Theorem mdslj2i 27808

Description: Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mdslle1.1	|- A e. CH
mdslle1.2	|- B e. CH
mdslle1.3	|- C e. CH
mdslle1.4	|- D e. CH

Assertion

Ref	Expression
mdslj2i	|- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ ( C i^i D ) /\ ( C vH D ) C_ B ) ) -> ( ( C i^i D ) vH A ) = ( ( C vH A ) i^i ( D vH A ) ) )

Proof of Theorem mdslj2i

Step	Hyp	Ref	Expression
1		mdslle1.3	. . . 4 |- C e. CH
2		mdslle1.4	. . . 4 |- D e. CH
3		mdslle1.1	. . . 4 |- A e. CH
4	1, 2, 3	lejdiri 27027	. . 3 |- ( ( C i^i D ) vH A ) C_ ( ( C vH A ) i^i ( D vH A ) )
5	4	a1i 11	. 2 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ ( C i^i D ) /\ ( C vH D ) C_ B ) ) -> ( ( C i^i D ) vH A ) C_ ( ( C vH A ) i^i ( D vH A ) ) )
6		ssin 3690	. . . . 5 |- ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) <-> ( A i^i B ) C_ ( C i^i D ) )
7	6	bicomi 205	. . . 4 |- ( ( A i^i B ) C_ ( C i^i D ) <-> ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) )
8		mdslle1.2	. . . . . 6 |- B e. CH
9	1, 2, 8	chlubi 26959	. . . . 5 |- ( ( C C_ B /\ D C_ B ) <-> ( C vH D ) C_ B )
10	9	bicomi 205	. . . 4 |- ( ( C vH D ) C_ B <-> ( C C_ B /\ D C_ B ) )
11	7, 10	anbi12i 701	. . 3 |- ( ( ( A i^i B ) C_ ( C i^i D ) /\ ( C vH D ) C_ B ) <-> ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) )
12		simpr 462	. . . . . 6 |- ( ( A MH B /\ B MH* A ) -> B MH* A )
13	3, 1	chub2i 26958	. . . . . . . 8 |- A C_ ( C vH A )
14	3, 2	chub2i 26958	. . . . . . . 8 |- A C_ ( D vH A )
15	13, 14	ssini 3691	. . . . . . 7 |- A C_ ( ( C vH A ) i^i ( D vH A ) )
16	15	a1i 11	. . . . . 6 |- ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) -> A C_ ( ( C vH A ) i^i ( D vH A ) ) )
17	1, 8, 3	chlej1i 26961	. . . . . . . . 9 |- ( C C_ B -> ( C vH A ) C_ ( B vH A ) )
18	8, 3	chjcomi 26956	. . . . . . . . 9 |- ( B vH A ) = ( A vH B )
19	17, 18	syl6sseq 3516	. . . . . . . 8 |- ( C C_ B -> ( C vH A ) C_ ( A vH B ) )
20		ssinss1 3696	. . . . . . . 8 |- ( ( C vH A ) C_ ( A vH B ) -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( A vH B ) )
21	19, 20	syl 17	. . . . . . 7 |- ( C C_ B -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( A vH B ) )
22	21	adantr 466	. . . . . 6 |- ( ( C C_ B /\ D C_ B ) -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( A vH B ) )
23	1, 3	chjcli 26945	. . . . . . . . 9 |- ( C vH A ) e. CH
24	2, 3	chjcli 26945	. . . . . . . . 9 |- ( D vH A ) e. CH
25	23, 24	chincli 26948	. . . . . . . 8 |- ( ( C vH A ) i^i ( D vH A ) ) e. CH
26	3, 8, 25	3pm3.2i 1183	. . . . . . 7 |- ( A e. CH /\ B e. CH /\ ( ( C vH A ) i^i ( D vH A ) ) e. CH )
27		dmdsl3 27803	. . . . . . 7 |- ( ( ( A e. CH /\ B e. CH /\ ( ( C vH A ) i^i ( D vH A ) ) e. CH ) /\ ( B MH* A /\ A C_ ( ( C vH A ) i^i ( D vH A ) ) /\ ( ( C vH A ) i^i ( D vH A ) ) C_ ( A vH B ) ) ) -> ( ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) vH A ) = ( ( C vH A ) i^i ( D vH A ) ) )
28	26, 27	mpan 674	. . . . . 6 |- ( ( B MH* A /\ A C_ ( ( C vH A ) i^i ( D vH A ) ) /\ ( ( C vH A ) i^i ( D vH A ) ) C_ ( A vH B ) ) -> ( ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) vH A ) = ( ( C vH A ) i^i ( D vH A ) ) )
29	12, 16, 22, 28	syl3an 1306	. . . . 5 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) -> ( ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) vH A ) = ( ( C vH A ) i^i ( D vH A ) ) )
30		inss1 3688	. . . . . . . . 9 |- ( ( C vH A ) i^i ( D vH A ) ) C_ ( C vH A )
31		ssrin 3693	. . . . . . . . 9 |- ( ( ( C vH A ) i^i ( D vH A ) ) C_ ( C vH A ) -> ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) C_ ( ( C vH A ) i^i B ) )
32	30, 31	ax-mp 5	. . . . . . . 8 |- ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) C_ ( ( C vH A ) i^i B )
33		simpl 458	. . . . . . . . 9 |- ( ( A MH B /\ B MH* A ) -> A MH B )
34		simpl 458	. . . . . . . . 9 |- ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) -> ( A i^i B ) C_ C )
35		simpl 458	. . . . . . . . 9 |- ( ( C C_ B /\ D C_ B ) -> C C_ B )
36	3, 8, 1	3pm3.2i 1183	. . . . . . . . . 10 |- ( A e. CH /\ B e. CH /\ C e. CH )
37		mdsl3 27804	. . . . . . . . . 10 |- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH B /\ ( A i^i B ) C_ C /\ C C_ B ) ) -> ( ( C vH A ) i^i B ) = C )
38	36, 37	mpan 674	. . . . . . . . 9 |- ( ( A MH B /\ ( A i^i B ) C_ C /\ C C_ B ) -> ( ( C vH A ) i^i B ) = C )
39	33, 34, 35, 38	syl3an 1306	. . . . . . . 8 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) -> ( ( C vH A ) i^i B ) = C )
40	32, 39	syl5sseq 3518	. . . . . . 7 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) -> ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) C_ C )
41		inss2 3689	. . . . . . . . 9 |- ( ( C vH A ) i^i ( D vH A ) ) C_ ( D vH A )
42		ssrin 3693	. . . . . . . . 9 |- ( ( ( C vH A ) i^i ( D vH A ) ) C_ ( D vH A ) -> ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) C_ ( ( D vH A ) i^i B ) )
43	41, 42	ax-mp 5	. . . . . . . 8 |- ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) C_ ( ( D vH A ) i^i B )
44		simpr 462	. . . . . . . . 9 |- ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) -> ( A i^i B ) C_ D )
45		simpr 462	. . . . . . . . 9 |- ( ( C C_ B /\ D C_ B ) -> D C_ B )
46	3, 8, 2	3pm3.2i 1183	. . . . . . . . . 10 |- ( A e. CH /\ B e. CH /\ D e. CH )
47		mdsl3 27804	. . . . . . . . . 10 |- ( ( ( A e. CH /\ B e. CH /\ D e. CH ) /\ ( A MH B /\ ( A i^i B ) C_ D /\ D C_ B ) ) -> ( ( D vH A ) i^i B ) = D )
48	46, 47	mpan 674	. . . . . . . . 9 |- ( ( A MH B /\ ( A i^i B ) C_ D /\ D C_ B ) -> ( ( D vH A ) i^i B ) = D )
49	33, 44, 45, 48	syl3an 1306	. . . . . . . 8 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) -> ( ( D vH A ) i^i B ) = D )
50	43, 49	syl5sseq 3518	. . . . . . 7 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) -> ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) C_ D )
51	40, 50	ssind 3692	. . . . . 6 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) -> ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) C_ ( C i^i D ) )
52	25, 8	chincli 26948	. . . . . . 7 |- ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) e. CH
53	1, 2	chincli 26948	. . . . . . 7 |- ( C i^i D ) e. CH
54	52, 53, 3	chlej1i 26961	. . . . . 6 |- ( ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) C_ ( C i^i D ) -> ( ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) vH A ) C_ ( ( C i^i D ) vH A ) )
55	51, 54	syl 17	. . . . 5 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) -> ( ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) vH A ) C_ ( ( C i^i D ) vH A ) )
56	29, 55	eqsstr3d 3505	. . . 4 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( ( C i^i D ) vH A ) )
57	56	3expb 1206	. . 3 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) /\ ( C C_ B /\ D C_ B ) ) ) -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( ( C i^i D ) vH A ) )
58	11, 57	sylan2b 477	. 2 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ ( C i^i D ) /\ ( C vH D ) C_ B ) ) -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( ( C i^i D ) vH A ) )
59	5, 58	eqssd 3487	1 |- ( ( ( A MH B /\ B MH* A ) /\ ( ( A i^i B ) C_ ( C i^i D ) /\ ( C vH D ) C_ B ) ) -> ( ( C i^i D ) vH A ) = ( ( C vH A ) i^i ( D vH A ) ) )

