mdslmd2i - Hilbert Space Explorer

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Theorem mdslmd2i 32358

Description: Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (join version). (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

**Assertion**
Ref	Expression
mdslmd2i	⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵)) → (𝐶 𝑀_ℋ 𝐷 ↔ (𝐶 ∨_ℋ 𝐴) 𝑀_ℋ (𝐷 ∨_ℋ 𝐴)))

**Proof of Theorem mdslmd2i**
Step	Hyp	Ref	Expression
1		mdslmd.3	. . . . . . . 8 ⊢ 𝐶 ∈ C_ℋ
2		mdslmd.4	. . . . . . . 8 ⊢ 𝐷 ∈ C_ℋ
3	1, 2	chjcli 31485	. . . . . . 7 ⊢ (𝐶 ∨_ℋ 𝐷) ∈ C_ℋ
4		mdslmd.2	. . . . . . 7 ⊢ 𝐵 ∈ C_ℋ
5		mdslmd.1	. . . . . . 7 ⊢ 𝐴 ∈ C_ℋ
6	3, 4, 5	chlej1i 31501	. . . . . 6 ⊢ ((𝐶 ∨_ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨_ℋ 𝐷) ∨_ℋ 𝐴) ⊆ (𝐵 ∨_ℋ 𝐴))
7	1, 2, 5	chjjdiri 31552	. . . . . 6 ⊢ ((𝐶 ∨_ℋ 𝐷) ∨_ℋ 𝐴) = ((𝐶 ∨_ℋ 𝐴) ∨_ℋ (𝐷 ∨_ℋ 𝐴))
8	4, 5	chjcomi 31496	. . . . . 6 ⊢ (𝐵 ∨_ℋ 𝐴) = (𝐴 ∨_ℋ 𝐵)
9	6, 7, 8	3sstr3g 4039	. . . . 5 ⊢ ((𝐶 ∨_ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨_ℋ 𝐴) ∨_ℋ (𝐷 ∨_ℋ 𝐴)) ⊆ (𝐴 ∨_ℋ 𝐵))
10	9	adantl 481	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨_ℋ 𝐴) ∨_ℋ (𝐷 ∨_ℋ 𝐴)) ⊆ (𝐴 ∨_ℋ 𝐵))
11	5, 1	chub2i 31498	. . . . 5 ⊢ 𝐴 ⊆ (𝐶 ∨_ℋ 𝐴)
12	5, 2	chub2i 31498	. . . . 5 ⊢ 𝐴 ⊆ (𝐷 ∨_ℋ 𝐴)
13	11, 12	ssini 4247	. . . 4 ⊢ 𝐴 ⊆ ((𝐶 ∨_ℋ 𝐴) ∩ (𝐷 ∨_ℋ 𝐴))
14	10, 13	jctil 519	. . 3 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵) → (𝐴 ⊆ ((𝐶 ∨_ℋ 𝐴) ∩ (𝐷 ∨_ℋ 𝐴)) ∧ ((𝐶 ∨_ℋ 𝐴) ∨_ℋ (𝐷 ∨_ℋ 𝐴)) ⊆ (𝐴 ∨_ℋ 𝐵)))
15	1, 5	chjcli 31485	. . . 4 ⊢ (𝐶 ∨_ℋ 𝐴) ∈ C_ℋ
16	2, 5	chjcli 31485	. . . 4 ⊢ (𝐷 ∨_ℋ 𝐴) ∈ C_ℋ
17	5, 4, 15, 16	mdslmd1i 32357	. . 3 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ (𝐴 ⊆ ((𝐶 ∨_ℋ 𝐴) ∩ (𝐷 ∨_ℋ 𝐴)) ∧ ((𝐶 ∨_ℋ 𝐴) ∨_ℋ (𝐷 ∨_ℋ 𝐴)) ⊆ (𝐴 ∨_ℋ 𝐵))) → ((𝐶 ∨_ℋ 𝐴) 𝑀_ℋ (𝐷 ∨_ℋ 𝐴) ↔ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) 𝑀_ℋ ((𝐷 ∨_ℋ 𝐴) ∩ 𝐵)))
18	14, 17	sylan2 593	. 2 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵)) → ((𝐶 ∨_ℋ 𝐴) 𝑀_ℋ (𝐷 ∨_ℋ 𝐴) ↔ ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) 𝑀_ℋ ((𝐷 ∨_ℋ 𝐴) ∩ 𝐵)))
19		id 22	. . . . . 6 ⊢ (𝐴 𝑀_ℋ 𝐵 → 𝐴 𝑀_ℋ 𝐵)
20		inss1 4244	. . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐶
21		sstr 4003	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶)
22	20, 21	mpan2 691	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐶)
23	1, 2	chub1i 31497	. . . . . . 7 ⊢ 𝐶 ⊆ (𝐶 ∨_ℋ 𝐷)
24		sstr 4003	. . . . . . 7 ⊢ ((𝐶 ⊆ (𝐶 ∨_ℋ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵) → 𝐶 ⊆ 𝐵)
25	23, 24	mpan 690	. . . . . 6 ⊢ ((𝐶 ∨_ℋ 𝐷) ⊆ 𝐵 → 𝐶 ⊆ 𝐵)
26	5, 4, 1	3pm3.2i 1338	. . . . . . 7 ⊢ (𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ )
27		mdsl3 32344	. . . . . . 7 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) ∧ (𝐴 𝑀_ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) → ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = 𝐶)
28	26, 27	mpan 690	. . . . . 6 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = 𝐶)
29	19, 22, 25, 28	syl3an 1159	. . . . 5 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) = 𝐶)
