atcvat4i - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > atcvat4i		Structured version Visualization version GIF version

Theorem atcvat4i 30660

Description: A condition implying existence of an atom with the properties shown. Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
atcvat3.1	⊢ 𝐴 ∈ C_ℋ

**Assertion**
Ref	Expression
atcvat4i	⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶

**Proof of Theorem atcvat4i**
Step	Hyp	Ref	Expression
1		atcvat3.1	. . . . . . . . 9 ⊢ 𝐴 ∈ C_ℋ
2	1	hatomici 30622	. . . . . . . 8 ⊢ (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴)
3		atelch 30607	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ HAtoms → 𝐶 ∈ C_ℋ )
4		atelch 30607	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
5		chub1 29770	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥))
6	3, 4, 5	syl2an 595	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ HAtoms ∧ 𝑥 ∈ HAtoms) → 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥))
7		sseq1 3942	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝐶 → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥)))
8	6, 7	syl5ibr 245	. . . . . . . . . . . . 13 ⊢ (𝐵 = 𝐶 → ((𝐶 ∈ HAtoms ∧ 𝑥 ∈ HAtoms) → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
9	8	expd 415	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐶 → (𝐶 ∈ HAtoms → (𝑥 ∈ HAtoms → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
10	9	impcom 407	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (𝑥 ∈ HAtoms → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
11	10	anim2d 611	. . . . . . . . . 10 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ∈ HAtoms) → (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
12	11	expcomd 416	. . . . . . . . 9 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (𝑥 ∈ HAtoms → (𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
13	12	reximdvai 3199	. . . . . . . 8 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
14	2, 13	syl5 34	. . . . . . 7 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
15	14	ex 412	. . . . . 6 ⊢ (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
16	15	a1i 11	. . . . 5 ⊢ (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) → (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))))
17	16	com4l 92	. . . 4 ⊢ (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → (𝐴 ≠ 0_ℋ → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))))
18	17	imp4a 422	. . 3 ⊢ (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
19	18	adantl 481	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐵 = 𝐶 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
20		atelch 30607	. . . . . . . 8 ⊢ (𝐵 ∈ HAtoms → 𝐵 ∈ C_ℋ )
21		chlejb2 29776	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝐶 ⊆ 𝐴 ↔ (𝐴 ∨_ℋ 𝐶) = 𝐴))
22	1, 21	mpan2 687	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ C_ℋ → (𝐶 ⊆ 𝐴 ↔ (𝐴 ∨_ℋ 𝐶) = 𝐴))
23	22	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∨_ℋ 𝐶) = 𝐴)
24	23	sseq2d 3949	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) ↔ 𝐵 ⊆ 𝐴))
25	24	biimpa 476	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴)
26	25	expl 457	. . . . . . . . . 10 ⊢ (𝐶 ∈ C_ℋ → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴))
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴))
28		chub2 29771	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))
29	27, 28	jctird 526	. . . . . . . 8 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
30	20, 3, 29	syl2an 595	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
31		simpl 482	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → 𝐵 ∈ HAtoms)
32	30, 31	jctild 525	. . . . . 6 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵)))))
33	32	impl 455	. . . . 5 ⊢ ((((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
34		sseq1 3942	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
35		oveq2 7263	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ 𝐵))
36	35	sseq2d 3949	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵)))
37	34, 36	anbi12d 630	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
38	37	rspcev 3552	. . . . 5 ⊢ ((𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
39	33, 38	syl 17	. . . 4 ⊢ ((((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
40	39	adantrl 712	. . 3 ⊢ ((((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
41	40	exp31 419	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐶 ⊆ 𝐴 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
42		simpr 484	. . 3 ⊢ ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))
43		ioran 980	. . . 4 ⊢ (¬ (𝐵 = 𝐶 ∨ 𝐶 ⊆ 𝐴) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴))
44	1	atcvat3i 30659	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms))
45	3	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → 𝐶 ∈ C_ℋ )
46	44	imp 406	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms)
47		simpll 763	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → 𝐵 ∈ HAtoms)
48	45, 46, 47	3jca 1126	. . . . . . . . . 10 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐶 ∈ C_ℋ ∧ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms ∧ 𝐵 ∈ HAtoms))
49		inss2 4160	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐵 ∨_ℋ 𝐶)
50		chjcom 29769	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝐶) = (𝐶 ∨_ℋ 𝐵))
51	20, 3, 50	syl2an 595	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐵 ∨_ℋ 𝐶) = (𝐶 ∨_ℋ 𝐵))
52	49, 51	sseqtrid 3969	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵))
53	52	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵))
54		atnssm0 30639	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
55	1, 54	mpan 686	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ HAtoms → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
56	55	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
57		inss1 4159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴
58		sslin 4165	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐶 ∩ 𝐴))
59	57, 58	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐶 ∩ 𝐴)
60		incom 4131	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
61	59, 60	sseqtri 3953	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐴 ∩ 𝐶)
62		sseq2 3943	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝐶) = 0_ℋ → ((𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐴 ∩ 𝐶) ↔ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ))
63	61, 62	mpbii 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝐶) = 0_ℋ → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ)
64		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → 𝐶 ∈ C_ℋ )
65		chjcl 29620	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝐶) ∈ C_ℋ )
66		chincl 29762	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ C_ℋ ∧ (𝐵 ∨_ℋ 𝐶) ∈ C_ℋ ) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ )
67	1, 65, 66	sylancr 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ )
68		chincl 29762	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ C_ℋ ∧ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ ) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
69	64, 67, 68	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
70	20, 3, 69	syl2an 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
71		chle0 29706	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ → ((𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ ↔ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
72	70, 71	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ ↔ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
73	63, 72	syl5ib 243	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ∩ 𝐶) = 0_ℋ → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
74	56, 73	sylbid 239	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
75	74	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ¬ 𝐶 ⊆ 𝐴) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
76	75	adantrl 712	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴)) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
77	76	adantrr 713	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
78	53, 77	jca 511	. . . . . . . . . 10 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
79		atexch 30644	. . . . . . . . . 10 ⊢ ((𝐶 ∈ C_ℋ ∧ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ) → 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)))))
80	48, 78, 79	sylc 65	. . . . . . . . 9 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))
81	80, 57	jctil 519	. . . . . . . 8 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)))))
82	81	ex 412	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))))
83	44, 82	jcad 512	. . . . . 6 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms ∧ ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)))))))
84		sseq1 3942	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝑥 ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴))
85		oveq2 7263	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))
86	85	sseq2d 3949	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)))))
87	84, 86	anbi12d 630	. . . . . . 7 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → ((𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)) ↔ ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))))
88	87	rspcev 3552	. . . . . 6 ⊢ (((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms ∧ ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
89	83, 88	syl6 35	. . . . 5 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
90	89	expd 415	. . . 4 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
91	43, 90	syl5bi 241	. . 3 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ (𝐵 = 𝐶 ∨ 𝐶 ⊆ 𝐴) → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
92	42, 91	syl7 74	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ (𝐵 = 𝐶 ∨ 𝐶 ⊆ 𝐴) → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
93	19, 41, 92	ecase3d 1030	1 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 ∩ cin 3882 ⊆ wss 3883 (class class class)co 7255 C_ℋ cch 29192 ∨_ℋ chj 29196 0_ℋc0h 29198 HAtomscat 29228

