mdslmd1lem2 - Hilbert Space Explorer

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Theorem mdslmd1lem2 30688

Description: Lemma for mdslmd1i 30691. (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 4167	. . . 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 773	. . . . . 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 29832	. . . . . . . . . . 11 ⊢ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)
8		sstr 3929	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
9	7, 8	mpan2 688	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
10	9	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
11	10	ad2antlr 724	. . . . . . . 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 766	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ 𝐷)
13	12	ad2antlr 724	. . . . . . . 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 4166	. . . . . . 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 4164	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ↔ 𝐴 ⊆ (𝐶 ∩ 𝐷))
16		mdslmd.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ C_ℋ
17	5, 16	chincli 29822	. . . . . . . . . . . 12 ⊢ (𝐶 ∩ 𝐷) ∈ C_ℋ
18	17, 6	chub2i 29832	. . . . . . . . . . 11 ⊢ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))
19		sstr 3929	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
20	18, 19	mpan2 688	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐶 ∩ 𝐷) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
21	15, 20	sylbi 216	. . . . . . . . 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 724	. . . . . . 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 4166	. . . . . 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 4163	. . . . . . . . . . 11 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷
26		sstr 3929	. . . . . . . . . . 11 ⊢ ((((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
27	25, 26	mpan 687	. . . . . . . . . 10 ⊢ (𝐷 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
28	27	ad2antll 726	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
29	28	ad2antlr 724	. . . . . . . 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 3929	. . . . . . . . . . . . . 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 743	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
33	32	adantll 711	. . . . . . . . . . 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 711	. . . . . . . . . 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 4171	. . . . . . . . . . . 12 ⊢ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
36	35	ad2antrl 725	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
37	36	ad2antlr 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)) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
38	34, 37	jca 512	. . . . . . . . 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 29819	. . . . . . . . . 10 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
42	6, 17, 41	chlubi 29833	. . . . . . . . 9 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
43	38, 42	sylib 217	. . . . . . . 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 512	. . . . . . 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 29819	. . . . . . . . 9 ⊢ (𝑅 ∨_ℋ 𝐶) ∈ C_ℋ
46	45, 16	chincli 29822	. . . . . . . 8 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∈ C_ℋ
47	6, 17	chjcli 29819	. . . . . . . 8 ⊢ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∈ C_ℋ
48	46, 47, 41	chlubi 29833	. . . . . . 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 217	. . . . . 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 30679	. . . . . 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 1370	. . . . 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 4161	. . . . . . 7 ⊢ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ 𝐵) = (((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))
53		sstr 3929	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
54	15, 53	sylanb 581	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
55	54	ad2ant2r 744	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝑅)
56		simplll 772	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝐶)
57	55, 56	ssind 4166	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ (𝑅 ∩ 𝐶))
58		simplrl 774	. . . . . . . . . . . . 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 512	. . . . . . . . . . . 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 29833	. . . . . . . . . . . 12 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐶 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ 𝐶) ⊆ (𝐴 ∨_ℋ 𝐵))
61	59, 60	sylib 217	. . . . . . . . . . 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 512	. . . . . . . . . 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 30681	. . . . . . . . . 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 593	. . . . . . . . 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 4145	. . . . . . 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 2791	. . . . . 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	biimpi 215	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) → 𝐴 ⊆ (𝐶 ∩ 𝐷))
69	68	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝐶 ∩ 𝐷))
70	54, 69	ssind 4166	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝑅 ∩ (𝐶 ∩ 𝐷)))
71	31	adantll 711	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
72	35	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
73	71, 72	jca 512	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)))
74	73, 42	sylib 217	. . . . . . . . . . 11 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
75	70, 74	anim12i 613	. . . . . . . . . 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)))
76	75	an4s 657	. . . . . . . . 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)))
77	39, 40, 6, 17	mdslj1i 30681	. . . . . . . . 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)))
78	76, 77	sylan2 593	. . . . . . . 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)))
79	78	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)))
80		inindir 4161	. . . . . . . . 9 ⊢ ((𝐶 ∩ 𝐷) ∩ 𝐵) = ((𝐶 ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))
81	80	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)))
82	81	oveq2d 7291	. . . . . . 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))))
83	79, 82	eqtr2d 2779	. . . . . 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))
84	67, 83	sseq12d 3954	. . . . 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)))
85	51, 84	bitr4d 281	. . . 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)))))
86	85	exbiri 808	. . 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)))))
87	86	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)))))
88	3, 87	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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 (class class class)co 7275 C_ℋ cch 29291 ∨_ℋ chj 29295 𝑀_ℋ cmd 29328 𝑀_ℋ^* cdmd 29329

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 ax-hilex 29361 ax-hfvadd 29362 ax-hvcom 29363 ax-hvass 29364 ax-hv0cl 29365 ax-hvaddid 29366 ax-hfvmul 29367 ax-hvmulid 29368 ax-hvmulass 29369 ax-hvdistr1 29370 ax-hvdistr2 29371 ax-hvmul0 29372 ax-hfi 29441 ax-his1 29444 ax-his2 29445 ax-his3 29446 ax-his4 29447 ax-hcompl 29564

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-cn 22378 df-cnp 22379 df-lm 22380 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cfil 24419 df-cau 24420 df-cmet 24421 df-grpo 28855 df-gid 28856 df-ginv 28857 df-gdiv 28858 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-vs 28961 df-nmcv 28962 df-ims 28963 df-dip 29063 df-ssp 29084 df-ph 29175 df-cbn 29225 df-hnorm 29330 df-hba 29331 df-hvsub 29333 df-hlim 29334 df-hcau 29335 df-sh 29569 df-ch 29583 df-oc 29614 df-ch0 29615 df-shs 29670 df-chj 29672 df-md 30642 df-dmd 30643

This theorem is referenced by: mdslmd1lem4 30690

W3C validator