mdsl2i - Hilbert Space Explorer

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Theorem mdsl2i 32258

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 31443	. . . . . . . . . . . 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 4205	. . . . . . . . . . . 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 31444	. . . . . . . . . . . 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 4210	. . . . . . . . . 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 31293	. . . . . . . . . . . 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 31435	. . . . . . . . . . . 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 31396	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
19		chlub 31445	. . . . . . . . . . 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 3965	. . . . . . . . 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 31406	. . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐵)
29		sstr 3958	. . . . . . . . . 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 3074	. 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 32257	. 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 1540 ∈ wcel 2109 ∀wral 3045 ∩ cin 3916 ⊆ wss 3917 class class class wbr 5110 (class class class)co 7390 C_ℋ cch 30865 ∨_ℋ chj 30869 𝑀_ℋ cmd 30902

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cc 10395 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155 ax-hilex 30935 ax-hfvadd 30936 ax-hvcom 30937 ax-hvass 30938 ax-hv0cl 30939 ax-hvaddid 30940 ax-hfvmul 30941 ax-hvmulid 30942 ax-hvmulass 30943 ax-hvdistr1 30944 ax-hvdistr2 30945 ax-hvmul0 30946 ax-hfi 31015 ax-his1 31018 ax-his2 31019 ax-his3 31020 ax-his4 31021 ax-hcompl 31138

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-acn 9902 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-cn 23121 df-cnp 23122 df-lm 23123 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cfil 25162 df-cau 25163 df-cmet 25164 df-grpo 30429 df-gid 30430 df-ginv 30431 df-gdiv 30432 df-ablo 30481 df-vc 30495 df-nv 30528 df-va 30531 df-ba 30532 df-sm 30533 df-0v 30534 df-vs 30535 df-nmcv 30536 df-ims 30537 df-dip 30637 df-ssp 30658 df-ph 30749 df-cbn 30799 df-hnorm 30904 df-hba 30905 df-hvsub 30907 df-hlim 30908 df-hcau 30909 df-sh 31143 df-ch 31157 df-oc 31188 df-ch0 31189 df-shs 31244 df-chj 31246 df-md 32216

This theorem is referenced by: mdsl2bi 32259 mdslmd1i 32265

W3C validator