Theorem ocsh 31635

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 31632	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
2		ssrab2 4034	. . . 4 ⊢ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} ⊆ ℋ
3	1, 2	eqsstrdi 3981	. . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
4		ssel 3931	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ))
5		hi01 31448	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → (0ℎ ·ih 𝑦) = 0)
6	4, 5	syl6 36	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ 𝐴 → (0ℎ ·ih 𝑦) = 0))
7	6	ralrimiv 3156	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0)
8		ax-hv0cl 31355	. . . . 5 ⊢ 0ℎ ∈ ℋ
9	7, 8	jctil 528	. . . 4 ⊢ (𝐴 ⊆ ℋ → (0ℎ ∈ ℋ ∧ ∀𝑦 ∈ 𝐴 (0ℎ ·ih 𝑦) = 0))
10		ocel 31633	. . . 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 520	. 2 ⊢ (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ ℋ ∧ 0ℎ ∈ (⊥‘𝐴)))
13		ssel2 3932	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℋ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℋ)
14		ax-his2 31435	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
15	14	3expa 1136	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 +ℎ 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
16		oveq12 7419	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = (0 + 0))
17		00id 11380	. . . . . . . . . . . . . 14 ⊢ (0 + 0) = 0
18	16, 17	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = 0)
19	15, 18	sylan9eq 2818	. . . . . . . . . . . 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 591	. . . . . . . . 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 664	. . . . . . . 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 3177	. . . . . . 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 581	. . . . . 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 31364	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ)
27	26	anim1i 626	. . . . . 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 36	. . . . 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 31633	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑥 ∈ (⊥‘𝐴) ↔ (𝑥 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑥 ·ih 𝑧) = 0)))
30		ocel 31633	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (𝑦 ∈ (⊥‘𝐴) ↔ (𝑦 ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
31	29, 30	anbi12d 643	. . . . . 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 668	. . . . . . 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 3125	. . . . . . . 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 634	. . . . . . 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 31633	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 +ℎ 𝑦) ·ih 𝑧) = 0)))
38	28, 36, 37	3imtr4d 297	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴)))
39	38	ralrimivv 3206	. . 3 ⊢ (𝐴 ⊆ ℋ → ∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴))
40		mul01 11384	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
41		oveq2 7418	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = (𝑥 · 0))
42	41	eqeq1d 2765	. . . . . . . . . . . . 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 740	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
45		ax-his3 31436	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = (𝑥 · (𝑦 ·ih 𝑧)))
46	45	eqeq1d 2765	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
47	46	3expa 1136	. . . . . . . . . . . 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 591	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℋ ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
51	50	an32s 664	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧 ∈ 𝐴) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
52	51	ralimdva 3177	. . . . . . 7 ⊢ ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0 → ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0))
53	52	imdistanda 581	. . . . . 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 31365	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
55	54	anim1i 626	. . . . . 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 36	. . . . 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 641	. . . . . 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 292	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧 ∈ 𝐴 (𝑦 ·ih 𝑧) = 0)))
60		ocel 31633	. . . . 5 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ ∀𝑧 ∈ 𝐴 ((𝑥 ·ℎ 𝑦) ·ih 𝑧) = 0)))
61	56, 59, 60	3imtr4d 297	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴)))
62	61	ralrimivv 3206	. . 3 ⊢ (𝐴 ⊆ ℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴))
63	39, 62	jca 520	. 2 ⊢ (𝐴 ⊆ ℋ → (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 +ℎ 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 ·ℎ 𝑦) ∈ (⊥‘𝐴)))
64		issh2 31561	. 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 594	1 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ )

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416   ⊆ wss 3905  ‘cfv 6536  (class class class)co 7410  ℂcc 11093  0cc0 11095   + caddc 11098   · cmul 11100   ℋchba 31271   +_ℎ cva 31272   ·_ℎ csm 31273   ·_ih csp 31274  0_ℎc0v 31276   S_ℋ csh 31280  ⊥cort 31282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-hilex 31351  ax-hfvadd 31352  ax-hv0cl 31355  ax-hfvmul 31357  ax-hvmul0 31362  ax-hfi 31431  ax-his2 31435  ax-his3 31436

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-ltxr 11243  df-sh 31559  df-oc 31604

This theorem is referenced by:  shocsh  31636  ocss  31637  occl  31656  spanssoc  31701  ssjo  31799  chscllem2  31990