30		inss2 4245	. . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐷
31		sstr 4003	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷)
32	30, 31	mpan2 691	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷)
33	2, 1	chub2i 31498	. . . . . . 7 ⊢ 𝐷 ⊆ (𝐶 ∨_ℋ 𝐷)
34		sstr 4003	. . . . . . 7 ⊢ ((𝐷 ⊆ (𝐶 ∨_ℋ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵) → 𝐷 ⊆ 𝐵)
35	33, 34	mpan 690	. . . . . 6 ⊢ ((𝐶 ∨_ℋ 𝐷) ⊆ 𝐵 → 𝐷 ⊆ 𝐵)
36	5, 4, 2	3pm3.2i 1338	. . . . . . 7 ⊢ (𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝐷 ∈ C_ℋ )
37		mdsl3 32344	. . . . . . 7 ⊢ (((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ 𝐷 ∈ C_ℋ ) ∧ (𝐴 𝑀_ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵)) → ((𝐷 ∨_ℋ 𝐴) ∩ 𝐵) = 𝐷)
38	36, 37	mpan 690	. . . . . 6 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵) → ((𝐷 ∨_ℋ 𝐴) ∩ 𝐵) = 𝐷)
39	19, 32, 35, 38	syl3an 1159	. . . . 5 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵) → ((𝐷 ∨_ℋ 𝐴) ∩ 𝐵) = 𝐷)
40	29, 39	breq12d 5160	. . . 4 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵) → (((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) 𝑀_ℋ ((𝐷 ∨_ℋ 𝐴) ∩ 𝐵) ↔ 𝐶 𝑀_ℋ 𝐷))
41	40	3expb 1119	. . 3 ⊢ ((𝐴 𝑀_ℋ 𝐵 ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵)) → (((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) 𝑀_ℋ ((𝐷 ∨_ℋ 𝐴) ∩ 𝐵) ↔ 𝐶 𝑀_ℋ 𝐷))
42	41	adantlr 715	. 2 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵)) → (((𝐶 ∨_ℋ 𝐴) ∩ 𝐵) 𝑀_ℋ ((𝐷 ∨_ℋ 𝐴) ∩ 𝐵) ↔ 𝐶 𝑀_ℋ 𝐷))
43	18, 42	bitr2d 280	1 ⊢ (((𝐴 𝑀_ℋ 𝐵 ∧ 𝐵 𝑀_ℋ^* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨_ℋ 𝐷) ⊆ 𝐵)) → (𝐶 𝑀_ℋ 𝐷 ↔ (𝐶 ∨_ℋ 𝐴) 𝑀_ℋ (𝐷 ∨_ℋ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∩ cin 3961 ⊆ wss 3962 class class class wbr 5147 (class class class)co 7430 C_ℋ cch 30957 ∨_ℋ chj 30961 𝑀_ℋ cmd 30994 𝑀_ℋ^* cdmd 30995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cc 10472 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232 ax-hilex 31027 ax-hfvadd 31028 ax-hvcom 31029 ax-hvass 31030 ax-hv0cl 31031 ax-hvaddid 31032 ax-hfvmul 31033 ax-hvmulid 31034 ax-hvmulass 31035 ax-hvdistr1 31036 ax-hvdistr2 31037 ax-hvmul0 31038 ax-hfi 31107 ax-his1 31110 ax-his2 31111 ax-his3 31112 ax-his4 31113 ax-hcompl 31230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-acn 9979 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-cn 23250 df-cnp 23251 df-lm 23252 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cfil 25302 df-cau 25303 df-cmet 25304 df-grpo 30521 df-gid 30522 df-ginv 30523 df-gdiv 30524 df-ablo 30573 df-vc 30587 df-nv 30620 df-va 30623 df-ba 30624 df-sm 30625 df-0v 30626 df-vs 30627 df-nmcv 30628 df-ims 30629 df-dip 30729 df-ssp 30750 df-ph 30841 df-cbn 30891 df-hnorm 30996 df-hba 30997 df-hvsub 30999 df-hlim 31000 df-hcau 31001 df-sh 31235 df-ch 31249 df-oc 31280 df-ch0 31281 df-shs 31336 df-chj 31338 df-md 32308 df-dmd 32309

This theorem is referenced by: mdsldmd1i 32359

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