mdslmd1lem2 - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > mdslmd1lem2		Structured version Visualization version GIF version

Theorem mdslmd1lem2 32415

Description: Lemma for mdslmd1i 32418. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
mdslmd.1	⊢ 𝐴 ∈ C_ℋ
mdslmd.2	⊢ 𝐵 ∈ C_ℋ
mdslmd.3	⊢ 𝐶 ∈ C_ℋ
mdslmd.4	⊢ 𝐷 ∈ C_ℋ
mdslmd1lem.5	⊢ 𝑅 ∈ C_ℋ

**Assertion**
Ref	Expression
mdslmd1lem2	⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) → (((𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵) → (((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))) → (((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))))

**Proof of Theorem mdslmd1lem2**
Step	Hyp	Ref	Expression
1		ssrin 4170	. . . 4 ⊢ (𝑅 ⊆ 𝐷 → (𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵))
2	1	adantl 482	. . 3 ⊢ (((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷) → (𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵))
3	2	imim1i 63	. 2 ⊢ (((𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵) → (((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))) → (((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷) → (((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))))
4		simpllr 781	. . . . . 6 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐵 𝑀_ℋ^* 𝐴)
5		mdslmd.3	. . . . . . . . . . . 12 ⊢ 𝐶 ∈ C_ℋ
6		mdslmd1lem.5	. . . . . . . . . . . 12 ⊢ 𝑅 ∈ C_ℋ
7	5, 6	chub2i 31559	. . . . . . . . . . 11 ⊢ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)
8		sstr 3923	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
9	7, 8	mpan2 697	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
10	9	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
11	10	ad2antlr 733	. . . . . . . 8 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
12		simplr 774	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ 𝐷)
13	12	ad2antlr 733	. . . . . . . 8 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝐷)
14	11, 13	ssind 4169	. . . . . . 7 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷))
15		ssin 4167	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ↔ 𝐴 ⊆ (𝐶 ∩ 𝐷))
16		mdslmd.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ C_ℋ
17	5, 16	chincli 31549	. . . . . . . . . . . 12 ⊢ (𝐶 ∩ 𝐷) ∈ C_ℋ
18	17, 6	chub2i 31559	. . . . . . . . . . 11 ⊢ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))
19		sstr 3923	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
20	18, 19	mpan2 697	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐶 ∩ 𝐷) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
21	15, 20	sylbi 218	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
22	21	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
23	22	ad2antlr 733	. . . . . . 7 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
24	14, 23	ssind 4169	. . . . . 6 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))))
25		inss2 4166	. . . . . . . . . . 11 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷
26		sstr 3923	. . . . . . . . . . 11 ⊢ ((((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
27	25, 26	mpan 696	. . . . . . . . . 10 ⊢ (𝐷 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
28	27	ad2antll 735	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
29	28	ad2antlr 733	. . . . . . . 8 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
30		sstr 3923	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
31	30	ancoms 459	. . . . . . . . . . . . 13 ⊢ ((𝐷 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝑅 ⊆ 𝐷) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
32	31	ad2ant2l 752	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
33	32	adantll 720	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
34	33	adantll 720	. . . . . . . . . 10 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
35		ssinss1 4174	. . . . . . . . . . . 12 ⊢ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
36	35	ad2antrl 734	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
37	36	ad2antlr 733	. . . . . . . . . 10 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
38	34, 37	jca 516	. . . . . . . . 9 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)))
39		mdslmd.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ C_ℋ
40		mdslmd.2	. . . . . . . . . . 11 ⊢ 𝐵 ∈ C_ℋ
41	39, 40	chjcli 31546	. . . . . . . . . 10 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
42	6, 17, 41	chlubi 31560	. . . . . . . . 9 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
43	38, 42	sylib 219	. . . . . . . 8 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
44	29, 43	jca 516	. . . . . . 7 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵)))
45	6, 5	chjcli 31546	. . . . . . . . 9 ⊢ (𝑅 ∨_ℋ 𝐶) ∈ C_ℋ
46	45, 16	chincli 31549	. . . . . . . 8 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∈ C_ℋ
47	6, 17	chjcli 31546	. . . . . . . 8 ⊢ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∈ C_ℋ
48	46, 47, 41	chlubi 31560	. . . . . . 7 ⊢ ((((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∨_ℋ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))) ⊆ (𝐴 ∨_ℋ 𝐵))
49	44, 48	sylib 219	. . . . . 6 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∨_ℋ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))) ⊆ (𝐴 ∨_ℋ 𝐵))
50	39, 40, 46, 47	mdslle1i 32406	. . . . . 6 ⊢ ((𝐵 𝑀_ℋ^* 𝐴 ∧ 𝐴 ⊆ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))) ∧ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∨_ℋ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))) ⊆ (𝐴 ∨_ℋ 𝐵)) → (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ↔ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ 𝐵) ⊆ ((𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∩ 𝐵)))
51	4, 24, 49, 50	syl3anc 1379	. . . . 5 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ↔ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ 𝐵) ⊆ ((𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∩ 𝐵)))
52		inindir 4164	. . . . . . 7 ⊢ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ 𝐵) = (((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))
53		sstr 3923	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
54	15, 53	sylanb 587	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
55	54	ad2ant2r 753	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝑅)
56		simplll 780	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝐶)
57	55, 56	ssind 4169	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ (𝑅 ∩ 𝐶))
58		simplrl 782	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐶 ⊆ (𝐴 ∨_ℋ 𝐵))
59	33, 58	jca 516	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐶 ⊆ (𝐴 ∨_ℋ 𝐵)))
60	6, 5, 41	chlubi 31560	. . . . . . . . . . . 12 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐶 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ 𝐶) ⊆ (𝐴 ∨_ℋ 𝐵))
61	59, 60	sylib 219	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (𝑅 ∨_ℋ 𝐶) ⊆ (𝐴 ∨_ℋ 𝐵))
62	57, 61	jca 516	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (𝐴 ⊆ (𝑅 ∩ 𝐶) ∧ (𝑅 ∨_ℋ 𝐶) ⊆ (𝐴 ∨_ℋ 𝐵)))
63	39, 40, 6, 5	mdslj1i 32408	. . . . . . . . . 10 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ (𝐴 ⊆ (𝑅 ∩ 𝐶) ∧ (𝑅 ∨_ℋ 𝐶) ⊆ (𝐴 ∨_ℋ 𝐵))) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) = ((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)))
64	62, 63	sylan2 599	. . . . . . . . 9 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷))) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) = ((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)))
65	64	anassrs 468	. . . . . . . 8 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) = ((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)))
66	65	ineq1d 4148	. . . . . . 7 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) = (((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)))
67	52, 66	eqtr2id 2787	. . . . . 6 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) = (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ 𝐵))
68	15	birani 504	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝐶 ∩ 𝐷))
69	54, 68	ssind 4169	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝑅 ∩ (𝐶 ∩ 𝐷)))
70	31	adantll 720	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
71	35	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
72	70, 71	jca 516	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)))
73	72, 42	sylib 219	. . . . . . . . . . 11 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
74	69, 73	anim12i 619	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) ∧ ((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷)) → (𝐴 ⊆ (𝑅 ∩ (𝐶 ∩ 𝐷)) ∧ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵)))
75	74	an4s 666	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (𝐴 ⊆ (𝑅 ∩ (𝐶 ∩ 𝐷)) ∧ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵)))
76	39, 40, 6, 17	mdslj1i 32408	. . . . . . . . 9 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ (𝐴 ⊆ (𝑅 ∩ (𝐶 ∩ 𝐷)) ∧ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))) → ((𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∩ 𝐵) = ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐷) ∩ 𝐵)))
77	75, 76	sylan2 599	. . . . . . . 8 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷))) → ((𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∩ 𝐵) = ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐷) ∩ 𝐵)))
78	77	anassrs 468	. . . . . . 7 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → ((𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∩ 𝐵) = ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐷) ∩ 𝐵)))
79		inindir 4164	. . . . . . . . 9 ⊢ ((𝐶 ∩ 𝐷) ∩ 𝐵) = ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))
80	79	a1i 11	. . . . . . . 8 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → ((𝐶 ∩ 𝐷) ∩ 𝐵) = ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))
81	80	oveq2d 7372	. . . . . . 7 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐷) ∩ 𝐵)) = ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))))
82	78, 81	eqtr2d 2775	. . . . . 6 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))) = ((𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∩ 𝐵))
83	67, 82	sseq12d 3948	. . . . 5 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → ((((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))) ↔ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ 𝐵) ⊆ ((𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∩ 𝐵)))
84	51, 83	bitr4d 283	. . . 4 ⊢ ((((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ↔ (((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))))
85	84	exbiri 816	. . 3 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) → (((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷) → ((((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))))
86	85	a2d 29	. 2 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) → ((((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷) → (((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))) → (((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))))
87	3, 86	syl5 34	1 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)))) → (((𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵) → (((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)) ⊆ ((𝑅 ∩ 𝐵) ∨_ℋ ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)))) → (((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∩ cin 3882 ⊆ wss 3883 class class class wbr 5072 (class class class)co 7356 C_ℋ cch 31018 ∨_ℋ chj 31022 𝑀_ℋ cmd 31055 𝑀_ℋ^* cdmd 31056