Syntax hints:   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   i^i cin 3441   C_ wss 3442   class class class wbr 4426  (class class class)co 6305   CHcch 26417   vH chj 26421   MH cmd 26454   MH* cdmd 26455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cc 8863  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618  ax-hilex 26487  ax-hfvadd 26488  ax-hvcom 26489  ax-hvass 26490  ax-hv0cl 26491  ax-hvaddid 26492  ax-hfvmul 26493  ax-hvmulid 26494  ax-hvmulass 26495  ax-hvdistr1 26496  ax-hvdistr2 26497  ax-hvmul0 26498  ax-hfi 26567  ax-his1 26570  ax-his2 26571  ax-his3 26572  ax-his4 26573  ax-hcompl 26690

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-omul 7195  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-acn 8375  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-cn 20174  df-cnp 20175  df-lm 20176  df-haus 20262  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cfil 22118  df-cau 22119  df-cmet 22120  df-grpo 25764  df-gid 25765  df-ginv 25766  df-gdiv 25767  df-ablo 25855  df-subgo 25875  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-vs 26063  df-nmcv 26064  df-ims 26065  df-dip 26182  df-ssp 26206  df-ph 26299  df-cbn 26350  df-hnorm 26456  df-hba 26457  df-hvsub 26459  df-hlim 26460  df-hcau 26461  df-sh 26695  df-ch 26709  df-oc 26740  df-ch0 26741  df-shs 26796  df-chj 26798  df-md 27768  df-dmd 27769

This theorem is referenced by: (None)