mdslmd1lem2 - Hilbert Space Explorer

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Theorem mdslmd1lem2 32412

Description: Lemma for mdslmd1i 32415. (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 4183	. . . 4 ⊢ (𝑅 ⊆ 𝐷 → (𝑅 ∩ 𝐵) ⊆ (𝐷 ∩ 𝐵))
2	1	adantl 481	. . 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 776	. . . . . 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 31556	. . . . . . . . . . 11 ⊢ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)
8		sstr 3931	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ (𝑅 ∨_ℋ 𝐶)) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
9	7, 8	mpan2 692	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
10	9	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ (𝑅 ∨_ℋ 𝐶))
11	10	ad2antlr 728	. . . . . . . 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 769	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ 𝐷)
13	12	ad2antlr 728	. . . . . . . 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 4182	. . . . . . 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 4180	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ↔ 𝐴 ⊆ (𝐶 ∩ 𝐷))
16		mdslmd.4	. . . . . . . . . . . . 13 ⊢ 𝐷 ∈ C_ℋ
17	5, 16	chincli 31546	. . . . . . . . . . . 12 ⊢ (𝐶 ∩ 𝐷) ∈ C_ℋ
18	17, 6	chub2i 31556	. . . . . . . . . . 11 ⊢ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))
19		sstr 3931	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷))) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
20	18, 19	mpan2 692	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐶 ∩ 𝐷) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
21	15, 20	sylbi 217	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
22	21	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → 𝐴 ⊆ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)))
23	22	ad2antlr 728	. . . . . . 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 4182	. . . . . 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 4179	. . . . . . . . . . 11 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷
26		sstr 3931	. . . . . . . . . . 11 ⊢ ((((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
27	25, 26	mpan 691	. . . . . . . . . 10 ⊢ (𝐷 ⊆ (𝐴 ∨_ℋ 𝐵) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
28	27	ad2antll 730	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
29	28	ad2antlr 728	. . . . . . . 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 3931	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
31	30	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝐷 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝑅 ⊆ 𝐷) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
32	31	ad2ant2l 747	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
33	32	adantll 715	. . . . . . . . . . 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 715	. . . . . . . . . 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 4187	. . . . . . . . . . . 12 ⊢ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
36	35	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
37	36	ad2antlr 728	. . . . . . . . . 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 511	. . . . . . . . 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 31543	. . . . . . . . . 10 ⊢ (𝐴 ∨_ℋ 𝐵) ∈ C_ℋ
42	6, 17, 41	chlubi 31557	. . . . . . . . 9 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
43	38, 42	sylib 218	. . . . . . . 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 511	. . . . . . 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 31543	. . . . . . . . 9 ⊢ (𝑅 ∨_ℋ 𝐶) ∈ C_ℋ
46	45, 16	chincli 31546	. . . . . . . 8 ⊢ ((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∈ C_ℋ
47	6, 17	chjcli 31543	. . . . . . . 8 ⊢ (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ∈ C_ℋ
48	46, 47, 41	chlubi 31557	. . . . . . 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 218	. . . . . 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 32403	. . . . . 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 1374	. . . . 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 4177	. . . . . . 7 ⊢ (((𝑅 ∨_ℋ 𝐶) ∩ 𝐷) ∩ 𝐵) = (((𝑅 ∨_ℋ 𝐶) ∩ 𝐵) ∩ (𝐷 ∩ 𝐵))
53		sstr 3931	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
54	15, 53	sylanb 582	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ 𝑅)
55	54	ad2ant2r 748	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝑅)
56		simplll 775	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ 𝐶)
57	55, 56	ssind 4182	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵))) ∧ ((𝐶 ∩ 𝐷) ⊆ 𝑅 ∧ 𝑅 ⊆ 𝐷)) → 𝐴 ⊆ (𝑅 ∩ 𝐶))
58		simplrl 777	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 31557	. . . . . . . . . . . 12 ⊢ ((𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐶 ⊆ (𝐴 ∨_ℋ 𝐵)) ↔ (𝑅 ∨_ℋ 𝐶) ⊆ (𝐴 ∨_ℋ 𝐵))
61	59, 60	sylib 218	. . . . . . . . . . 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 511	. . . . . . . . . 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 32405	. . . . . . . . . 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 594	. . . . . . . . 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 467	. . . . . . . 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 4160	. . . . . . 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 2785	. . . . . 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 216	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) → 𝐴 ⊆ (𝐶 ∩ 𝐷))
69	68	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝐶 ∩ 𝐷))
70	54, 69	ssind 4182	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝑅) → 𝐴 ⊆ (𝑅 ∩ (𝐶 ∩ 𝐷)))
71	31	adantll 715	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → 𝑅 ⊆ (𝐴 ∨_ℋ 𝐵))
72	35	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵))
73	71, 72	jca 511	. . . . . . . . . . . 12 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ (𝐶 ∩ 𝐷) ⊆ (𝐴 ∨_ℋ 𝐵)))
74	73, 42	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐶 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝐷 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑅 ⊆ 𝐷) → (𝑅 ∨_ℋ (𝐶 ∩ 𝐷)) ⊆ (𝐴 ∨_ℋ 𝐵))
75	70, 74	anim12i 614	. . . . . . . . . 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 661	. . . . . . . . 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 32405	. . . . . . . . 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 594	. . . . . . . 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 467	. . . . . . 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 4177	. . . . . . . . 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 7376	. . . . . . 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 2773	. . . . . 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 3956	. . . . 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 282	. . . 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 811	. . 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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5086 (class class class)co 7360 C_ℋ cch 31015 ∨_ℋ chj 31019 𝑀_ℋ cmd 31052 𝑀_ℋ^* cdmd 31053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 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 31085 ax-hfvadd 31086 ax-hvcom 31087 ax-hvass 31088 ax-hv0cl 31089 ax-hvaddid 31090 ax-hfvmul 31091 ax-hvmulid 31092 ax-hvmulass 31093 ax-hvdistr1 31094 ax-hvdistr2 31095 ax-hvmul0 31096 ax-hfi 31165 ax-his1 31168 ax-his2 31169 ax-his3 31170 ax-his4 31171 ax-hcompl 31288

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-oadd 8402 df-omul 8403 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 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 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-cn 23202 df-cnp 23203 df-lm 23204 df-haus 23290 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cfil 25232 df-cau 25233 df-cmet 25234 df-grpo 30579 df-gid 30580 df-ginv 30581 df-gdiv 30582 df-ablo 30631 df-vc 30645 df-nv 30678 df-va 30681 df-ba 30682 df-sm 30683 df-0v 30684 df-vs 30685 df-nmcv 30686 df-ims 30687 df-dip 30787 df-ssp 30808 df-ph 30899 df-cbn 30949 df-hnorm 31054 df-hba 31055 df-hvsub 31057 df-hlim 31058 df-hcau 31059 df-sh 31293 df-ch 31307 df-oc 31338 df-ch0 31339 df-shs 31394 df-chj 31396 df-md 32366 df-dmd 32367

This theorem is referenced by: mdslmd1lem4 32414

W3C validator