unoplin - Hilbert Space Explorer

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Theorem unoplin 31902

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 31898	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)
2		f1of 6768	. . 3 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3	1, 2	syl 17	. 2 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4		simplll 774	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑇 ∈ UniOp)
5		hvmulcl 30995	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ℎ 𝑦) ∈ ℋ)
6		hvaddcl 30994	. . . . . . . . . . 11 ⊢ (((𝑥 ·_ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
7	5, 6	sylan 580	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
8	7	adantll 714	. . . . . . . . 9 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
9	8	adantr 480	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ)
10		simpr 484	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑤 ∈ ℋ)
11		unopadj 31901	. . . . . . . 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 1373	. . . . . . 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 770	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℂ)
14	13	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑥 ∈ ℂ)
15		simprr 772	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
16	15	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑦 ∈ ℋ)
17		simplr 768	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑧 ∈ ℋ)
18		cnvunop 31900	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ UniOp → ^◡𝑇 ∈ UniOp)
19		unopf1o 31898	. . . . . . . . . . . 12 ⊢ (^◡𝑇 ∈ UniOp → ^◡𝑇: ℋ–1-1-onto→ ℋ)
20		f1of 6768	. . . . . . . . . . . 12 ⊢ (^◡𝑇: ℋ–1-1-onto→ ℋ → ^◡𝑇: ℋ⟶ ℋ)
21	18, 19, 20	3syl 18	. . . . . . . . . . 11 ⊢ (𝑇 ∈ UniOp → ^◡𝑇: ℋ⟶ ℋ)
22	21	ffvelcdmda 7023	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
23	22	adantlr 715	. . . . . . . . 9 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
24	23	adantllr 719	. . . . . . . 8 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (^◡𝑇‘𝑤) ∈ ℋ)
25		hiassdi 31073	. . . . . . . 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 838	. . . . . . 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	ffvelcdmda 7023	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
28	27	adantrl 716	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑦) ∈ ℋ)
29	28	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
30	3	ffvelcdmda 7023	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
31	30	adantr 480	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
32	31	adantllr 719	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
33		hiassdi 31073	. . . . . . . . 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 838	. . . . . . . 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 31901	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (^◡𝑇‘𝑤)))
36	35	3expa 1118	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑤) = (𝑦 ·_ih (^◡𝑇‘𝑤)))
37	36	oveq2d 7368	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
38	37	adantlrl 720	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
39	38	adantlr 715	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (𝑥 · ((𝑇‘𝑦) ·_ih 𝑤)) = (𝑥 · (𝑦 ·_ih (^◡𝑇‘𝑤))))
40		unopadj 31901	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
41	40	3expa 1118	. . . . . . . . . 10 ⊢ (((𝑇 ∈ UniOp ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
42	41	adantllr 719	. . . . . . . . 9 ⊢ ((((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((𝑇‘𝑧) ·_ih 𝑤) = (𝑧 ·_ih (^◡𝑇‘𝑤)))
43	39, 42	oveq12d 7370	. . . . . . . 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 2769	. . . . . . 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 2772	. . . . . 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 3125	. . . . 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		ffvelcdm 7020	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧) ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
48	7, 47	sylan2 593	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
49	48	anassrs 467	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) ∈ ℋ)
50		ffvelcdm 7020	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
51		hvmulcl 30995	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
52	50, 51	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
53	52	an12s 649	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
54	53	adantr 480	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ)
55		ffvelcdm 7020	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
56	55	adantlr 715	. . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝑇‘𝑧) ∈ ℋ)
57		hvaddcl 30994	. . . . . . . 8 ⊢ (((𝑥 ·_ℎ (𝑇‘𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
58	54, 56, 57	syl2anc 584	. . . . . . 7 ⊢ (((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)) ∈ ℋ)
59		hial2eq 31088	. . . . . . 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 584	. . . . . 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 680	. . . . 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 232	. . . 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 3125	. . 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 3176	. 2 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
65		ellnop 31840	. 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 583	1 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3048 ^◡ccnv 5618 ⟶wf 6482 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7352 ℂcc 11011 + caddc 11016 · cmul 11018 ℋchba 30901 +_ℎ cva 30902 ·_ℎ csm 30903 ·_ih csp 30904 LinOpclo 30929 UniOpcuo 30931

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-hilex 30981 ax-hfvadd 30982 ax-hvcom 30983 ax-hvass 30984 ax-hv0cl 30985 ax-hvaddid 30986 ax-hfvmul 30987 ax-hvmulid 30988 ax-hvdistr2 30991 ax-hvmul0 30992 ax-hfi 31061 ax-his1 31064 ax-his2 31065 ax-his3 31066 ax-his4 31067

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 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 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-cj 15008 df-re 15009 df-im 15010 df-hvsub 30953 df-lnop 31823 df-unop 31825

This theorem is referenced by: unopadj2 31920 idlnop 31974 elunop2 31995 nmopun 31996 unopbd 31997

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