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Theorem dmdbr5ati 32454

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 32339	. . . . . . 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 32376	. . . . . 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 3151	. . 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 31538	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
10	2, 5, 9	sylancr 586	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ HAtoms → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
11	10	ineq1d 4240	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴)))
12	1, 2	chjcomi 31500	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
13	12	ineq2i 4238	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴))
14	11, 13	eqtr4di 2798	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ HAtoms → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
15	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
16	12	sseq2i 4038	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴))
17	16	notbii 320	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ ¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴))
18	2, 1	atabs2i 32434	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵))
19	18	imp 406	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵)
20	17, 19	sylan2b 593	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵)
21	15, 20	eqtr3d 2782	. . . . . . . . . 10 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐵)
22		chjcl 31389	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
23	5, 2, 22	sylancl 585	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
24		chincl 31531	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∨_ℋ 𝐵) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
25	23, 1, 24	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
26		chub2 31540	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ C_ℋ ∧ ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ) → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
27	2, 25, 26	sylancr 586	. . . . . . . . . . 11 ⊢ (𝑥 ∈ HAtoms → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
28	27	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
29	21, 28	eqsstrd 4047	. . . . . . . . 9 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
30	29	ex 412	. . . . . . . 8 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
31	30	biantrud 531	. . . . . . 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 287	. . . . . 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 3097	. . . . 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 32451	. . . . 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 217	. . . 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 32452	. . . 4 ⊢ ((𝐴 +_ℋ 𝐵) = (𝐴 ∨_ℋ 𝐵) ↔ 𝐴 𝑀_ℋ^* 𝐵)
38	36, 37	sylib 218	. . 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 209	. 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 31502	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)
41	40	biantru 529	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)))
42	1, 2	chjcli 31489	. . . . . . . . . . . . 13 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
43		chlub 31541	. . . . . . . . . . . . 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 283	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
46		ssid 4031	. . . . . . . . . . . . 13 ⊢ (𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵)
47	46	biantrur 530	. . . . . . . . . . . 12 ⊢ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵) ↔ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵) ∧ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
48		ssin 4260	. . . . . . . . . . . 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 275	. . . . . . . . . . 11 ⊢ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
50	45, 49	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵))))
51	50	biimpa 476	. . . . . . . . 9 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
52		inss1 4258	. . . . . . . . 9 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (𝑥 ∨_ℋ 𝐵)
53	51, 52	jctil 519	. . . . . . . 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 4024	. . . . . . . 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 234	. . . . . . 7 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = (𝑥 ∨_ℋ 𝐵))
56	55	sseq1d 4040	. . . . . 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 690	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
58	57, 1, 24	sylancl 585	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
59	2, 58, 26	sylancr 586	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
60	59	biantrud 531	. . . . . . . 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 31389	. . . . . . . . . 10 ⊢ ((((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
62	58, 2, 61	sylancl 585	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
63		chlub 31541	. . . . . . . . . 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 686	. . . . . . . 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 279	. . . . . . 7 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
67	66	adantr 480	. . . . . 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 282	. . . . 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 803	. . . 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 3097	. 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 275	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 206 ∧ wa 395 = 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 cph 30963 ∨_ℋ chj 30965 HAtomscat 30997 𝑀_ℋ^* cdmd 30999

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-span 31341 df-chj 31342 df-chsup 31343 df-pjh 31427 df-cv 32311 df-md 32312 df-dmd 32313 df-at 32370

This theorem is referenced by: dmdbr6ati 32455 dmdbr7ati 32456

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