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Theorem dmdbr5ati 30457

Description: Dual modular pair property in terms of atoms. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
sumdmdi.1	⊢ 𝐴 ∈ C_ℋ
sumdmdi.2	⊢ 𝐵 ∈ C_ℋ

**Assertion**
Ref	Expression
dmdbr5ati	⊢ (𝐴 𝑀_ℋ^* 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem dmdbr5ati**
Step	Hyp	Ref	Expression
1		sumdmdi.1	. . . . . . 7 ⊢ 𝐴 ∈ C_ℋ
2		sumdmdi.2	. . . . . . 7 ⊢ 𝐵 ∈ C_ℋ
3		dmdi4 30342	. . . . . . 7 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐴 𝑀_ℋ^* 𝐵 → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
4	1, 2, 3	mp3an12 1453	. . . . . 6 ⊢ (𝑥 ∈ C_ℋ → (𝐴 𝑀_ℋ^* 𝐵 → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
5		atelch 30379	. . . . . 6 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
6	4, 5	syl11 33	. . . . 5 ⊢ (𝐴 𝑀_ℋ^* 𝐵 → (𝑥 ∈ HAtoms → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
7	6	a1dd 50	. . . 4 ⊢ (𝐴 𝑀_ℋ^* 𝐵 → (𝑥 ∈ HAtoms → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))))
8	7	ralrimiv 3094	. . 3 ⊢ (𝐴 𝑀_ℋ^* 𝐵 → ∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
9		chjcom 29541	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
10	2, 5, 9	sylancr 590	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ HAtoms → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
11	10	ineq1d 4112	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴)))
12	1, 2	chjcomi 29503	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
13	12	ineq2i 4110	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴))
14	11, 13	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ HAtoms → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
15	14	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
16	12	sseq2i 3916	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴))
17	16	notbii 323	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ ¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴))
18	2, 1	atabs2i 30437	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵))
19	18	imp 410	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵)
20	17, 19	sylan2b 597	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵)
21	15, 20	eqtr3d 2773	. . . . . . . . . 10 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐵)
22		chjcl 29392	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
23	5, 2, 22	sylancl 589	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
24		chincl 29534	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∨_ℋ 𝐵) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
25	23, 1, 24	sylancl 589	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
26		chub2 29543	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ C_ℋ ∧ ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ) → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
27	2, 25, 26	sylancr 590	. . . . . . . . . . 11 ⊢ (𝑥 ∈ HAtoms → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
28	27	adantr 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
29	21, 28	eqsstrd 3925	. . . . . . . . 9 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
30	29	ex 416	. . . . . . . 8 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
31	30	biantrud 535	. . . . . . 7 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ↔ ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ∧ (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))))
32		pm4.83 1025	. . . . . . 7 ⊢ (((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ∧ (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))) ↔ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
33	31, 32	bitrdi 290	. . . . . 6 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ↔ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
34	33	ralbiia 3077	. . . . 5 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ↔ ∀𝑥 ∈ HAtoms ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
35	1, 2	sumdmdlem2 30454	. . . . 5 ⊢ (∀𝑥 ∈ HAtoms ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) → (𝐴 +_ℋ 𝐵) = (𝐴 ∨_ℋ 𝐵))
36	34, 35	sylbi 220	. . . 4 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) → (𝐴 +_ℋ 𝐵) = (𝐴 ∨_ℋ 𝐵))
37	1, 2	sumdmdi 30455	. . . 4 ⊢ ((𝐴 +_ℋ 𝐵) = (𝐴 ∨_ℋ 𝐵) ↔ 𝐴 𝑀_ℋ^* 𝐵)
38	36, 37	sylib 221	. . 3 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) → 𝐴 𝑀_ℋ^* 𝐵)
39	8, 38	impbii 212	. 2 ⊢ (𝐴 𝑀_ℋ^* 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
40	2, 1	chub2i 29505	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)
41	40	biantru 533	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)))
42	1, 2	chjcli 29492	. . . . . . . . . . . . 13 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
43		chlub 29544	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ ) → ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
44	2, 42, 43	mp3an23 1455	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
