Theorem mdslj2i 26943

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 26161	. . 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 3720	. . . . 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 202	. . . 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 26093	. . . . 5 |- ( ( C C_ B /\ D C_ B ) <-> ( C vH D ) C_ B )
10	9	bicomi 202	. . . 4 |- ( ( C vH D ) C_ B <-> ( C C_ B /\ D C_ B ) )
11	7, 10	anbi12i 697	. . 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 461	. . . . . 6 |- ( ( A MH B /\ B MH* A ) -> B MH* A )
13	3, 1	chub2i 26092	. . . . . . . 8 |- A C_ ( C vH A )
14	3, 2	chub2i 26092	. . . . . . . 8 |- A C_ ( D vH A )
15	13, 14	ssini 3721	. . . . . . 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 26095	. . . . . . . . 9 |- ( C C_ B -> ( C vH A ) C_ ( B vH A ) )
18	8, 3	chjcomi 26090	. . . . . . . . 9 |- ( B vH A ) = ( A vH B )
19	17, 18	syl6sseq 3550	. . . . . . . 8 |- ( C C_ B -> ( C vH A ) C_ ( A vH B ) )
20		ssinss1 3726	. . . . . . . 8 |- ( ( C vH A ) C_ ( A vH B ) -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( A vH B ) )
21	19, 20	syl 16	. . . . . . 7 |- ( C C_ B -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( A vH B ) )
22	21	adantr 465	. . . . . 6 |- ( ( C C_ B /\ D C_ B ) -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( A vH B ) )
23	1, 3	chjcli 26079	. . . . . . . . 9 |- ( C vH A ) e. CH
24	2, 3	chjcli 26079	. . . . . . . . 9 |- ( D vH A ) e. CH
25	23, 24	chincli 26082	. . . . . . . 8 |- ( ( C vH A ) i^i ( D vH A ) ) e. CH
26	3, 8, 25	3pm3.2i 1174	. . . . . . 7 |- ( A e. CH /\ B e. CH /\ ( ( C vH A ) i^i ( D vH A ) ) e. CH )
27		dmdsl3 26938	. . . . . . 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 670	. . . . . 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 1270	. . . . 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 3718	. . . . . . . . 9 |- ( ( C vH A ) i^i ( D vH A ) ) C_ ( C vH A )
31		ssrin 3723	. . . . . . . . 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 457	. . . . . . . . 9 |- ( ( A MH B /\ B MH* A ) -> A MH B )
34		simpl 457	. . . . . . . . 9 |- ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) -> ( A i^i B ) C_ C )
35		simpl 457	. . . . . . . . 9 |- ( ( C C_ B /\ D C_ B ) -> C C_ B )
36	3, 8, 1	3pm3.2i 1174	. . . . . . . . . 10 |- ( A e. CH /\ B e. CH /\ C e. CH )
37		mdsl3 26939	. . . . . . . . . 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 670	. . . . . . . . 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 1270	. . . . . . . 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 3552	. . . . . . 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 3719	. . . . . . . . 9 |- ( ( C vH A ) i^i ( D vH A ) ) C_ ( D vH A )
42		ssrin 3723	. . . . . . . . 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 461	. . . . . . . . 9 |- ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) -> ( A i^i B ) C_ D )
45		simpr 461	. . . . . . . . 9 |- ( ( C C_ B /\ D C_ B ) -> D C_ B )
46	3, 8, 2	3pm3.2i 1174	. . . . . . . . . 10 |- ( A e. CH /\ B e. CH /\ D e. CH )
47		mdsl3 26939	. . . . . . . . . 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 670	. . . . . . . . 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 1270	. . . . . . . 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 3552	. . . . . . 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 3722	. . . . . 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 26082	. . . . . . 7 |- ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) e. CH
53	1, 2	chincli 26082	. . . . . . 7 |- ( C i^i D ) e. CH
54	52, 53, 3	chlej1i 26095	. . . . . 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 16	. . . . 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 3539	. . . 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 1197	. . 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 475	. 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 3521	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 369   /\ w3a 973   = wceq 1379   e. wcel 1767   i^i cin 3475   C_ wss 3476   class class class wbr 4447  (class class class)co 6284   CHcch 25550   vH chj 25554   MH cmd 25587   MH* cdmd 25588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cc 8815  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572  ax-hilex 25620  ax-hfvadd 25621  ax-hvcom 25622  ax-hvass 25623  ax-hv0cl 25624  ax-hvaddid 25625  ax-hfvmul 25626  ax-hvmulid 25627  ax-hvmulass 25628  ax-hvdistr1 25629  ax-hvdistr2 25630  ax-hvmul0 25631  ax-hfi 25700  ax-his1 25703  ax-his2 25704  ax-his3 25705  ax-his4 25706  ax-hcompl 25823

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-cn 19522  df-cnp 19523  df-lm 19524  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cfil 21457  df-cau 21458  df-cmet 21459  df-grpo 24897  df-gid 24898  df-ginv 24899  df-gdiv 24900  df-ablo 24988  df-subgo 25008  df-vc 25143  df-nv 25189  df-va 25192  df-ba 25193  df-sm 25194  df-0v 25195  df-vs 25196  df-nmcv 25197  df-ims 25198  df-dip 25315  df-ssp 25339  df-ph 25432  df-cbn 25483  df-hnorm 25589  df-hba 25590  df-hvsub 25592  df-hlim 25593  df-hcau 25594  df-sh 25828  df-ch 25843  df-oc 25874  df-ch0 25875  df-shs 25930  df-chj 25932  df-md 26903  df-dmd 26904

This theorem is referenced by: (None)