mdexchi - Hilbert Space Explorer

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Theorem mdexchi 31277

Description: An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
mdexch.1	⊢ 𝐴 ∈ C_ℋ
mdexch.2	⊢ 𝐵 ∈ C_ℋ
mdexch.3	⊢ 𝐶 ∈ C_ℋ

**Assertion**
Ref	Expression
mdexchi	⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ 𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐶 ∨_ℋ 𝐴) 𝑀_ℋ 𝐵 ∧ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = (𝐴 ∩ 𝐵)))

Proof of Theorem mdexchi Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdexch.3	. . . . . . . . . . . . . . 15 ⊢ 𝐶 ∈ C_ℋ
2		mdexch.1	. . . . . . . . . . . . . . 15 ⊢ 𝐴 ∈ C_ℋ
3		chjass 30475	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
4	1, 2, 3	mp3an12 1451	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
5	1, 2	chjcli 30399	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∨_ℋ 𝐴) ∈ C_ℋ
6		chjcom 30448	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∨_ℋ 𝐴) ∈ C_ℋ ) → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥))
7	5, 6	mpan2 689	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥))
8		chjcl 30299	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ )
9	2, 8	mpan 688	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ )
10		chjcom 30448	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
11	9, 1, 10	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
12	4, 7, 11	3eqtr4d 2786	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶))
13	12	ineq1d 4171	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵))
14		inass 4179	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
15		incom 4161	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵) = (𝐵 ∩ (𝐴 ∨_ℋ 𝐵))
16		mdexch.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 ∈ C_ℋ
17	2, 16	chjcomi 30410	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
18	17	ineq2i 4169	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = (𝐵 ∩ (𝐵 ∨_ℋ 𝐴))
19	16, 2	chabs2i 30461	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵
20	18, 19	eqtri 2764	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐵
21	15, 20	eqtri 2764	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵
22	21	ineq2i 4169	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵)
23	14, 22	eqtri 2764	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵)
24	13, 23	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) = ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵))
25	24	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴)) → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) = ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵))
26		chlej2 30453	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) ∧ 𝑥 ⊆ 𝐵) → (𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵))
27	26	ex 413	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝑥 ⊆ 𝐵 → (𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵)))
28	16, 2, 27	mp3an23 1453	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵)))
29	2, 16	chjcli 30399	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
30		mdi 31237	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐶 ∈ C_ℋ ∧ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ ∧ (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ) ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵))) → (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))
31	30	exp32 421	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ C_ℋ ∧ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ ∧ (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ) → (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) → ((𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵) → (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))))
32	1, 29, 31	mp3an12 1451	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∨_ℋ 𝑥) ∈ C_ℋ → (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) → ((𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵) → (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))))
33	9, 32	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ C_ℋ → (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) → ((𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵) → (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))))
34	33	com23 86	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵) → (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) → (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))))
35	28, 34	syld 47	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) → (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))))
36	35	imp31 418	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵)) → (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))
37	36	adantrr 715	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴)) → (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))
38	1, 29	chincli 30402	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∈ C_ℋ
39		chlej2 30453	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ∧ (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴))
40	39	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ∧ (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ) → ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴)))
41	38, 2, 40	mp3an12 1451	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∨_ℋ 𝑥) ∈ C_ℋ → ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴)))
42	9, 41	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴)))
43	42	imp 407	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴))
44		chjcom 30448	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
45	9, 2, 44	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
46	2	chjidmi 30463	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∨_ℋ 𝐴) = 𝐴
47	46	oveq1i 7367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝑥)
48		chjass 30475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → ((𝐴 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
49	2, 2, 48	mp3an12 1451	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
50		chjcom 30448	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐴 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐴))
51	2, 50	mpan 688	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐴))
52	47, 49, 51	3eqtr3a 2800	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)) = (𝑥 ∨_ℋ 𝐴))
53	45, 52	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝑥 ∨_ℋ 𝐴))
54	53	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝑥 ∨_ℋ 𝐴))
55	43, 54	sseqtrd 3984	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ (𝑥 ∨_ℋ 𝐴))
56	55	ad2ant2rl 747	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴)) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ (𝑥 ∨_ℋ 𝐴))
57	37, 56	eqsstrd 3982	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴)) → (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (𝑥 ∨_ℋ 𝐴))
58	57	ssrind 4195	. . . . . . . . . 10 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴)) → ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))
59	25, 58	eqsstrd 3982	. . . . . . . . 9 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴)) → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))
60	59	adantrl 714	. . . . . . . 8 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ (𝐴 𝑀_ℋ 𝐵 ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴))) → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))