45	41, 44	syl5bb 286	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
46		ssid 3909	. . . . . . . . . . . . 13 ⊢ (𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵)
47	46	biantrur 534	. . . . . . . . . . . 12 ⊢ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵) ↔ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵) ∧ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
48		ssin 4131	. . . . . . . . . . . 12 ⊢ (((𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵) ∧ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
49	47, 48	bitri 278	. . . . . . . . . . 11 ⊢ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
50	45, 49	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵))))
51	50	biimpa 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
52		inss1 4129	. . . . . . . . 9 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (𝑥 ∨_ℋ 𝐵)
53	51, 52	jctil 523	. . . . . . . 8 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (𝑥 ∨_ℋ 𝐵) ∧ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵))))
54		eqss 3902	. . . . . . . 8 ⊢ (((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = (𝑥 ∨_ℋ 𝐵) ↔ (((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (𝑥 ∨_ℋ 𝐵) ∧ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵))))
55	53, 54	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = (𝑥 ∨_ℋ 𝐵))
56	55	sseq1d 3918	. . . . . 6 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
57	2, 22	mpan2 691	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
58	57, 1, 24	sylancl 589	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
59	2, 58, 26	sylancr 590	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
60	59	biantrud 535	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ↔ (𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))))
61		chjcl 29392	. . . . . . . . . 10 ⊢ ((((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
62	58, 2, 61	sylancl 589	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
63		chlub 29544	. . . . . . . . . 10 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ ) → ((𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
64	2, 63	mp3an2 1451	. . . . . . . . 9 ⊢ ((𝑥 ∈ C_ℋ ∧ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ ) → ((𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
65	62, 64	mpdan 687	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
66	60, 65	bitrd 282	. . . . . . 7 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
67	66	adantr 484	. . . . . 6 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
68	56, 67	bitr4d 285	. . . . 5 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ↔ 𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
69	68	pm5.74da 804	. . . 4 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ↔ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))))
70	5, 69	syl 17	. . 3 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ↔ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))))
71	70	ralbiia 3077	. 2 ⊢ (∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)) ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
72	39, 71	bitri 278	1 ⊢ (𝐴 𝑀_ℋ^* 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∩ cin 3852 ⊆ wss 3853 class class class wbr 5039 (class class class)co 7191 C_ℋ cch 28964 +_ℋ cph 28966 ∨_ℋ chj 28968 HAtomscat 29000 𝑀_ℋ^* cdmd 29002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cc 10014 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 ax-hilex 29034 ax-hfvadd 29035 ax-hvcom 29036 ax-hvass 29037 ax-hv0cl 29038 ax-hvaddid 29039 ax-hfvmul 29040 ax-hvmulid 29041 ax-hvmulass 29042 ax-hvdistr1 29043 ax-hvdistr2 29044 ax-hvmul0 29045 ax-hfi 29114 ax-his1 29117 ax-his2 29118 ax-his3 29119 ax-his4 29120 ax-hcompl 29237

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-oadd 8184 df-omul 8185 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-acn 9523 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-rlim 15015 df-sum 15215 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-mulg 18443 df-cntz 18665 df-cmn 19126 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-cn 22078 df-cnp 22079 df-lm 22080 df-haus 22166 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-xms 23172 df-ms 23173 df-tms 23174 df-cfil 24106 df-cau 24107 df-cmet 24108 df-grpo 28528 df-gid 28529 df-ginv 28530 df-gdiv 28531 df-ablo 28580 df-vc 28594 df-nv 28627 df-va 28630 df-ba 28631 df-sm 28632 df-0v 28633 df-vs 28634 df-nmcv 28635 df-ims 28636 df-dip 28736 df-ssp 28757 df-ph 28848 df-cbn 28898 df-hnorm 29003 df-hba 29004 df-hvsub 29006 df-hlim 29007 df-hcau 29008 df-sh 29242 df-ch 29256 df-oc 29287 df-ch0 29288 df-shs 29343 df-span 29344 df-chj 29345 df-chsup 29346 df-pjh 29430 df-cv 30314 df-md 30315 df-dmd 30316 df-at 30373

This theorem is referenced by: dmdbr6ati 30458 dmdbr7ati 30459

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