mdslmd1lem2 - Hilbert Space Explorer

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Theorem mdslmd1lem2 32974

Description: Lemma for mdslmd1i 32977. (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 4141	. . . 4 ⊢ (𝑅 ⊆ 𝐷 → (𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵))
2	1	adantl 487	. . 3 ⊢ (((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷) → (𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵))
3	2	imim1i 64	. 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 788	. . . . . 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 32118	. . . . . . . . . . 11 ⊢ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)
8		sstr 3921	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
9	7, 8	mpan2 704	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
10	9	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
11	10	ad2antlr 740	. . . . . . . 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 781	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ 𝐷)
13	12	ad2antlr 740	. . . . . . . 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 4140	. . . . . . 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 4138	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ↔ 𝐴 ⊆ (𝐶 ∩ 𝐷))
16		mdslmd.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ C_ℋ
17	5, 16	chincli 32108	. . . . . . . . . . . 12 ⊢ (𝐶 ∩ 𝐷) ∈ C_ℋ
18	17, 6	chub2i 32118	. . . . . . . . . . 11 ⊢ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))
19		sstr 3921	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
20	18, 19	mpan2 704	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐶 ∩ 𝐷) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
21	15, 20	sylbi 220	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
22	21	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
23	22	ad2antlr 740	. . . . . . 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 4140	. . . . . 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 4137	. . . . . . . . . . 11 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷
26		sstr 3921	. . . . . . . . . . 11 ⊢ ((((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
27	25, 26	mpan 703	. . . . . . . . . 10 ⊢ (𝐷 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
28	27	ad2antll 742	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
29	28	ad2antlr 740	. . . . . . . 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 3921	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
31	30	ancoms 464	. . . . . . . . . . . . 13 ⊢ ((𝐷 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝑅 ⊆ 𝐷) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
32	31	ad2ant2l 759	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
33	32	adantll 727	. . . . . . . . . . 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 727	. . . . . . . . . 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 4145	. . . . . . . . . . . 12 ⊢ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
36	35	ad2antrl 741	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
37	36	ad2antlr 740	. . . . . . . . . 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 521	. . . . . . . . 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 32105	. . . . . . . . . 10 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
42	6, 17, 41	chlubi 32119	. . . . . . . . 9 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
43	38, 42	sylib 221	. . . . . . . 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 521	. . . . . . 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 32105	. . . . . . . . 9 ⊢ (𝑅 ∨_ℋ 𝐶) ∈ C_ℋ
46	45, 16	chincli 32108	. . . . . . . 8 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∈ C_ℋ
47	6, 17	chjcli 32105	. . . . . . . 8 ⊢ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∈ C_ℋ
48	46, 47, 41	chlubi 32119	. . . . . . 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 221	. . . . . 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 32965	. . . . . 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 1399	. . . . 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 4135	. . . . . . 7 ⊢ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ 𝐵) = (((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))
53		sstr 3921	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
54	15, 53	sylanb 593	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
55	54	ad2ant2r 760	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝑅)
56		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝐶)
57	55, 56	ssind 4140	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ (𝑅 ∩ 𝐶))
58		simplrl 789	. . . . . . . . . . . . 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 521	. . . . . . . . . . . 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 32119	. . . . . . . . . . . 12 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐶 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ 𝐶) ⊆ (𝐴 ∨_ℋ 𝐵))
61	59, 60	sylib 221	. . . . . . . . . . 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 521	. . . . . . . . . 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 32967	. . . . . . . . . 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 605	. . . . . . . . 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 473	. . . . . . . 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 4119	. . . . . . 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 2811	. . . . . 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 509	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝐶 ∩ 𝐷))
69	54, 68	ssind 4140	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝑅 ∩ (𝐶 ∩ 𝐷)))
70	31	adantll 727	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
71	35	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
72	70, 71	jca 521	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)))
73	72, 42	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
74	69, 73	anim12i 625	. . . . . . . . . 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 673	. . . . . . . . 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 32967	. . . . . . . . 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 605	. . . . . . . 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 473	. . . . . . 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 4135	. . . . . . . . 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 7465	. . . . . . 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 2799	. . . . . 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 3946	. . . . 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 285	. . . 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 823	. . 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 30	. 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 35	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 ∩ cin 4092 class class class wbr 5106 (class class class)co 7449 C_ℋ cch 31577 ∨_ℋ chj 31581 𝑀_ℋ cmd 31614 𝑀_ℋ^* cdmd 31615

