Theorem ocsh 31358

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 31355	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
2		ssrab2 4032	. . . 4 ⊢ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} ⊆ ℋ
3	1, 2	eqsstrdi 3978	. . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
4		ssel 3927	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ))
5		hi01 31171	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → (0ℎ ·ih 𝑦) = 0)
6	4, 5	syl6 35	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ 𝐴 → (0ℎ ·ih 𝑦) = 0))
7	6	ralrimiv 3127	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0)
8		ax-hv0cl 31078	. . . . 5 ⊢ 0ℎ ∈ ℋ
9	7, 8	jctil 519	. . . 4 ⊢ (𝐴 ⊆ ℋ → (0ℎ ∈ ℋ ∧ ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0))
10		ocel 31356	. . . 4 ⊢ (𝐴 ⊆ ℋ → (0ℎ ∈ (⊥‘𝐴) ↔ (0ℎ ∈ ℋ ∧ ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0)))
11	9, 10	mpbird 257	. . 3 ⊢ (𝐴 ⊆ ℋ → 0ℎ ∈ (⊥‘𝐴))
12	3, 11	jca 511	. 2 ⊢ (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ ℋ ∧ 0ℎ ∈ (⊥‘𝐴)))
13		ssel2 3928	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℋ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℋ)
14		ax-his2 31158	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
15	14	3expa 1118	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
16		oveq12 7367	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = (0 + 0))
17		00id 11308	. . . . . . . . . . . . . 14 ⊢ (0 + 0) = 0
18	16, 17	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = 0)
19	15, 18	sylan9eq 2791	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0)
20	19	ex 412	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0))
21	20	ancoms 458	. . . . . . . . . 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 580	. . . . . . . . 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 3148	. . . . . . 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 571	. . . . . 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 31087	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ)
27	26	anim1i 615	. . . . . 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 31356	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑥 ∈ (⊥‘𝐴) ↔ (𝑥 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0)))
30		ocel 31356	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ (⊥‘𝐴) ↔ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
31	29, 30	anbi12d 632	. . . . . 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 3096	. . . . . . . 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 623	. . . . . . 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 278	. . . . . 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 287	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0))))
37		ocel 31356	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0)))
38	28, 36, 37	3imtr4d 294	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴)))
39	38	ralrimivv 3177	. . 3 ⊢ (𝐴 ⊆ ℋ → ∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴))
40		mul01 11312	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
41		oveq2 7366	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = (𝑥 · 0))
42	41	eqeq1d 2738	. . . . . . . . . . . . 13 ⊢ ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · (𝑦 ·ih 𝑧)) = 0 ↔ (𝑥 · 0) = 0))
43	40, 42	syl5ibrcom 247	. . . . . . . . . . . 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 31159	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = (𝑥 · (𝑦 ·ih 𝑧)))
46	45	eqeq1d 2738	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
47	46	3expa 1118	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
48	47	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
49	44, 48	sylibrd 259	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
50	13, 49	sylan 580	. . . . . . . . 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 3148	. . . . . . 7 ⊢ ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0 → ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
53	52	imdistanda 571	. . . . . 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 31088	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
55	54	anim1i 615	. . . . . 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 630	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0))))
58		anass 468	. . . . . 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 289	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
60		ocel 31356	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0)))
61	56, 59, 60	3imtr4d 294	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴)))
62	61	ralrimivv 3177	. . 3 ⊢ (𝐴 ⊆ ℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴))
63	39, 62	jca 511	. 2 ⊢ (𝐴 ⊆ ℋ → (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴)))
64		issh2 31284	. 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 583	1 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399   ⊆ wss 3901  ‘cfv 6492  (class class class)co 7358  ℂcc 11024  0cc0 11026   + caddc 11029   · cmul 11031   ℋchba 30994   +_ℎ cva 30995   ·_ℎ csm 30996   ·_ih csp 30997  0_ℎc0v 30999   S_ℋ csh 31003  ⊥cort 31005

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-hilex 31074  ax-hfvadd 31075  ax-hv0cl 31078  ax-hfvmul 31080  ax-hvmul0 31085  ax-hfi 31154  ax-his2 31158  ax-his3 31159

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-ltxr 11171  df-sh 31282  df-oc 31327

This theorem is referenced by:  shocsh  31359  ocss  31360  occl  31379  spanssoc  31424  ssjo  31522  chscllem2  31713