61		mdi 31237	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) ∧ (𝐴 𝑀_ℋ 𝐵 ∧ 𝑥 ⊆ 𝐵)) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))
62	61	exp32 421	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐴 𝑀_ℋ 𝐵 → (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
63	2, 16, 62	mp3an12 1451	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → (𝐴 𝑀_ℋ 𝐵 → (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
64	63	com23 86	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (𝐴 𝑀_ℋ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
65	64	imp31 418	. . . . . . . . . 10 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐴 𝑀_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))
66	2, 1	chub2i 30412	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐶 ∨_ℋ 𝐴)
67		ssrin 4193	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ (𝐶 ∨_ℋ 𝐴) → (𝐴 ∩ 𝐵) ⊆ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵))
68	66, 67	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐵) ⊆ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)
69	2, 16	chincli 30402	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
70	5, 16	chincli 30402	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ
71		chlej2 30453	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)) → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
72	71	ex 413	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → ((𝐴 ∩ 𝐵) ⊆ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵))))
73	69, 70, 72	mp3an12 1451	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∩ 𝐵) ⊆ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵))))
74	68, 73	mpi 20	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
75	74	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐴 𝑀_ℋ 𝐵) → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
76	65, 75	eqsstrd 3982	. . . . . . . . 9 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐴 𝑀_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
77	76	adantrr 715	. . . . . . . 8 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ (𝐴 𝑀_ℋ 𝐵 ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴))) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
78	60, 77	sstrd 3954	. . . . . . 7 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ (𝐴 𝑀_ℋ 𝐵 ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴))) → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
79	78	exp31 420	. . . . . 6 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → ((𝐴 𝑀_ℋ 𝐵 ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴)) → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))))
80	79	com3r 87	. . . . 5 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴)) → (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))))
81	80	3impb 1115	. . . 4 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ 𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))))
82	81	ralrimiv 3142	. . 3 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ 𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ∀𝑥 ∈ C_ℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵))))
83		mdbr2 31238	. . . 4 ⊢ (((𝐶 ∨_ℋ 𝐴) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ((𝐶 ∨_ℋ 𝐴) 𝑀_ℋ 𝐵 ↔ ∀𝑥 ∈ C_ℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))))
84	5, 16, 83	mp2an 690	. . 3 ⊢ ((𝐶 ∨_ℋ 𝐴) 𝑀_ℋ 𝐵 ↔ ∀𝑥 ∈ C_ℋ (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵))))
85	82, 84	sylibr 233	. 2 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ 𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → (𝐶 ∨_ℋ 𝐴) 𝑀_ℋ 𝐵)
86	1, 2	chjcomi 30410	. . . . 5 ⊢ (𝐶 ∨_ℋ 𝐴) = (𝐴 ∨_ℋ 𝐶)
87		incom 4161	. . . . . 6 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)
88	18, 87, 19	3eqtr3ri 2773	. . . . 5 ⊢ 𝐵 = ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)
89	86, 88	ineq12i 4170	. . . 4 ⊢ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = ((𝐴 ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
90		inass 4179	. . . . 5 ⊢ (((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = ((𝐴 ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
91	2, 16	chub1i 30411	. . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∨_ℋ 𝐵)
92		mdi 31237	. . . . . . . . . 10 ⊢ (((𝐶 ∈ C_ℋ ∧ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) ∧ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ 𝐴 ⊆ (𝐴 ∨_ℋ 𝐵))) → ((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))
93	92	exp32 421	. . . . . . . . 9 ⊢ ((𝐶 ∈ C_ℋ ∧ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) → (𝐴 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))))
94	1, 29, 2, 93	mp3an 1461	. . . . . . . 8 ⊢ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) → (𝐴 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)))))
95	91, 94	mpi 20	. . . . . . 7 ⊢ (𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) → ((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))))
96	2, 38	chjcomi 30410	. . . . . . . 8 ⊢ (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) = ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∨_ℋ 𝐴)
97	38, 2	chlejb1i 30418	. . . . . . . . 9 ⊢ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 ↔ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∨_ℋ 𝐴) = 𝐴)
98	97	biimpi 215	. . . . . . . 8 ⊢ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 → ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∨_ℋ 𝐴) = 𝐴)
99	96, 98	eqtrid 2788	. . . . . . 7 ⊢ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 → (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) = 𝐴)
100	95, 99	sylan9eq 2796	. . . . . 6 ⊢ ((𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐴)
101	100	ineq1d 4171	. . . . 5 ⊢ ((𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → (((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = (𝐴 ∩ 𝐵))
102	90, 101	eqtr3id 2790	. . . 4 ⊢ ((𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)) = (𝐴 ∩ 𝐵))
103	89, 102	eqtrid 2788	. . 3 ⊢ ((𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = (𝐴 ∩ 𝐵))
104	103	3adant1 1130	. 2 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ 𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = (𝐴 ∩ 𝐵))
105	85, 104	jca 512	1 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ 𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐶 ∨_ℋ 𝐴) 𝑀_ℋ 𝐵 ∧ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = (𝐴 ∩ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∩ cin 3909 ⊆ wss 3910 class class class wbr 5105 (class class class)co 7357 C_ℋ cch 29871 ∨_ℋ chj 29875 𝑀_ℋ cmd 29908

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 ax-hilex 29941 ax-hfvadd 29942 ax-hvcom 29943 ax-hvass 29944 ax-hv0cl 29945 ax-hvaddid 29946 ax-hfvmul 29947 ax-hvmulid 29948 ax-hvmulass 29949 ax-hvdistr1 29950 ax-hvdistr2 29951 ax-hvmul0 29952 ax-hfi 30021 ax-his1 30024 ax-his2 30025 ax-his3 30026 ax-his4 30027 ax-hcompl 30144

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-cn 22578 df-cnp 22579 df-lm 22580 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cfil 24619 df-cau 24620 df-cmet 24621 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-dip 29643 df-ssp 29664 df-ph 29755 df-cbn 29805 df-hnorm 29910 df-hba 29911 df-hvsub 29913 df-hlim 29914 df-hcau 29915 df-sh 30149 df-ch 30163 df-oc 30194 df-ch0 30195 df-shs 30250 df-chj 30252 df-md 31222

This theorem is referenced by: (None)

W3C validator