mdexchi - Hilbert Space Explorer

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Theorem mdexchi 31588

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 30786	. . . . . . . . . . . . . . 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 1452	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
5	1, 2	chjcli 30710	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∨_ℋ 𝐴) ∈ C_ℋ
6		chjcom 30759	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∨_ℋ 𝐴) ∈ C_ℋ ) → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥))
7	5, 6	mpan2 690	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐶 ∨_ℋ 𝐴) ∨_ℋ 𝑥))
8		chjcl 30610	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ )
9	2, 8	mpan 689	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ 𝑥) ∈ C_ℋ )
10		chjcom 30759	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
11	9, 1, 10	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
12	4, 7, 11	3eqtr4d 2783	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶))
13	12	ineq1d 4212	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵))
14		inass 4220	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
15		incom 4202	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵) = (𝐵 ∩ (𝐴 ∨_ℋ 𝐵))
16		mdexch.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 ∈ C_ℋ
17	2, 16	chjcomi 30721	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
18	17	ineq2i 4210	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = (𝐵 ∩ (𝐵 ∨_ℋ 𝐴))
19	16, 2	chabs2i 30772	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵
20	18, 19	eqtri 2761	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐵
21	15, 20	eqtri 2761	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵
22	21	ineq2i 4210	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵)
23	14, 22	eqtri 2761	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵)
24	13, 23	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) = ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵))
25	24	ad2antrr 725	. . . . . . . . . 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 30764	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) ∧ 𝑥 ⊆ 𝐵) → (𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵))
27	26	ex 414	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝑥 ⊆ 𝐵 → (𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵)))
28	16, 2, 27	mp3an23 1454	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵)))
29	2, 16	chjcli 30710	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
30		mdi 31548	. . . . . . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . . . . . 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 1452	. . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . 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 716	. . . . . . . . . . . 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 30713	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∈ C_ℋ
39		chlej2 30764	. . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . 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 1452	. . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴))
44		chjcom 30759	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
45	9, 2, 44	sylancl 587	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
46	2	chjidmi 30774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∨_ℋ 𝐴) = 𝐴
47	46	oveq1i 7419	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝑥)
48		chjass 30786	. . . . . . . . . . . . . . . . . 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 1452	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
50		chjcom 30759	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐴 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐴))
51	2, 50	mpan 689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐴))
52	47, 49, 51	3eqtr3a 2797	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)) = (𝑥 ∨_ℋ 𝐴))
53	45, 52	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝑥 ∨_ℋ 𝐴))
54	53	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝑥 ∨_ℋ 𝐴))
55	43, 54	sseqtrd 4023	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ (𝑥 ∨_ℋ 𝐴))
56	55	ad2ant2rl 748	. . . . . . . . . . . 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 4021	. . . . . . . . . . 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 4236	. . . . . . . . . 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 4021	. . . . . . . . 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 715	. . . . . . . 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 31548	. . . . . . . . . . . . . 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 422	. . . . . . . . . . . . 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 1452	. . . . . . . . . . . 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 419	. . . . . . . . . 10 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐴 𝑀_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))
66	2, 1	chub2i 30723	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐶 ∨_ℋ 𝐴)
67		ssrin 4234	. . . . . . . . . . . . 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 30713	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
70	5, 16	chincli 30713	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ
71		chlej2 30764	. . . . . . . . . . . . . 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 414	. . . . . . . . . . . . 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 1452	. . . . . . . . . . . 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 725	. . . . . . . . . 10 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐴 𝑀_ℋ 𝐵) → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
76	65, 75	eqsstrd 4021	. . . . . . . . 9 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐴 𝑀_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵)))
77	76	adantrr 716	. . . . . . . 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 3993	. . . . . . 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 421	. . . . . 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 1116	. . . 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 3146	. . 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 31549	. . . 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 691	. . 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 30721	. . . . 5 ⊢ (𝐶 ∨_ℋ 𝐴) = (𝐴 ∨_ℋ 𝐶)
87		incom 4202	. . . . . 6 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)
88	18, 87, 19	3eqtr3ri 2770	. . . . 5 ⊢ 𝐵 = ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)
89	86, 88	ineq12i 4211	. . . 4 ⊢ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = ((𝐴 ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
90		inass 4220	. . . . 5 ⊢ (((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = ((𝐴 ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
91	2, 16	chub1i 30722	. . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∨_ℋ 𝐵)
92		mdi 31548	. . . . . . . . . 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 422	. . . . . . . . 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 1462	. . . . . . . 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 30721	. . . . . . . 8 ⊢ (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) = ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∨_ℋ 𝐴)
97	38, 2	chlejb1i 30729	. . . . . . . . 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 2785	. . . . . . 7 ⊢ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 → (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) = 𝐴)
100	95, 99	sylan9eq 2793	. . . . . 6 ⊢ ((𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐴)
101	100	ineq1d 4212	. . . . 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 2787	. . . 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 2785	. . 3 ⊢ ((𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = (𝐴 ∩ 𝐵))
104	103	3adant1 1131	. 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 513	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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∩ cin 3948 ⊆ wss 3949 class class class wbr 5149 (class class class)co 7409 C_ℋ cch 30182 ∨_ℋ chj 30186 𝑀_ℋ cmd 30219

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 ax-hilex 30252 ax-hfvadd 30253 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvmulass 30260 ax-hvdistr1 30261 ax-hvdistr2 30262 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his2 30336 ax-his3 30337 ax-his4 30338 ax-hcompl 30455

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-cn 22731 df-cnp 22732 df-lm 22733 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cfil 24772 df-cau 24773 df-cmet 24774 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-vs 29852 df-nmcv 29853 df-ims 29854 df-dip 29954 df-ssp 29975 df-ph 30066 df-cbn 30116 df-hnorm 30221 df-hba 30222 df-hvsub 30224 df-hlim 30225 df-hcau 30226 df-sh 30460 df-ch 30474 df-oc 30505 df-ch0 30506 df-shs 30561 df-chj 30563 df-md 31533

This theorem is referenced by: (None)

W3C validator