HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdsl0 Unicode version

Theorem mdsl0 22850
Description: A sublattice condition that transfers the modular pair property. Exercise 12 of [Kalmbach] p. 103. Also Lemma 1.5.3 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdsl0  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH )
)  ->  ( (
( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H )  /\  A  MH  B )  ->  C  MH  D ) )

Proof of Theorem mdsl0
StepHypRef Expression
1 sstr2 3161 . . . . . . . 8  |-  ( x 
C_  D  ->  ( D  C_  B  ->  x  C_  B ) )
21com12 29 . . . . . . 7  |-  ( D 
C_  B  ->  (
x  C_  D  ->  x 
C_  B ) )
32ad2antlr 710 . . . . . 6  |-  ( ( ( C  C_  A  /\  D  C_  B )  /\  ( A  i^i  B )  =  0H )  ->  ( x  C_  D  ->  x  C_  B
) )
43ad2antlr 710 . . . . 5  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  /\  x  e.  CH )  ->  (
x  C_  D  ->  x 
C_  B ) )
5 chlej2 22050 . . . . . . . . . . . . . 14  |-  ( ( ( C  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  /\  C  C_  A )  ->  ( x  vH  C )  C_  (
x  vH  A )
)
6 ss2in 3371 . . . . . . . . . . . . . . 15  |-  ( ( ( x  vH  C
)  C_  ( x  vH  A )  /\  D  C_  B )  ->  (
( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
) )
76ex 425 . . . . . . . . . . . . . 14  |-  ( ( x  vH  C ) 
C_  ( x  vH  A )  ->  ( D  C_  B  ->  (
( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
) ) )
85, 7syl 17 . . . . . . . . . . . . 13  |-  ( ( ( C  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  /\  C  C_  A )  ->  ( D  C_  B  ->  ( ( x  vH  C )  i^i 
D )  C_  (
( x  vH  A
)  i^i  B )
) )
98ex 425 . . . . . . . . . . . 12  |-  ( ( C  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  ->  ( C  C_  A  ->  ( D  C_  B  ->  (
( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
) ) ) )
1093expia 1158 . . . . . . . . . . 11  |-  ( ( C  e.  CH  /\  A  e.  CH )  ->  ( x  e.  CH  ->  ( C  C_  A  ->  ( D  C_  B  ->  ( ( x  vH  C )  i^i  D
)  C_  ( (
x  vH  A )  i^i  B ) ) ) ) )
1110ancoms 441 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( x  e.  CH  ->  ( C  C_  A  ->  ( D  C_  B  ->  ( ( x  vH  C )  i^i  D
)  C_  ( (
x  vH  A )  i^i  B ) ) ) ) )
1211ad2ant2r 730 . . . . . . . . 9  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH )
)  ->  ( x  e.  CH  ->  ( C  C_  A  ->  ( D  C_  B  ->  ( (
x  vH  C )  i^i  D )  C_  (
( x  vH  A
)  i^i  B )
) ) ) )
1312imp43 581 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( C  C_  A  /\  D  C_  B ) )  ->  ( (
x  vH  C )  i^i  D )  C_  (
( x  vH  A
)  i^i  B )
)
1413adantrr 700 . . . . . . 7  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  (
( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
) )
15 oveq2 5800 . . . . . . . . . . . . 13  |-  ( ( A  i^i  B )  =  0H  ->  (
x  vH  ( A  i^i  B ) )  =  ( x  vH  0H ) )
16 chj0 22036 . . . . . . . . . . . . 13  |-  ( x  e.  CH  ->  (
x  vH  0H )  =  x )
1715, 16sylan9eqr 2312 . . . . . . . . . . . 12  |-  ( ( x  e.  CH  /\  ( A  i^i  B )  =  0H )  -> 
( x  vH  ( A  i^i  B ) )  =  x )
1817adantl 454 . . . . . . . . . . 11  |-  ( ( ( C  e.  CH  /\  D  e.  CH )  /\  ( x  e.  CH  /\  ( A  i^i  B
)  =  0H ) )  ->  ( x  vH  ( A  i^i  B
) )  =  x )
19 chincl 22038 . . . . . . . . . . . . 13  |-  ( ( C  e.  CH  /\  D  e.  CH )  ->  ( C  i^i  D
)  e.  CH )
20 chub1 22046 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CH  /\  ( C  i^i  D )  e.  CH )  ->  x  C_  ( x  vH  ( C  i^i  D ) ) )
2120ancoms 441 . . . . . . . . . . . . 13  |-  ( ( ( C  i^i  D
)  e.  CH  /\  x  e.  CH )  ->  x  C_  ( x  vH  ( C  i^i  D
) ) )