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-inf2 9677 ax-cc 10550 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314 ax-pre-sup 11315 ax-addf 11316 ax-mulf 11317 ax-hilex 31647 ax-hfvadd 31648 ax-hvcom 31649 ax-hvass 31650 ax-hv0cl 31651 ax-hvaddid 31652 ax-hfvmul 31653 ax-hvmulid 31654 ax-hvmulass 31655 ax-hvdistr1 31656 ax-hvdistr2 31657 ax-hvmul0 31658 ax-hfi 31727 ax-his1 31730 ax-his2 31731 ax-his3 31732 ax-his4 31733 ax-hcompl 31850

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-isom 6573 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-of 7723 df-om 7910 df-1st 8033 df-2nd 8034 df-supp 8213 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-1o 8512 df-2o 8513 df-oadd 8516 df-omul 8517 df-er 8753 df-map 8885 df-pm 8886 df-ixp 8962 df-en 9010 df-dom 9011 df-sdom 9012 df-fin 9013 df-fsupp 9390 df-fi 9438 df-sup 9469 df-inf 9470 df-oi 9539 df-card 10055 df-acn 10058 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-div 12012 df-nn 12374 df-2 12443 df-3 12444 df-4 12445 df-5 12446 df-6 12447 df-7 12448 df-8 12449 df-9 12450 df-n0 12645 df-z 12732 df-dec 12853 df-uz 13004 df-q 13114 df-rp 13159 df-xneg 13279 df-xadd 13280 df-xmul 13281 df-ioo 13518 df-ico 13520 df-icc 13521 df-fz 13678 df-fzo 13827 df-fl 13970 df-seq 14183 df-exp 14243 df-hash 14514 df-cj 15307 df-re 15308 df-im 15309 df-sqrt 15443 df-abs 15444 df-clim 15696 df-rlim 15697 df-sum 15895 df-struct 17366 df-sets 17383 df-slot 17401 df-ndx 17413 df-base 17429 df-ress 17450 df-plusg 17482 df-mulr 17483 df-starv 17484 df-sca 17485 df-vsca 17486 df-ip 17487 df-tset 17488 df-ple 17489 df-ds 17491 df-unif 17492 df-hom 17493 df-cco 17494 df-rest 17634 df-topn 17635 df-0g 17653 df-gsum 17654 df-topgen 17655 df-pt 17656 df-prds 17659 df-xrs 17715 df-qtop 17720 df-imas 17721 df-xps 17723 df-mre 17797 df-mrc 17798 df-acs 17800 df-mgm 18857 df-sgrp 18949 df-mnd 18965 df-submnd 19020 df-mulg 19319 df-cntz 19573 df-cmn 20038 df-psmet 21712 df-xmet 21713 df-met 21714 df-bl 21715 df-mopn 21716 df-fbas 21717 df-fg 21718 df-cnfld 21721 df-top 23256 df-topon 23273 df-topsp 23295 df-bases 23308 df-cld 23381 df-ntr 23382 df-cls 23383 df-nei 23460 df-cn 23589 df-cnp 23590 df-lm 23591 df-haus 23677 df-tx 23925 df-hmeo 24118 df-fil 24209 df-fm 24301 df-flim 24302 df-flf 24303 df-xms 24683 df-ms 24684 df-tms 24685 df-cfil 25620 df-cau 25621 df-cmet 25622 df-grpo 31141 df-gid 31142 df-ginv 31143 df-gdiv 31144 df-ablo 31193 df-vc 31207 df-nv 31240 df-va 31243 df-ba 31244 df-sm 31245 df-0v 31246 df-vs 31247 df-nmcv 31248 df-ims 31249 df-dip 31349 df-ssp 31370 df-ph 31461 df-cbn 31511 df-hnorm 31616 df-hba 31617 df-hvsub 31619 df-hlim 31620 df-hcau 31621 df-sh 31855 df-ch 31869 df-oc 31900 df-ch0 31901 df-shs 31956 df-chj 31958 df-md 32928 df-dmd 32929

This theorem is used by: mdslmd1lem4 32976

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