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Theorem dmdbr5ati 32625

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 32510	. . . . . . 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 1472	. . . . . 6 ⊢ (𝑥 ∈ C_ℋ → (𝐴 𝑀_ℋ^* 𝐵 → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
5		atelch 32547	. . . . . 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 3153	. . 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 31709	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
10	2, 5, 9	sylancr 596	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ HAtoms → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
11	10	ineq1d 4171	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴)))
12	1, 2	chjcomi 31671	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
13	12	ineq2i 4169	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴))
14	11, 13	eqtr4di 2815	. . . . . . . . . . . 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 3965	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴))
17	16	notbii 322	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ ¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴))
18	2, 1	atabs2i 32605	. . . . . . . . . . . . 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 603	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵)
21	15, 20	eqtr3d 2799	. . . . . . . . . 10 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐵)
22		chjcl 31560	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
23	5, 2, 22	sylancl 595	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
24		chincl 31702	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∨_ℋ 𝐵) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
25	23, 1, 24	sylancl 595	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
26		chub2 31711	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ C_ℋ ∧ ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ) → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
27	2, 25, 26	sylancr 596	. . . . . . . . . . 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 3970	. . . . . . . . 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 539	. . . . . . 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 1038	. . . . . . 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 289	. . . . . 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 3106	. . . . 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 32622	. . . . 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 219	. . . 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 32623	. . . 4 ⊢ ((𝐴 +_ℋ 𝐵) = (𝐴 ∨_ℋ 𝐵) ↔ 𝐴 𝑀_ℋ^* 𝐵)
38	36, 37	sylib 220	. . 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 211	. 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 31673	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)
41	40	biantru 537	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)))
42	1, 2	chjcli 31660	. . . . . . . . . . . . 13 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
43		chlub 31712	. . . . . . . . . . . . 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 1474	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
45	41, 44	bitrid 285	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
46		ssid 3958	. . . . . . . . . . . . 13 ⊢ (𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵)
47	46	biantrur 538	. . . . . . . . . . . 12 ⊢ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵) ↔ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵) ∧ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
48		ssin 4190	. . . . . . . . . . . 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 277	. . . . . . . . . . 11 ⊢ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
50	45, 49	bitrdi 289	. . . . . . . . . 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 4188	. . . . . . . . 9 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (𝑥 ∨_ℋ 𝐵)
53	51, 52	jctil 527	. . . . . . . 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 3951	. . . . . . . 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 236	. . . . . . 7 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = (𝑥 ∨_ℋ 𝐵))
56	55	sseq1d 3967	. . . . . 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 701	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
58	57, 1, 24	sylancl 595	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
59	2, 58, 26	sylancr 596	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
60	59	biantrud 539	. . . . . . . 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 31560	. . . . . . . . . 10 ⊢ ((((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
62	58, 2, 61	sylancl 595	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
63		chlub 31712	. . . . . . . . . 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 1470	. . . . . . . . 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 697	. . . . . . . 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 281	. . . . . . 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 284	. . . . 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 813	. . . 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 3106	. 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 277	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 208 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ∀wral 3076 ∩ cin 3903 ⊆ wss 3904 class class class wbr 5100 (class class class)co 7396 C_ℋ cch 31132 +_ℋ cph 31134 ∨_ℋ chj 31136 HAtomscat 31168 𝑀_ℋ^* cdmd 31170

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cc 10392 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152 ax-mulf 11153 ax-hilex 31202 ax-hfvadd 31203 ax-hvcom 31204 ax-hvass 31205 ax-hv0cl 31206 ax-hvaddid 31207 ax-hfvmul 31208 ax-hvmulid 31209 ax-hvmulass 31210 ax-hvdistr1 31211 ax-hvdistr2 31212 ax-hvmul0 31213 ax-hfi 31282 ax-his1 31285 ax-his2 31286 ax-his3 31287 ax-his4 31288 ax-hcompl 31405

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-acn 9900 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-clim 15515 df-rlim 15516 df-sum 15714 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-fbas 21421 df-fg 21422 df-cnfld 21425 df-top 22954 df-topon 22971 df-topsp 22993 df-bases 23006 df-cld 23079 df-ntr 23080 df-cls 23081 df-nei 23158 df-cn 23287 df-cnp 23288 df-lm 23289 df-haus 23375 df-tx 23622 df-hmeo 23815 df-fil 23906 df-fm 23998 df-flim 23999 df-flf 24000 df-xms 24380 df-ms 24381 df-tms 24382 df-cfil 25317 df-cau 25318 df-cmet 25319 df-grpo 30696 df-gid 30697 df-ginv 30698 df-gdiv 30699 df-ablo 30748 df-vc 30762 df-nv 30795 df-va 30798 df-ba 30799 df-sm 30800 df-0v 30801 df-vs 30802 df-nmcv 30803 df-ims 30804 df-dip 30904 df-ssp 30925 df-ph 31016 df-cbn 31066 df-hnorm 31171 df-hba 31172 df-hvsub 31174 df-hlim 31175 df-hcau 31176 df-sh 31410 df-ch 31424 df-oc 31455 df-ch0 31456 df-shs 31511 df-span 31512 df-chj 31513 df-chsup 31514 df-pjh 31598 df-cv 32482 df-md 32483 df-dmd 32484 df-at 32541

This theorem is referenced by: dmdbr6ati 32626 dmdbr7ati 32627

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