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) ↔ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
60	49, 58, 59	syl2anc 586	. . . . . 6 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (∀𝑤 ∈ ℋ ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) ↔ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
61	3, 60	sylanl1 678	. . . . 5 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (∀𝑤 ∈ ℋ ((𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ·_ih 𝑤) = (((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ·_ih 𝑤) ↔ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
62	46, 61	mpbid 234	. . . 4 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
63	62	ralrimiva 3184	. . 3 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
64	63	ralrimivva 3193	. 2 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
65		ellnop 29637	. 2 ⊢ (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
66	3, 64, 65	sylanbrc 585	1 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ^◡ccnv 5556 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 + caddc 10542 · cmul 10544 ℋchba 28698 +_ℎ cva 28699 ·_ℎ csm 28700 ·_ih csp 28701 LinOpclo 28726 UniOpcuo 28728

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 df-hvsub 28750 df-lnop 29620 df-unop 29622

This theorem is referenced by: unopadj2 29717 idlnop 29771 elunop2 29792 nmopun 29793 unopbd 29794

W3C validator