mdexchi - Hilbert Space Explorer

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Theorem mdexchi 32367

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 31565	. . . . . . . . . . . . . . 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 31489	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∨_ℋ 𝐴) ∈ C_ℋ
6		chjcom 31538	. . . . . . . . . . . . . . 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 31389	. . . . . . . . . . . . . . . 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 31538	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
11	9, 1, 10	sylancl 585	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) = (𝐶 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
12	4, 7, 11	3eqtr4d 2790	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) = ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶))
13	12	ineq1d 4240	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ (𝐶 ∨_ℋ 𝐴)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵))
14		inass 4249	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
15		incom 4230	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵) = (𝐵 ∩ (𝐴 ∨_ℋ 𝐵))
16		mdexch.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 ∈ C_ℋ
17	2, 16	chjcomi 31500	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
18	17	ineq2i 4238	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = (𝐵 ∩ (𝐵 ∨_ℋ 𝐴))
19	16, 2	chabs2i 31551	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵
20	18, 19	eqtri 2768	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐵
21	15, 20	eqtri 2768	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵
22	21	ineq2i 4238	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵)
23	14, 22	eqtri 2768	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = (((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐶) ∩ 𝐵)
24	13, 23	eqtr4di 2798	. . . . . . . . . . 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 31543	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) ∧ 𝑥 ⊆ 𝐵) → (𝐴 ∨_ℋ 𝑥) ⊆ (𝐴 ∨_ℋ 𝐵))
27	26	ex 412	. . . . . . . . . . . . . . . 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 31489	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
30		mdi 32327	. . . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . 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 31492	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∈ C_ℋ
39		chlej2 31543	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) ⊆ ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴))
44		chjcom 31538	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∨_ℋ 𝑥) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
45	9, 2, 44	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)))
46	2	chjidmi 31553	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∨_ℋ 𝐴) = 𝐴
47	46	oveq1i 7458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∨_ℋ 𝐴) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝑥)
48		chjass 31565	. . . . . . . . . . . . . . . . . 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 31538	. . . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ (𝐴 ∨_ℋ 𝑥)) = (𝑥 ∨_ℋ 𝐴))
53	45, 52	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝑥 ∨_ℋ 𝐴))
54	53	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝑥) ∨_ℋ 𝐴) = (𝑥 ∨_ℋ 𝐴))
55	43, 54	sseqtrd 4049	. . . . . . . . . . . . 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 4047	. . . . . . . . . . 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 4265	. . . . . . . . . 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 4047	. . . . . . . . 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 32327	. . . . . . . . . . . . . 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 420	. . . . . . . . . . . . 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 417	. . . . . . . . . 10 ⊢ (((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ 𝐵) ∧ 𝐴 𝑀_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))
66	2, 1	chub2i 31502	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐶 ∨_ℋ 𝐴)
67		ssrin 4263	. . . . . . . . . . . . 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 31492	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
70	5, 16	chincli 31492	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ
71		chlej2 31543	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 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 4047	. . . . . . . . 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 4019	. . . . . . 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 419	. . . . . 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 3151	. . 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 32328	. . . 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 234	. 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 31500	. . . . 5 ⊢ (𝐶 ∨_ℋ 𝐴) = (𝐴 ∨_ℋ 𝐶)
87		incom 4230	. . . . . 6 ⊢ (𝐵 ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)
88	18, 87, 19	3eqtr3ri 2777	. . . . 5 ⊢ 𝐵 = ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵)
89	86, 88	ineq12i 4239	. . . 4 ⊢ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = ((𝐴 ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
90		inass 4249	. . . . 5 ⊢ (((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) ∩ 𝐵) = ((𝐴 ∨_ℋ 𝐶) ∩ ((𝐴 ∨_ℋ 𝐵) ∩ 𝐵))
91	2, 16	chub1i 31501	. . . . . . . 8 ⊢ 𝐴 ⊆ (𝐴 ∨_ℋ 𝐵)
92		mdi 32327	. . . . . . . . . 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 420	. . . . . . . . 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 31500	. . . . . . . 8 ⊢ (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) = ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∨_ℋ 𝐴)
97	38, 2	chlejb1i 31508	. . . . . . . . 9 ⊢ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 ↔ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∨_ℋ 𝐴) = 𝐴)
98	97	biimpi 216	. . . . . . . 8 ⊢ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 → ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ∨_ℋ 𝐴) = 𝐴)
99	96, 98	eqtrid 2792	. . . . . . 7 ⊢ ((𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴 → (𝐴 ∨_ℋ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵))) = 𝐴)
100	95, 99	sylan9eq 2800	. . . . . 6 ⊢ ((𝐶 𝑀_ℋ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ 𝐴) → ((𝐴 ∨_ℋ 𝐶) ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐴)
101	100	ineq1d 4240	. . . . 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 2794	. . . 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 2792	. . 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 511	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∩ cin 3975 ⊆ wss 3976 class class class wbr 5166 (class class class)co 7448 C_ℋ cch 30961 ∨_ℋ chj 30965 𝑀_ℋ cmd 30998

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264 ax-hilex 31031 ax-hfvadd 31032 ax-hvcom 31033 ax-hvass 31034 ax-hv0cl 31035 ax-hvaddid 31036 ax-hfvmul 31037 ax-hvmulid 31038 ax-hvmulass 31039 ax-hvdistr1 31040 ax-hvdistr2 31041 ax-hvmul0 31042 ax-hfi 31111 ax-his1 31114 ax-his2 31115 ax-his3 31116 ax-his4 31117 ax-hcompl 31234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-cn 23256 df-cnp 23257 df-lm 23258 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cfil 25308 df-cau 25309 df-cmet 25310 df-grpo 30525 df-gid 30526 df-ginv 30527 df-gdiv 30528 df-ablo 30577 df-vc 30591 df-nv 30624 df-va 30627 df-ba 30628 df-sm 30629 df-0v 30630 df-vs 30631 df-nmcv 30632 df-ims 30633 df-dip 30733 df-ssp 30754 df-ph 30845 df-cbn 30895 df-hnorm 31000 df-hba 31001 df-hvsub 31003 df-hlim 31004 df-hcau 31005 df-sh 31239 df-ch 31253 df-oc 31284 df-ch0 31285 df-shs 31340 df-chj 31342 df-md 32312

This theorem is referenced by: (None)

W3C validator