mdslmd1lem2 - Hilbert Space Explorer

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Theorem mdslmd1lem2 32486

Description: Lemma for mdslmd1i 32489. (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 4191	. . . 4 ⊢ (𝑅 ⊆ 𝐷 → (𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵))
2	1	adantl 485	. . 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 785	. . . . . 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 31630	. . . . . . . . . . 11 ⊢ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)
8		sstr 3942	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
9	7, 8	mpan2 701	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
10	9	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
11	10	ad2antlr 737	. . . . . . . 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 778	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ 𝐷)
13	12	ad2antlr 737	. . . . . . . 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 4190	. . . . . . 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 4188	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ↔ 𝐴 ⊆ (𝐶 ∩ 𝐷))
16		mdslmd.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ C_ℋ
17	5, 16	chincli 31620	. . . . . . . . . . . 12 ⊢ (𝐶 ∩ 𝐷) ∈ C_ℋ
18	17, 6	chub2i 31630	. . . . . . . . . . 11 ⊢ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))
19		sstr 3942	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
20	18, 19	mpan2 701	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐶 ∩ 𝐷) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
21	15, 20	sylbi 219	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
22	21	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
23	22	ad2antlr 737	. . . . . . 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 4190	. . . . . 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 4187	. . . . . . . . . . 11 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷
26		sstr 3942	. . . . . . . . . . 11 ⊢ ((((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
27	25, 26	mpan 700	. . . . . . . . . 10 ⊢ (𝐷 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
28	27	ad2antll 739	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
29	28	ad2antlr 737	. . . . . . . 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 3942	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
31	30	ancoms 462	. . . . . . . . . . . . 13 ⊢ ((𝐷 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝑅 ⊆ 𝐷) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
32	31	ad2ant2l 756	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
33	32	adantll 724	. . . . . . . . . . 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 724	. . . . . . . . . 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 4195	. . . . . . . . . . . 12 ⊢ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
36	35	ad2antrl 738	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
37	36	ad2antlr 737	. . . . . . . . . 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 519	. . . . . . . . 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 31617	. . . . . . . . . 10 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
42	6, 17, 41	chlubi 31631	. . . . . . . . 9 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
43	38, 42	sylib 220	. . . . . . . 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 519	. . . . . . 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 31617	. . . . . . . . 9 ⊢ (𝑅 ∨_ℋ 𝐶) ∈ C_ℋ
46	45, 16	chincli 31620	. . . . . . . 8 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∈ C_ℋ
47	6, 17	chjcli 31617	. . . . . . . 8 ⊢ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∈ C_ℋ
48	46, 47, 41	chlubi 31631	. . . . . . 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 220	. . . . . 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 32477	. . . . . 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 1389	. . . . 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 4185	. . . . . . 7 ⊢ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ 𝐵) = (((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))
53		sstr 3942	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
54	15, 53	sylanb 590	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
55	54	ad2ant2r 757	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝑅)
56		simplll 784	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝐶)
57	55, 56	ssind 4190	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ (𝑅 ∩ 𝐶))
58		simplrl 786	. . . . . . . . . . . . 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 519	. . . . . . . . . . . 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 31631	. . . . . . . . . . . 12 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐶 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ 𝐶) ⊆ (𝐴 ∨_ℋ 𝐵))
61	59, 60	sylib 220	. . . . . . . . . . 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 519	. . . . . . . . . 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 32479	. . . . . . . . . 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 602	. . . . . . . . 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 471	. . . . . . . 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 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)) → (((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) ∩ (𝐷 ∩ 𝐵)) = (((𝑅 ∩ 𝐵) ∨_ℋ (𝐶 ∩ 𝐵)) ∩ (𝐷 ∩ 𝐵)))
67	52, 66	eqtr2id 2809	. . . . . 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 507	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝐶 ∩ 𝐷))
69	54, 68	ssind 4190	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝑅 ∩ (𝐶 ∩ 𝐷)))
70	31	adantll 724	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
71	35	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
72	70, 71	jca 519	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)))
73	72, 42	sylib 220	. . . . . . . . . . 11 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
74	69, 73	anim12i 622	. . . . . . . . . 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 670	. . . . . . . . 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 32479	. . . . . . . . 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 602	. . . . . . . 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 471	. . . . . . 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 4185	. . . . . . . . 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 7407	. . . . . . 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 2797	. . . . . 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 3967	. . . . 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 284	. . . 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 820	. . 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 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∩ cin 3901 ⊆ wss 3902 class class class wbr 5097 (class class class)co 7391 C_ℋ cch 31089 ∨_ℋ chj 31093 𝑀_ℋ cmd 31126 𝑀_ℋ^* cdmd 31127

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-inf2 9590 ax-cc 10386 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 ax-addf 11146 ax-mulf 11147 ax-hilex 31159 ax-hfvadd 31160 ax-hvcom 31161 ax-hvass 31162 ax-hv0cl 31163 ax-hvaddid 31164 ax-hfvmul 31165 ax-hvmulid 31166 ax-hvmulass 31167 ax-hvdistr1 31168 ax-hvdistr2 31169 ax-hvmul0 31170 ax-hfi 31239 ax-his1 31242 ax-his2 31243 ax-his3 31244 ax-his4 31245 ax-hcompl 31362

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-oadd 8435 df-omul 8436 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-fi 9351 df-sup 9382 df-inf 9383 df-oi 9452 df-card 9891 df-acn 9894 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-ioo 13347 df-ico 13349 df-icc 13350 df-fz 13507 df-fzo 13654 df-fl 13796 df-seq 14009 df-exp 14069 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-clim 15506 df-rlim 15507 df-sum 15705 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-rest 17442 df-topn 17443 df-0g 17461 df-gsum 17462 df-topgen 17463 df-pt 17464 df-prds 17467 df-xrs 17523 df-qtop 17528 df-imas 17529 df-xps 17531 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19101 df-cntz 19348 df-cmn 19813 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-fbas 21409 df-fg 21410 df-cnfld 21413 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-cld 23067 df-ntr 23068 df-cls 23069 df-nei 23146 df-cn 23275 df-cnp 23276 df-lm 23277 df-haus 23363 df-tx 23610 df-hmeo 23803 df-fil 23894 df-fm 23986 df-flim 23987 df-flf 23988 df-xms 24368 df-ms 24369 df-tms 24370 df-cfil 25305 df-cau 25306 df-cmet 25307 df-grpo 30653 df-gid 30654 df-ginv 30655 df-gdiv 30656 df-ablo 30705 df-vc 30719 df-nv 30752 df-va 30755 df-ba 30756 df-sm 30757 df-0v 30758 df-vs 30759 df-nmcv 30760 df-ims 30761 df-dip 30861 df-ssp 30882 df-ph 30973 df-cbn 31023 df-hnorm 31128 df-hba 31129 df-hvsub 31131 df-hlim 31132 df-hcau 31133 df-sh 31367 df-ch 31381 df-oc 31412 df-ch0 31413 df-shs 31468 df-chj 31470 df-md 32440 df-dmd 32441

This theorem is referenced by: mdslmd1lem4 32488

W3C validator