2219, 21sylan 459 . . . . . . . . . . . 12  |-  ( ( ( C  e.  CH  /\  D  e.  CH )  /\  x  e.  CH )  ->  x  C_  ( x  vH  ( C  i^i  D
) ) )
2322adantrr 700 . . . . . . . . . . 11  |-  ( ( ( C  e.  CH  /\  D  e.  CH )  /\  ( x  e.  CH  /\  ( A  i^i  B
)  =  0H ) )  ->  x  C_  (
x  vH  ( C  i^i  D ) ) )
2418, 23eqsstrd 3187 . . . . . . . . . 10  |-  ( ( ( C  e.  CH  /\  D  e.  CH )  /\  ( x  e.  CH  /\  ( A  i^i  B
)  =  0H ) )  ->  ( x  vH  ( A  i^i  B
) )  C_  (
x  vH  ( C  i^i  D ) ) )
2524adantll 697 . . . . . . . . 9  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( x  e.  CH  /\  ( A  i^i  B )  =  0H ) )  ->  ( x  vH  ( A  i^i  B ) )  C_  ( x  vH  ( C  i^i  D
) ) )
2625anassrs 632 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( A  i^i  B
)  =  0H )  ->  ( x  vH  ( A  i^i  B ) )  C_  ( x  vH  ( C  i^i  D
) ) )
2726adantrl 699 . . . . . . 7  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  (
x  vH  ( A  i^i  B ) )  C_  ( x  vH  ( C  i^i  D ) ) )
28 sstr2 3161 . . . . . . . . 9  |-  ( ( ( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
)  ->  ( (
( x  vH  A
)  i^i  B )  C_  ( x  vH  ( A  i^i  B ) )  ->  ( ( x  vH  C )  i^i 
D )  C_  (
x  vH  ( A  i^i  B ) ) ) )
29 sstr2 3161 . . . . . . . . 9  |-  ( ( ( x  vH  C
)  i^i  D )  C_  ( x  vH  ( A  i^i  B ) )  ->  ( ( x  vH  ( A  i^i  B ) )  C_  (
x  vH  ( C  i^i  D ) )  -> 
( ( x  vH  C )  i^i  D
)  C_  ( x  vH  ( C  i^i  D
) ) ) )
3028, 29syl6 31 . . . . . . . 8  |-  ( ( ( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
)  ->  ( (
( x  vH  A
)  i^i  B )  C_  ( x  vH  ( A  i^i  B ) )  ->  ( ( x  vH  ( A  i^i  B ) )  C_  (
x  vH  ( C  i^i  D ) )  -> 
( ( x  vH  C )  i^i  D
)  C_  ( x  vH  ( C  i^i  D
) ) ) ) )
3130com23 74 . . . . . . 7  |-  ( ( ( x  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  i^i  B
)  ->  ( (
x  vH  ( A  i^i  B ) )  C_  ( x  vH  ( C  i^i  D ) )  ->  ( ( ( x  vH  A )  i^i  B )  C_  ( x  vH  ( A  i^i  B ) )  ->  ( ( x  vH  C )  i^i 
D )  C_  (
x  vH  ( C  i^i  D ) ) ) ) )
3214, 27, 31sylc 58 . . . . . 6  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  x  e.  CH )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  (
( ( x  vH  A )  i^i  B
)  C_  ( x  vH  ( A  i^i  B
) )  ->  (
( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) )
3332an32s 782 . . . . 5  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  /\  x  e.  CH )  ->  (
( ( x  vH  A )  i^i  B
)  C_  ( x  vH  ( A  i^i  B
) )  ->  (
( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) )
344, 33imim12d 70 . . . 4  |-  ( ( ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  /\  x  e.  CH )  ->  (
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  C_  ( x  vH  ( A  i^i  B
) ) )  -> 
( x  C_  D  ->  ( ( x  vH  C )  i^i  D
)  C_  ( x  vH  ( C  i^i  D
) ) ) ) )
3534ralimdva 2596 . . 3  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  C_  ( x  vH  ( A  i^i  B ) ) )  ->  A. x  e.  CH  ( x  C_  D  ->  ( ( x  vH  C )  i^i 
D )  C_  (
x  vH  ( C  i^i  D ) ) ) ) )
36 mdbr2 22836 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  C_  ( x  vH  ( A  i^i  B ) ) ) ) )
3736ad2antrr 709 . . 3  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( A  MH  B  <->  A. x  e.  CH  ( x  C_  B  ->  ( ( x  vH  A )  i^i 
B )  C_  (
x  vH  ( A  i^i  B ) ) ) ) )
38 mdbr2 22836 . . . 4  |-  ( ( C  e.  CH  /\  D  e.  CH )  ->  ( C  MH  D  <->  A. x  e.  CH  (
x  C_  D  ->  ( ( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) ) )
3938ad2antlr 710 . . 3  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( C  MH  D  <->  A. x  e.  CH  ( x  C_  D  ->  ( ( x  vH  C )  i^i 
D )  C_  (
x  vH  ( C  i^i  D ) ) ) ) )
4035, 37, 393imtr4d 261 . 2  |-  ( ( ( ( A  e. 
CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH ) )  /\  ( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( A  MH  B  ->  C  MH  D ) )
4140expimpd 589 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( C  e.  CH  /\  D  e.  CH )
)  ->  ( (
( ( C  C_  A  /\  D  C_  B
)  /\  ( A  i^i  B )  =  0H )  /\  A  MH  B )  ->  C  MH  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2518    i^i cin 3126    C_ wss 3127   class class class wbr 3997  (class class class)co 5792   CHcch 21469    vH chj 21473   0Hc0h 21475    MH cmd 21506
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cc 8029  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785  ax-hilex 21539  ax-hfvadd 21540  ax-hvcom 21541  ax-hvass 21542  ax-hv0cl 21543  ax-hvaddid 21544  ax-hfvmul 21545  ax-hvmulid 21546  ax-hvmulass 21547  ax-hvdistr1 21548  ax-hvdistr2 21549  ax-hvmul0 21550  ax-hfi 21618  ax-his1 21621  ax-his2 21622  ax-his3 21623  ax-his4 21624  ax-hcompl 21741
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-omul 6452  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-acn 7543  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9933  df-z 9992  df-dec 10092  df-uz 10198  df-q 10284  df-rp 10322  df-xneg 10419  df-xadd 10420  df-xmul 10421  df-ioo 10626  df-ico 10628  df-icc 10629  df-fz 10749  df-fzo 10837  df-fl 10891  df-seq 11013  df-exp 11071  df-hash 11304  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-clim 11927  df-rlim 11928  df-sum 12124  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-starv 13185  df-sca 13186  df-vsca 13187  df-tset 13189  df-ple 13190  df-ds 13192  df-hom 13194  df-cco 13195  df-rest 13289  df-topn 13290  df-topgen 13306  df-pt 13307  df-prds 13310  df-xrs 13365  df-0g 13366  df-gsum 13367  df-qtop 13372  df-imas 13373  df-xps 13375  df-mre 13450  df-mrc 13451  df-acs 13453  df-mnd 14329  df-submnd 14378  df-mulg 14454  df-cntz 14755  df-cmn 15053  df-xmet 16335  df-met 16336  df-bl 16337  df-mopn 16338  df-cnfld 16340  df-top 16598  df-bases 16600  df-topon 16601  df-topsp 16602  df-cld 16718  df-ntr 16719  df-cls 16720  df-nei 16797  df-cn 16919  df-cnp 16920  df-lm 16921  df-haus 17005  df-tx 17219  df-hmeo 17408  df-fbas 17482  df-fg 17483  df-fil 17503  df-fm 17595  df-flim 17596  df-flf 17597  df-xms 17847  df-ms 17848  df-tms 17849  df-cfil 18643  df-cau 18644  df-cmet 18645  df-grpo 20818  df-gid 20819  df-ginv 20820  df-gdiv 20821  df-ablo 20909  df-subgo 20929  df-vc 21062  df-nv 21108  df-va 21111  df-ba 21112  df-sm 21113  df-0v 21114  df-vs 21115  df-nmcv 21116  df-ims 21117  df-dip 21234  df-ssp 21258  df-ph 21351  df-cbn 21402  df-hnorm 21508  df-hba 21509  df-hvsub 21511  df-hlim 21512  df-hcau 21513  df-sh 21746  df-ch 21761  df-oc 21791  df-ch0 21792  df-shs 21847  df-chj 21849  df-md 22820
  Copyright terms: Public domain W3C validator