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Theorem dmdbr5ati 31662

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 31547	. . . . . . 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 1451	. . . . . 6 ⊢ (𝑥 ∈ C_ℋ → (𝐴 𝑀_ℋ^* 𝐵 → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
5		atelch 31584	. . . . . 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 3145	. . 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 30746	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
10	2, 5, 9	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ HAtoms → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
11	10	ineq1d 4210	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴)))
12	1, 2	chjcomi 30708	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
13	12	ineq2i 4208	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴))
14	11, 13	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ HAtoms → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
15	14	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
16	12	sseq2i 4010	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴))
17	16	notbii 319	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ ¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴))
18	2, 1	atabs2i 31642	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵))
19	18	imp 407	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵)
20	17, 19	sylan2b 594	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵)
21	15, 20	eqtr3d 2774	. . . . . . . . . 10 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐵)
22		chjcl 30597	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
23	5, 2, 22	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
24		chincl 30739	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∨_ℋ 𝐵) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
25	23, 1, 24	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
26		chub2 30748	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ C_ℋ ∧ ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ) → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
27	2, 25, 26	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑥 ∈ HAtoms → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
28	27	adantr 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
29	21, 28	eqsstrd 4019	. . . . . . . . 9 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
30	29	ex 413	. . . . . . . 8 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
31	30	biantrud 532	. . . . . . 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 1023	. . . . . . 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 286	. . . . . 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 3091	. . . . 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 31659	. . . . 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 216	. . . 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 31660	. . . 4 ⊢ ((𝐴 +_ℋ 𝐵) = (𝐴 ∨_ℋ 𝐵) ↔ 𝐴 𝑀_ℋ^* 𝐵)
38	36, 37	sylib 217	. . 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 208	. 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 30710	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)
41	40	biantru 530	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)))
42	1, 2	chjcli 30697	. . . . . . . . . . . . 13 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
43		chlub 30749	. . . . . . . . . . . . 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 1453	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
45	41, 44	bitrid 282	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
46		ssid 4003	. . . . . . . . . . . . 13 ⊢ (𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵)
47	46	biantrur 531	. . . . . . . . . . . 12 ⊢ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵) ↔ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵) ∧ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
48		ssin 4229	. . . . . . . . . . . 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 274	. . . . . . . . . . 11 ⊢ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
50	45, 49	bitrdi 286	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵))))
51	50	biimpa 477	. . . . . . . . 9 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
52		inss1 4227	. . . . . . . . 9 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (𝑥 ∨_ℋ 𝐵)
53	51, 52	jctil 520	. . . . . . . 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 3996	. . . . . . . 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 233	. . . . . . 7 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = (𝑥 ∨_ℋ 𝐵))
56	55	sseq1d 4012	. . . . . 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 689	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
58	57, 1, 24	sylancl 586	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
59	2, 58, 26	sylancr 587	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
60	59	biantrud 532	. . . . . . . 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 30597	. . . . . . . . . 10 ⊢ ((((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
62	58, 2, 61	sylancl 586	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
63		chlub 30749	. . . . . . . . . 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 1449	. . . . . . . . 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 685	. . . . . . . 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 278	. . . . . . 7 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
67	66	adantr 481	. . . . . 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 281	. . . . 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 802	. . . 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 3091	. 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 274	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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 (class class class)co 7405 C_ℋ cch 30169 +_ℋ cph 30171 ∨_ℋ chj 30173 HAtomscat 30205 𝑀_ℋ^* cdmd 30207

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 ax-hilex 30239 ax-hfvadd 30240 ax-hvcom 30241 ax-hvass 30242 ax-hv0cl 30243 ax-hvaddid 30244 ax-hfvmul 30245 ax-hvmulid 30246 ax-hvmulass 30247 ax-hvdistr1 30248 ax-hvdistr2 30249 ax-hvmul0 30250 ax-hfi 30319 ax-his1 30322 ax-his2 30323 ax-his3 30324 ax-his4 30325 ax-hcompl 30442

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-cn 22722 df-cnp 22723 df-lm 22724 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cfil 24763 df-cau 24764 df-cmet 24765 df-grpo 29733 df-gid 29734 df-ginv 29735 df-gdiv 29736 df-ablo 29785 df-vc 29799 df-nv 29832 df-va 29835 df-ba 29836 df-sm 29837 df-0v 29838 df-vs 29839 df-nmcv 29840 df-ims 29841 df-dip 29941 df-ssp 29962 df-ph 30053 df-cbn 30103 df-hnorm 30208 df-hba 30209 df-hvsub 30211 df-hlim 30212 df-hcau 30213 df-sh 30447 df-ch 30461 df-oc 30492 df-ch0 30493 df-shs 30548 df-span 30549 df-chj 30550 df-chsup 30551 df-pjh 30635 df-cv 31519 df-md 31520 df-dmd 31521 df-at 31578

This theorem is referenced by: dmdbr6ati 31663 dmdbr7ati 31664

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