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109 ax-hilex 31088 ax-hfvadd 31089 ax-hvcom 31090 ax-hvass 31091 ax-hv0cl 31092 ax-hvaddid 31093 ax-hfvmul 31094 ax-hvmulid 31095 ax-hvmulass 31096 ax-hvdistr1 31097 ax-hvdistr2 31098 ax-hvmul0 31099 ax-hfi 31168 ax-his1 31171 ax-his2 31172 ax-his3 31173 ax-his4 31174 ax-hcompl 31291

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-cn 23210 df-cnp 23211 df-lm 23212 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cfil 25240 df-cau 25241 df-cmet 25242 df-grpo 30582 df-gid 30583 df-ginv 30584 df-gdiv 30585 df-ablo 30634 df-vc 30648 df-nv 30681 df-va 30684 df-ba 30685 df-sm 30686 df-0v 30687 df-vs 30688 df-nmcv 30689 df-ims 30690 df-dip 30790 df-ssp 30811 df-ph 30902 df-cbn 30952 df-hnorm 31057 df-hba 31058 df-hvsub 31060 df-hlim 31061 df-hcau 31062 df-sh 31296 df-ch 31310 df-oc 31341 df-ch0 31342 df-shs 31397 df-chj 31399 df-md 32369 df-dmd 32370

This theorem is referenced by: mdslmd1lem4 32417

W3C validator