Theorem mdslj2i 28054

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 27273	. . 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 3645	. . . . 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 207	. . . 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 27205	. . . . 5 |- ( ( C C_ B /\ D C_ B ) <-> ( C vH D ) C_ B )
10	9	bicomi 207	. . . 4 |- ( ( C vH D ) C_ B <-> ( C C_ B /\ D C_ B ) )
11	7, 10	anbi12i 711	. . 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 468	. . . . . 6 |- ( ( A MH B /\ B MH* A ) -> B MH* A )
13	3, 1	chub2i 27204	. . . . . . . 8 |- A C_ ( C vH A )
14	3, 2	chub2i 27204	. . . . . . . 8 |- A C_ ( D vH A )
15	13, 14	ssini 3646	. . . . . . 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 27207	. . . . . . . . 9 |- ( C C_ B -> ( C vH A ) C_ ( B vH A ) )
18	8, 3	chjcomi 27202	. . . . . . . . 9 |- ( B vH A ) = ( A vH B )
19	17, 18	syl6sseq 3464	. . . . . . . 8 |- ( C C_ B -> ( C vH A ) C_ ( A vH B ) )
20		ssinss1 3651	. . . . . . . 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 472	. . . . . 6 |- ( ( C C_ B /\ D C_ B ) -> ( ( C vH A ) i^i ( D vH A ) ) C_ ( A vH B ) )
23	1, 3	chjcli 27191	. . . . . . . . 9 |- ( C vH A ) e. CH
24	2, 3	chjcli 27191	. . . . . . . . 9 |- ( D vH A ) e. CH
25	23, 24	chincli 27194	. . . . . . . 8 |- ( ( C vH A ) i^i ( D vH A ) ) e. CH
26	3, 8, 25	3pm3.2i 1208	. . . . . . 7 |- ( A e. CH /\ B e. CH /\ ( ( C vH A ) i^i ( D vH A ) ) e. CH )
27		dmdsl3 28049	. . . . . . 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 684	. . . . . 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 1334	. . . . 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 3643	. . . . . . . . 9 |- ( ( C vH A ) i^i ( D vH A ) ) C_ ( C vH A )
31		ssrin 3648	. . . . . . . . 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 464	. . . . . . . . 9 |- ( ( A MH B /\ B MH* A ) -> A MH B )
34		simpl 464	. . . . . . . . 9 |- ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) -> ( A i^i B ) C_ C )
35		simpl 464	. . . . . . . . 9 |- ( ( C C_ B /\ D C_ B ) -> C C_ B )
36	3, 8, 1	3pm3.2i 1208	. . . . . . . . . 10 |- ( A e. CH /\ B e. CH /\ C e. CH )
37		mdsl3 28050	. . . . . . . . . 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 684	. . . . . . . . 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 1334	. . . . . . . 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 3466	. . . . . . 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 3644	. . . . . . . . 9 |- ( ( C vH A ) i^i ( D vH A ) ) C_ ( D vH A )
42		ssrin 3648	. . . . . . . . 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 468	. . . . . . . . 9 |- ( ( ( A i^i B ) C_ C /\ ( A i^i B ) C_ D ) -> ( A i^i B ) C_ D )
45		simpr 468	. . . . . . . . 9 |- ( ( C C_ B /\ D C_ B ) -> D C_ B )
46	3, 8, 2	3pm3.2i 1208	. . . . . . . . . 10 |- ( A e. CH /\ B e. CH /\ D e. CH )
47		mdsl3 28050	. . . . . . . . . 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 684	. . . . . . . . 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 1334	. . . . . . . 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 3466	. . . . . . 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 3647	. . . . . 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 27194	. . . . . . 7 |- ( ( ( C vH A ) i^i ( D vH A ) ) i^i B ) e. CH
53	1, 2	chincli 27194	. . . . . . 7 |- ( C i^i D ) e. CH
54	52, 53, 3	chlej1i 27207	. . . . . 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 3453	. . . 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 1232	. . 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 483	. 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 3435	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 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   i^i cin 3389   C_ wss 3390   class class class wbr 4395  (class class class)co 6308   CHcch 26663   vH chj 26667   MH cmd 26700   MH* cdmd 26701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637  ax-hilex 26733  ax-hfvadd 26734  ax-hvcom 26735  ax-hvass 26736  ax-hv0cl 26737  ax-hvaddid 26738  ax-hfvmul 26739  ax-hvmulid 26740  ax-hvmulass 26741  ax-hvdistr1 26742  ax-hvdistr2 26743  ax-hvmul0 26744  ax-hfi 26813  ax-his1 26816  ax-his2 26817  ax-his3 26818  ax-his4 26819  ax-hcompl 26936

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-cnp 20321  df-lm 20322  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cfil 22303  df-cau 22304  df-cmet 22305  df-grpo 26000  df-gid 26001  df-ginv 26002  df-gdiv 26003  df-ablo 26091  df-subgo 26111  df-vc 26246  df-nv 26292  df-va 26295  df-ba 26296  df-sm 26297  df-0v 26298  df-vs 26299  df-nmcv 26300  df-ims 26301  df-dip 26418  df-ssp 26442  df-ph 26535  df-cbn 26586  df-hnorm 26702  df-hba 26703  df-hvsub 26705  df-hlim 26706  df-hcau 26707  df-sh 26941  df-ch 26955  df-oc 26986  df-ch0 26987  df-shs 27042  df-chj 27044  df-md 28014  df-dmd 28015

This theorem is referenced by: (None)