Theorem ocsh 31372

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 31369	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
2		ssrab2 4011	. . . 4 ⊢ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} ⊆ ℋ
3	1, 2	eqsstrdi 3959	. . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
4		ssel 3909	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ))
5		hi01 31185	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → (0ℎ ·ih 𝑦) = 0)
6	4, 5	syl6 35	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ 𝐴 → (0ℎ ·ih 𝑦) = 0))
7	6	ralrimiv 3130	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0)
8		ax-hv0cl 31092	. . . . 5 ⊢ 0ℎ ∈ ℋ
9	7, 8	jctil 524	. . . 4 ⊢ (𝐴 ⊆ ℋ → (0ℎ ∈ ℋ ∧ ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0))
10		ocel 31370	. . . 4 ⊢ (𝐴 ⊆ ℋ → (0ℎ ∈ (⊥‘𝐴) ↔ (0ℎ ∈ ℋ ∧ ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0)))
11	9, 10	mpbird 258	. . 3 ⊢ (𝐴 ⊆ ℋ → 0ℎ ∈ (⊥‘𝐴))
12	3, 11	jca 516	. 2 ⊢ (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ ℋ ∧ 0ℎ ∈ (⊥‘𝐴)))
13		ssel2 3910	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℋ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℋ)
14		ax-his2 31172	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
15	14	3expa 1124	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
16		oveq12 7365	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = (0 + 0))
17		00id 11312	. . . . . . . . . . . . . 14 ⊢ (0 + 0) = 0
18	16, 17	eqtrdi 2790	. . . . . . . . . . . . 13 ⊢ (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = 0)
19	15, 18	sylan9eq 2794	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0)
20	19	ex 413	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0))
21	20	ancoms 459	. . . . . . . . . 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 586	. . . . . . . . 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 658	. . . . . . . 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 3151	. . . . . . 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 31101	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ)
27	26	anim1i 621	. . . . . 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 31370	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑥 ∈ (⊥‘𝐴) ↔ (𝑥 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0)))
30		ocel 31370	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ (⊥‘𝐴) ↔ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
31	29, 30	anbi12d 638	. . . . . 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 662	. . . . . . 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 3099	. . . . . . . 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 629	. . . . . . 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 279	. . . . . 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 288	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0))))
37		ocel 31370	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0)))
38	28, 36, 37	3imtr4d 295	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴)))
39	38	ralrimivv 3180	. . 3 ⊢ (𝐴 ⊆ ℋ → ∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴))
40		mul01 11316	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
41		oveq2 7364	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = (𝑥 · 0))
42	41	eqeq1d 2741	. . . . . . . . . . . . 13 ⊢ ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · (𝑦 ·ih 𝑧)) = 0 ↔ (𝑥 · 0) = 0))
43	40, 42	syl5ibrcom 248	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
44	43	ad2antrl 734	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
45		ax-his3 31173	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = (𝑥 · (𝑦 ·ih 𝑧)))
46	45	eqeq1d 2741	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
47	46	3expa 1124	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
48	47	ancoms 459	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
49	44, 48	sylibrd 260	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
50	13, 49	sylan 586	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℋ ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
51	50	an32s 658	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ 𝐴) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
52	51	ralimdva 3151	. . . . . . 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 31102	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
55	54	anim1i 621	. . . . . 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 636	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0))))
58		anass 469	. . . . . 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 290	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
60		ocel 31370	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0)))
61	56, 59, 60	3imtr4d 295	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴)))
62	61	ralrimivv 3180	. . 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 31298	. 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 589	1 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ )

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391   ⊆ wss 3883  ‘cfv 6485  (class class class)co 7356  ℂcc 11027  0cc0 11029   + caddc 11032   · cmul 11034   ℋchba 31008   +_ℎ cva 31009   ·_ℎ csm 31010   ·_ih csp 31011  0_ℎc0v 31013   S_ℋ csh 31017  ⊥cort 31019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-hilex 31088  ax-hfvadd 31089  ax-hv0cl 31092  ax-hfvmul 31094  ax-hvmul0 31099  ax-hfi 31168  ax-his2 31172  ax-his3 31173

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-ltxr 11175  df-sh 31296  df-oc 31341

This theorem is referenced by:  shocsh  31373  ocss  31374  occl  31393  spanssoc  31438  ssjo  31536  chscllem2  31727