Theorem ocsh 29166

Description: The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)

Assertion

Ref	Expression
ocsh	⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ )

Proof of Theorem ocsh

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ocval 29163	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
2		ssrab2 3985	. . . 4 ⊢ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} ⊆ ℋ
3	1, 2	eqsstrdi 3947	. . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
4		ssel 3886	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ))
5		hi01 28979	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → (0ℎ ·ih 𝑦) = 0)
6	4, 5	syl6 35	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ 𝐴 → (0ℎ ·ih 𝑦) = 0))
7	6	ralrimiv 3113	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0)
8		ax-hv0cl 28886	. . . . 5 ⊢ 0ℎ ∈ ℋ
9	7, 8	jctil 524	. . . 4 ⊢ (𝐴 ⊆ ℋ → (0ℎ ∈ ℋ ∧ ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0))
10		ocel 29164	. . . 4 ⊢ (𝐴 ⊆ ℋ → (0ℎ ∈ (⊥‘𝐴) ↔ (0ℎ ∈ ℋ ∧ ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0)))
11	9, 10	mpbird 260	. . 3 ⊢ (𝐴 ⊆ ℋ → 0ℎ ∈ (⊥‘𝐴))
12	3, 11	jca 516	. 2 ⊢ (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ ℋ ∧ 0ℎ ∈ (⊥‘𝐴)))
13		ssel2 3888	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℋ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℋ)
14		ax-his2 28966	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
15	14	3expa 1116	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
16		oveq12 7160	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = (0 + 0))
17		00id 10854	. . . . . . . . . . . . . 14 ⊢ (0 + 0) = 0
18	16, 17	eqtrdi 2810	. . . . . . . . . . . . 13 ⊢ (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = 0)
19	15, 18	sylan9eq 2814	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0)
20	19	ex 417	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0))
21	20	ancoms 463	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0))
22	13, 21	sylan 584	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℋ ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0))
23	22	an32s 652	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ 𝐴) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0))
24	23	ralimdva 3109	. . . . . . 7 ⊢ ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (∀𝑧 ∈ 𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ∀𝑧 ∈ 𝐴 ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0))
25	24	imdistanda 576	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0)))
26		hvaddcl 28895	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ)
27	26	anim1i 618	. . . . . 6 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0) → ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0))
28	25, 27	syl6 35	. . . . 5 ⊢ (𝐴 ⊆ ℋ → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0)))
29		ocel 29164	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑥 ∈ (⊥‘𝐴) ↔ (𝑥 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0)))
30		ocel 29164	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ (⊥‘𝐴) ↔ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
31	29, 30	anbi12d 634	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0))))
32		an4 656	. . . . . . 7 ⊢ (((𝑥 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
33		r19.26 3102	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) ↔ (∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0))
34	33	anbi2i 626	. . . . . . 7 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
35	32, 34	bitr4i 281	. . . . . 6 ⊢ (((𝑥 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)))
36	31, 35	bitrdi 290	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0))))
37		ocel 29164	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0)))
38	28, 36, 37	3imtr4d 298	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴)))
39	38	ralrimivv 3120	. . 3 ⊢ (𝐴 ⊆ ℋ → ∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴))
40		mul01 10858	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
41		oveq2 7159	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = (𝑥 · 0))
42	41	eqeq1d 2761	. . . . . . . . . . . . 13 ⊢ ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · (𝑦 ·ih 𝑧)) = 0 ↔ (𝑥 · 0) = 0))
43	40, 42	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
44	43	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
45		ax-his3 28967	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = (𝑥 · (𝑦 ·ih 𝑧)))
46	45	eqeq1d 2761	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
47	46	3expa 1116	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
48	47	ancoms 463	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
49	44, 48	sylibrd 262	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
50	13, 49	sylan 584	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℋ ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
51	50	an32s 652	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ 𝐴) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
52	51	ralimdva 3109	. . . . . . 7 ⊢ ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0 → ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
53	52	imdistanda 576	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0) → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0)))
54		hvmulcl 28896	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
55	54	anim1i 618	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0) → ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
56	53, 55	syl6 35	. . . . 5 ⊢ (𝐴 ⊆ ℋ → (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0)))
57	30	anbi2d 632	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0))))
58		anass 473	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
59	57, 58	bitr4di 293	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
60		ocel 29164	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0)))
61	56, 59, 60	3imtr4d 298	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴)))
62	61	ralrimivv 3120	. . 3 ⊢ (𝐴 ⊆ ℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴))
63	39, 62	jca 516	. 2 ⊢ (𝐴 ⊆ ℋ → (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴)))
64		issh2 29092	. 2 ⊢ ((⊥‘𝐴) ∈ Sℋ ↔ (((⊥‘𝐴) ⊆ ℋ ∧ 0ℎ ∈ (⊥‘𝐴)) ∧ (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴))))
65	12, 63, 64	sylanbrc 587	1 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ )

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075   ⊆ wss 3859  ‘cfv 6336  (class class class)co 7151  ℂcc 10574  0cc0 10576   + caddc 10579   · cmul 10581   ℋchba 28802   +_ℎ cva 28803   ·_ℎ csm 28804   ·_ih csp 28805  0_ℎc0v 28807   S_ℋ csh 28811  ⊥cort 28813

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-resscn 10633  ax-1cn 10634  ax-icn 10635  ax-addcl 10636  ax-addrcl 10637  ax-mulcl 10638  ax-mulrcl 10639  ax-mulcom 10640  ax-addass 10641  ax-mulass 10642  ax-distr 10643  ax-i2m1 10644  ax-1ne0 10645  ax-1rid 10646  ax-rnegex 10647  ax-rrecex 10648  ax-cnre 10649  ax-pre-lttri 10650  ax-pre-lttrn 10651  ax-pre-ltadd 10652  ax-hilex 28882  ax-hfvadd 28883  ax-hv0cl 28886  ax-hfvmul 28888  ax-hvmul0 28893  ax-hfi 28962  ax-his2 28966  ax-his3 28967

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-po 5444  df-so 5445  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-pnf 10716  df-mnf 10717  df-ltxr 10719  df-sh 29090  df-oc 29135

This theorem is referenced by:  shocsh  29167  ocss  29168  occl  29187  spanssoc  29232  ssjo  29330  chscllem2  29521