mdsl2i - Hilbert Space Explorer

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Theorem mdsl2i 32302

Description: If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 28-Apr-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
mdsl.1	⊢ 𝐴 ∈ C_ℋ
mdsl.2	⊢ 𝐵 ∈ C_ℋ

**Assertion**
Ref	Expression
mdsl2i	⊢ (𝐴 𝑀_ℋ 𝐵 ↔ ∀𝑥 ∈ C_ℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem mdsl2i**
Step	Hyp	Ref	Expression
1		mdsl.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ C_ℋ
2		chub1 31487	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → 𝑥 ⊆ (𝑥 ∨_ℋ 𝐴))
3	1, 2	mpan2 691	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → 𝑥 ⊆ (𝑥 ∨_ℋ 𝐴))
4		iba 527	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ↔ (𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵)))
5		ssin 4186	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))
6	4, 5	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ↔ 𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
7	3, 6	syl5ibcom 245	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → 𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
8		chub2 31488	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → 𝐴 ⊆ (𝑥 ∨_ℋ 𝐴))
9	1, 8	mpan 690	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → 𝐴 ⊆ (𝑥 ∨_ℋ 𝐴))
10	9	ssrind 4191	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))
11	7, 10	jctird 526	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))))
12		chjcl 31337	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝑥 ∨_ℋ 𝐴) ∈ C_ℋ )
13	1, 12	mpan2 691	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ 𝐴) ∈ C_ℋ )
14		mdsl.2	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ C_ℋ
15		chincl 31479	. . . . . . . . . . . 12 ⊢ (((𝑥 ∨_ℋ 𝐴) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ )
16	14, 15	mpan2 691	. . . . . . . . . . 11 ⊢ ((𝑥 ∨_ℋ 𝐴) ∈ C_ℋ → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ )
17	13, 16	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ )
18	1, 14	chincli 31440	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
19		chlub 31489	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ C_ℋ ∧ (𝐴 ∩ 𝐵) ∈ C_ℋ ∧ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ ) → ((𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)) ↔ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
20	18, 19	mp3an2 1451	. . . . . . . . . 10 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ ) → ((𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)) ↔ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
21	17, 20	mpdan 687	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)) ↔ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
22	11, 21	sylibd 239	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
23		eqss 3945	. . . . . . . . 9 ⊢ (((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ↔ (((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ∧ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
24	23	rbaib 538	. . . . . . . 8 ⊢ ((𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) → (((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))))
25	22, 24	syl6 35	. . . . . . 7 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
26	25	adantld 490	. . . . . 6 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → (((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
27	26	pm5.74d 273	. . . . 5 ⊢ (𝑥 ∈ C_ℋ → ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
28	14, 1	chub2i 31450	. . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)
29		sstr 3938	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵))
30	28, 29	mpan2 691	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵))
31	30	pm4.71ri 560	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 ↔ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵))
32	31	anbi2i 623	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵)))
33		anass 468	. . . . . . 7 ⊢ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵)))
34	32, 33	bitr4i 278	. . . . . 6 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵))
35	34	imbi1i 349	. . . . 5 ⊢ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))) ↔ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))))
36	27, 35	bitr3di 286	. . . 4 ⊢ (𝑥 ∈ C_ℋ → ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))) ↔ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
37		impexp 450	. . . 4 ⊢ (((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
38	36, 37	bitrdi 287	. . 3 ⊢ (𝑥 ∈ C_ℋ → ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))))))
39	38	ralbiia 3076	. 2 ⊢ (∀𝑥 ∈ C_ℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))) ↔ ∀𝑥 ∈ C_ℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))))
40	1, 14	mdsl1i 32301	. 2 ⊢ (∀𝑥 ∈ C_ℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)))) ↔ 𝐴 𝑀_ℋ 𝐵)
41	39, 40	bitr2i 276	1 ⊢ (𝐴 𝑀_ℋ 𝐵 ↔ ∀𝑥 ∈ C_ℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨_ℋ (𝐴 ∩ 𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∩ cin 3896 ⊆ wss 3897 class class class wbr 5089 (class class class)co 7346 C_ℋ cch 30909 ∨_ℋ chj 30913 𝑀_ℋ cmd 30946

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cc 10326 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086 ax-hilex 30979 ax-hfvadd 30980 ax-hvcom 30981 ax-hvass 30982 ax-hv0cl 30983 ax-hvaddid 30984 ax-hfvmul 30985 ax-hvmulid 30986 ax-hvmulass 30987 ax-hvdistr1 30988 ax-hvdistr2 30989 ax-hvmul0 30990 ax-hfi 31059 ax-his1 31062 ax-his2 31063 ax-his3 31064 ax-his4 31065 ax-hcompl 31182

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-acn 9835 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-cn 23142 df-cnp 23143 df-lm 23144 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cfil 25182 df-cau 25183 df-cmet 25184 df-grpo 30473 df-gid 30474 df-ginv 30475 df-gdiv 30476 df-ablo 30525 df-vc 30539 df-nv 30572 df-va 30575 df-ba 30576 df-sm 30577 df-0v 30578 df-vs 30579 df-nmcv 30580 df-ims 30581 df-dip 30681 df-ssp 30702 df-ph 30793 df-cbn 30843 df-hnorm 30948 df-hba 30949 df-hvsub 30951 df-hlim 30952 df-hcau 30953 df-sh 31187 df-ch 31201 df-oc 31232 df-ch0 31233 df-shs 31288 df-chj 31290 df-md 32260

This theorem is referenced by: mdsl2bi 32303 mdslmd1i 32309

W3C validator