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cc 10122 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882 ax-hilex 29262 ax-hfvadd 29263 ax-hvcom 29264 ax-hvass 29265 ax-hv0cl 29266 ax-hvaddid 29267 ax-hfvmul 29268 ax-hvmulid 29269 ax-hvmulass 29270 ax-hvdistr1 29271 ax-hvdistr2 29272 ax-hvmul0 29273 ax-hfi 29342 ax-his1 29345 ax-his2 29346 ax-his3 29347 ax-his4 29348 ax-hcompl 29465

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-cn 22286 df-cnp 22287 df-lm 22288 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cfil 24324 df-cau 24325 df-cmet 24326 df-grpo 28756 df-gid 28757 df-ginv 28758 df-gdiv 28759 df-ablo 28808 df-vc 28822 df-nv 28855 df-va 28858 df-ba 28859 df-sm 28860 df-0v 28861 df-vs 28862 df-nmcv 28863 df-ims 28864 df-dip 28964 df-ssp 28985 df-ph 29076 df-cbn 29126 df-hnorm 29231 df-hba 29232 df-hvsub 29234 df-hlim 29235 df-hcau 29236 df-sh 29470 df-ch 29484 df-oc 29515 df-ch0 29516 df-shs 29571 df-span 29572 df-chj 29573 df-chsup 29574 df-pjh 29658 df-cv 30542 df-at 30601

This theorem is referenced by: mdsymlem3 30668

W3C validator