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Theorem mdexchi 30697

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 29895	. . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
5	1, 2	chjcli 29819	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∨_ℋ 𝐴) ∈ C_ℋ
6		chjcom 29868	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∨_ℋ 𝐴) ∈ C_ℋ ) → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥))
7	5, 6	mpan2 688	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥))
8		chjcl 29719	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ )
9	2, 8	mpan 687	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ )
10		chjcom 29868	. . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶))
13	12	ineq1d 4145	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵))
14		inass 4153	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
15		incom 4135	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵) = (𝐵 ∩ (𝐴 ∨_ℋ 𝐵))
16		mdexch.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 ∈ C_ℋ
17	2, 16	chjcomi 29830	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
18	17	ineq2i 4143	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = (𝐵 ∩ (𝐵 ∨_ℋ 𝐴))
19	16, 2	chabs2i 29881	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵
20	18, 19	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐵
21	15, 20	eqtri 2766	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵
22	21	ineq2i 4143	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵)
23	14, 22	eqtri 2766	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵)
24	13, 23	eqtr4di 2796	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) = ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵))
25	24	ad2antrr 723	. . . . . . . . . 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 29873	. . . . . . . . . . . . . . . . 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 1452	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵)))
29	2, 16	chjcli 29819	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
30		mdi 30657	. . . . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . 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 29822	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∈ C_ℋ
39		chlej2 29873	. . . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . . . 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 29868	. . . . . . . . . . . . . . . . 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 29883	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∨_ℋ 𝐴) = 𝐴
47	46	oveq1i 7285	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝑥)
48		chjass 29895	. . . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
50		chjcom 29868	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐴 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐴))
51	2, 50	mpan 687	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐴))
52	47, 49, 51	3eqtr3a 2802	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)) = (𝑥 ∨_ℋ 𝐴))
53	45, 52	eqtrd 2778	. . . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ (𝑥 ∨_ℋ 𝐴))
56	55	ad2ant2rl 746	. . . . . . . . . . . 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 3959	. . . . . . . . . . 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 4169	. . . . . . . . . 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 3959	. . . . . . . . 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 713	. . . . . . . 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 30657	. . . . . . . . . . . . . 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 1450	. . . . . . . . . . . 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 29832	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐶 ∨_ℋ 𝐴)
67		ssrin 4167	. . . . . . . . . . . . 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 29822	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
70	5, 16	chincli 29822	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ
71		chlej2 29873	. . . . . . . . . . . . . 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 1450	. . . . . . . . . . . 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 723	. . . . . . . . . 10 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐴 𝑀_ℋ 𝐵) → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
76	65, 75	eqsstrd 3959	. . . . . . . . 9 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐴 𝑀_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
77	76	adantrr 714	. . . . . . . 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 3931	. . . . . . 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 1114	. . . 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 3102	. . 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 30658	. . . 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 689	. . 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 29830	. . . . 5 ⊢ (𝐶 ∨_ℋ 𝐴) = (𝐴 ∨_ℋ 𝐶)
87		incom 4135	. . . . . 6 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)
88	18, 87, 19	3eqtr3ri 2775	. . . . 5 ⊢ 𝐵 = ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)
89	86, 88	ineq12i 4144	. . . 4 ⊢ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = ((𝐴 ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
90		inass 4153	. . . . 5 ⊢ (((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = ((𝐴 ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
91	2, 16	chub1i 29831	. . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∨_ℋ 𝐵)
92		mdi 30657	. . . . . . . . . 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 1460	. . . . . . . 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 29830	. . . . . . . 8 ⊢ (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) = ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∨_ℋ 𝐴)
97	38, 2	chlejb1i 29838	. . . . . . . . 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 2790	. . . . . . 7 ⊢ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 → (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) = 𝐴)
100	95, 99	sylan9eq 2798	. . . . . 6 ⊢ ((𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐴)
101	100	ineq1d 4145	. . . . 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 2792	. . . 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 2790	. . 3 ⊢ ((𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = (𝐴 ∩ 𝐵))
104	103	3adant1 1129	. 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 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 (class class class)co 7275 C_ℋ cch 29291 ∨_ℋ chj 29295 𝑀_ℋ cmd 29328

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 ax-hilex 29361 ax-hfvadd 29362 ax-hvcom 29363 ax-hvass 29364 ax-hv0cl 29365 ax-hvaddid 29366 ax-hfvmul 29367 ax-hvmulid 29368 ax-hvmulass 29369 ax-hvdistr1 29370 ax-hvdistr2 29371 ax-hvmul0 29372 ax-hfi 29441 ax-his1 29444 ax-his2 29445 ax-his3 29446 ax-his4 29447 ax-hcompl 29564

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-cn 22378 df-cnp 22379 df-lm 22380 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cfil 24419 df-cau 24420 df-cmet 24421 df-grpo 28855 df-gid 28856 df-ginv 28857 df-gdiv 28858 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-vs 28961 df-nmcv 28962 df-ims 28963 df-dip 29063 df-ssp 29084 df-ph 29175 df-cbn 29225 df-hnorm 29330 df-hba 29331 df-hvsub 29333 df-hlim 29334 df-hcau 29335 df-sh 29569 df-ch 29583 df-oc 29614 df-ch0 29615 df-shs 29670 df-chj 29672 df-md 30642

This theorem is referenced by: (None)

W3C validator