mdsl2i - Hilbert Space Explorer

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Theorem mdsl2i 31327

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 30512	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → 𝑥 ⊆ (𝑥 ∨_ℋ 𝐴))
3	1, 2	mpan2 689	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → 𝑥 ⊆ (𝑥 ∨_ℋ 𝐴))
4		iba 528	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ↔ (𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵)))
5		ssin 4195	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))
6	4, 5	bitrdi 286	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ (𝑥 ∨_ℋ 𝐴) ↔ 𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
7	3, 6	syl5ibcom 244	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → 𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
8		chub2 30513	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → 𝐴 ⊆ (𝑥 ∨_ℋ 𝐴))
9	1, 8	mpan 688	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → 𝐴 ⊆ (𝑥 ∨_ℋ 𝐴))
10	9	ssrind 4200	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))
11	7, 10	jctird 527	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (𝑥 ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵))))
12		chjcl 30362	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝑥 ∨_ℋ 𝐴) ∈ C_ℋ )
13	1, 12	mpan2 689	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ∨_ℋ 𝐴) ∈ C_ℋ )
14		mdsl.2	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ C_ℋ
15		chincl 30504	. . . . . . . . . . . 12 ⊢ (((𝑥 ∨_ℋ 𝐴) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵) ∈ C_ℋ )
16	14, 15	mpan2 689	. . . . . . . . . . 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 30465	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
19		chlub 30514	. . . . . . . . . . 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 1449	. . . . . . . . . 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 685	. . . . . . . . 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 238	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 → (𝑥 ∨_ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨_ℋ 𝐴) ∩ 𝐵)))
23		eqss 3962	. . . . . . . . 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 539	. . . . . . . 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 491	. . . . . 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 272	. . . . 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 30475	. . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)
29		sstr 3955	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)) → 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵))
30	28, 29	mpan2 689	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵))
31	30	pm4.71ri 561	. . . . . . . 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 469	. . . . . . 7 ⊢ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨_ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑥 ⊆ (𝐴 ∨_ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵)))
34	32, 33	bitr4i 277	. . . . . 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 285	. . . 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 451	. . . 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 286	. . 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 3090	. 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 31326	. 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 275	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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 ∩ cin 3912 ⊆ wss 3913 class class class wbr 5110 (class class class)co 7362 C_ℋ cch 29934 ∨_ℋ chj 29938 𝑀_ℋ cmd 29971

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9586 ax-cc 10380 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138 ax-addf 11139 ax-mulf 11140 ax-hilex 30004 ax-hfvadd 30005 ax-hvcom 30006 ax-hvass 30007 ax-hv0cl 30008 ax-hvaddid 30009 ax-hfvmul 30010 ax-hvmulid 30011 ax-hvmulass 30012 ax-hvdistr1 30013 ax-hvdistr2 30014 ax-hvmul0 30015 ax-hfi 30084 ax-his1 30087 ax-his2 30088 ax-his3 30089 ax-his4 30090 ax-hcompl 30207

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-omul 8422 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9356 df-sup 9387 df-inf 9388 df-oi 9455 df-card 9884 df-acn 9887 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12423 df-z 12509 df-dec 12628 df-uz 12773 df-q 12883 df-rp 12925 df-xneg 13042 df-xadd 13043 df-xmul 13044 df-ioo 13278 df-ico 13280 df-icc 13281 df-fz 13435 df-fzo 13578 df-fl 13707 df-seq 13917 df-exp 13978 df-hash 14241 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-clim 15382 df-rlim 15383 df-sum 15583 df-struct 17030 df-sets 17047 df-slot 17065 df-ndx 17077 df-base 17095 df-ress 17124 df-plusg 17160 df-mulr 17161 df-starv 17162 df-sca 17163 df-vsca 17164 df-ip 17165 df-tset 17166 df-ple 17167 df-ds 17169 df-unif 17170 df-hom 17171 df-cco 17172 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18511 df-sgrp 18560 df-mnd 18571 df-submnd 18616 df-mulg 18887 df-cntz 19111 df-cmn 19578 df-psmet 20825 df-xmet 20826 df-met 20827 df-bl 20828 df-mopn 20829 df-fbas 20830 df-fg 20831 df-cnfld 20834 df-top 22280 df-topon 22297 df-topsp 22319 df-bases 22333 df-cld 22407 df-ntr 22408 df-cls 22409 df-nei 22486 df-cn 22615 df-cnp 22616 df-lm 22617 df-haus 22703 df-tx 22950 df-hmeo 23143 df-fil 23234 df-fm 23326 df-flim 23327 df-flf 23328 df-xms 23710 df-ms 23711 df-tms 23712 df-cfil 24656 df-cau 24657 df-cmet 24658 df-grpo 29498 df-gid 29499 df-ginv 29500 df-gdiv 29501 df-ablo 29550 df-vc 29564 df-nv 29597 df-va 29600 df-ba 29601 df-sm 29602 df-0v 29603 df-vs 29604 df-nmcv 29605 df-ims 29606 df-dip 29706 df-ssp 29727 df-ph 29818 df-cbn 29868 df-hnorm 29973 df-hba 29974 df-hvsub 29976 df-hlim 29977 df-hcau 29978 df-sh 30212 df-ch 30226 df-oc 30257 df-ch0 30258 df-shs 30313 df-chj 30315 df-md 31285

This theorem is referenced by: mdsl2bi 31328 mdslmd1i 31334

W3C validator