dmdbr5ati - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > dmdbr5ati		Structured version Visualization version GIF version

Theorem dmdbr5ati 30193

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 30078	. . . . . . 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 1447	. . . . . 6 ⊢ (𝑥 ∈ C_ℋ → (𝐴 𝑀_ℋ^* 𝐵 → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
5		atelch 30115	. . . . . 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 3181	. . 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 29277	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
10	2, 5, 9	sylancr 589	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ HAtoms → (𝐵 ∨_ℋ 𝑥) = (𝑥 ∨_ℋ 𝐵))
11	10	ineq1d 4188	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴)))
12	1, 2	chjcomi 29239	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
13	12	ineq2i 4186	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐵 ∨_ℋ 𝐴))
14	11, 13	syl6eqr 2874	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ HAtoms → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
15	14	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
16	12	sseq2i 3996	. . . . . . . . . . . . 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 30173	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵))
19	18	imp 409	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐵 ∨_ℋ 𝐴)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵)
20	17, 19	sylan2b 595	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝐵 ∨_ℋ 𝑥) ∩ (𝐵 ∨_ℋ 𝐴)) = 𝐵)
21	15, 20	eqtr3d 2858	. . . . . . . . . 10 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) = 𝐵)
22		chjcl 29128	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
23	5, 2, 22	sylancl 588	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ HAtoms → (𝑥 ∨_ℋ 𝐵) ∈ C_ℋ )
24		chincl 29270	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∨_ℋ 𝐵) ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
25	23, 1, 24	sylancl 588	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
26		chub2 29279	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ C_ℋ ∧ ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ) → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
27	2, 25, 26	sylancr 589	. . . . . . . . . . 11 ⊢ (𝑥 ∈ HAtoms → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
28	27	adantr 483	. . . . . . . . . 10 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
29	21, 28	eqsstrd 4005	. . . . . . . . 9 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
30	29	ex 415	. . . . . . . 8 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
31	30	biantrud 534	. . . . . . 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 1021	. . . . . . 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	syl6bb 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 3164	. . . . 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 30190	. . . . 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 30191	. . . 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 29241	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)
41	40	biantru 532	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)))
42	1, 2	chjcli 29228	. . . . . . . . . . . . 13 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
43		chlub 29280	. . . . . . . . . . . . 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 1449	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
45	41, 44	syl5bb 285	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
46		ssid 3989	. . . . . . . . . . . . 13 ⊢ (𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵)
47	46	biantrur 533	. . . . . . . . . . . 12 ⊢ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵) ↔ ((𝑥 ∨_ℋ 𝐵) ⊆ (𝑥 ∨_ℋ 𝐵) ∧ (𝑥 ∨_ℋ 𝐵) ⊆ (𝐴 ∨_ℋ 𝐵)))
48		ssin 4207	. . . . . . . . . . . 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	syl6bb 289	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵))))
51	50	biimpa 479	. . . . . . . . 9 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (𝑥 ∨_ℋ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)))
52		inss1 4205	. . . . . . . . 9 ⊢ ((𝑥 ∨_ℋ 𝐵) ∩ (𝐴 ∨_ℋ 𝐵)) ⊆ (𝑥 ∨_ℋ 𝐵)
53	51, 52	jctil 522	. . . . . . . 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 3982	. . . . . . . 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 3998	. . . . . 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 588	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ )
59	2, 58, 26	sylancr 589	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → 𝐵 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵))
60	59	biantrud 534	. . . . . . . 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 29128	. . . . . . . . . 10 ⊢ ((((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
62	58, 2, 61	sylancl 588	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ∈ C_ℋ )
63		chlub 29280	. . . . . . . . . 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 1445	. . . . . . . . 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 281	. . . . . . 7 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵) ↔ (𝑥 ∨_ℋ 𝐵) ⊆ (((𝑥 ∨_ℋ 𝐵) ∩ 𝐴) ∨_ℋ 𝐵)))
67	66	adantr 483	. . . . . 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 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 3164	. 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 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5059 (class class class)co 7150 C_ℋ cch 28700 +_ℋ cph 28702 ∨_ℋ chj 28704 HAtomscat 28736 𝑀_ℋ^* cdmd 28738

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cc 9851 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvmulass 28778 ax-hvdistr1 28779 ax-hvdistr2 28780 ax-hvmul0 28781 ax-hfi 28850 ax-his1 28853 ax-his2 28854 ax-his3 28855 ax-his4 28856 ax-hcompl 28973

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-cn 21829 df-cnp 21830 df-lm 21831 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cfil 23852 df-cau 23853 df-cmet 23854 df-grpo 28264 df-gid 28265 df-ginv 28266 df-gdiv 28267 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-vs 28370 df-nmcv 28371 df-ims 28372 df-dip 28472 df-ssp 28493 df-ph 28584 df-cbn 28634 df-hnorm 28739 df-hba 28740 df-hvsub 28742 df-hlim 28743 df-hcau 28744 df-sh 28978 df-ch 28992 df-oc 29023 df-ch0 29024 df-shs 29079 df-span 29080 df-chj 29081 df-chsup 29082 df-pjh 29166 df-cv 30050 df-md 30051 df-dmd 30052 df-at 30109

This theorem is referenced by: dmdbr6ati 30194 dmdbr7ati 30195

W3C validator