HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdslj1i Structured version   Unicode version

Theorem mdslj1i 27365
Description: Join preservation of the one-to-one onto 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
mdslj1i  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )

Proof of Theorem mdslj1i
StepHypRef Expression
1 ssin 3716 . . . . 5  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
21bicomi 202 . . . 4  |-  ( A 
C_  ( C  i^i  D )  <->  ( A  C_  C  /\  A  C_  D
) )
3 mdslle1.3 . . . . . 6  |-  C  e. 
CH
4 mdslle1.4 . . . . . 6  |-  D  e. 
CH
5 mdslle1.1 . . . . . . 7  |-  A  e. 
CH
6 mdslle1.2 . . . . . . 7  |-  B  e. 
CH
75, 6chjcli 26502 . . . . . 6  |-  ( A  vH  B )  e. 
CH
83, 4, 7chlubi 26516 . . . . 5  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  <->  ( C  vH  D )  C_  ( A  vH  B ) )
98bicomi 202 . . . 4  |-  ( ( C  vH  D ) 
C_  ( A  vH  B )  <->  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )
102, 9anbi12i 697 . . 3  |-  ( ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) )  <->  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )
11 simpr 461 . . . . . . . . . 10  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  B  MH*  A )
12 simpl 457 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  C )
13 simpl 457 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  C  C_  ( A  vH  B
) )
145, 6, 33pm3.2i 1174 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  C  e. 
CH )
15 dmdsl3 27361 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )
1614, 15mpan 670 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B
) )  ->  (
( C  i^i  B
)  vH  A )  =  C )
1711, 12, 13, 16syl3an 1270 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  =  C )
183, 6chincli 26505 . . . . . . . . . . 11  |-  ( C  i^i  B )  e. 
CH
194, 6chincli 26505 . . . . . . . . . . 11  |-  ( D  i^i  B )  e. 
CH
2018, 19chub1i 26514 . . . . . . . . . 10  |-  ( C  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
2118, 19chjcli 26502 . . . . . . . . . . 11  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH
2218, 21, 5chlej1i 26518 . . . . . . . . . 10  |-  ( ( C  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  ->  ( ( C  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
2320, 22mp1i 12 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
2417, 23eqsstr3d 3534 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  C  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
25 simpr 461 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  D )
26 simpr 461 . . . . . . . . . 10  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  D  C_  ( A  vH  B
) )
275, 6, 43pm3.2i 1174 . . . . . . . . . . 11  |-  ( A  e.  CH  /\  B  e.  CH  /\  D  e. 
CH )
28 dmdsl3 27361 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  D  e.  CH )  /\  ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
2927, 28mpan 670 . . . . . . . . . 10  |-  ( ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B
) )  ->  (
( D  i^i  B
)  vH  A )  =  D )
3011, 25, 26, 29syl3an 1270 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
3119, 18chub2i 26515 . . . . . . . . . 10  |-  ( D  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )
3219, 21, 5chlej1i 26518 . . . . . . . . . 10  |-  ( ( D  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  ->  ( ( D  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3331, 32mp1i 12 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3430, 33eqsstr3d 3534 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  D  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3524, 34jca 532 . . . . . . 7  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( C  C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  /\  D  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) ) )
3621, 5chjcli 26502 . . . . . . . 8  |-  ( ( ( C  i^i  B
)  vH  ( D  i^i  B ) )  vH  A )  e.  CH
373, 4, 36chlubi 26516 . . . . . . 7  |-  ( ( C  C_  ( (
( C  i^i  B
)  vH  ( D  i^i  B ) )  vH  A )  /\  D  C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )  <->  ( C  vH  D )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
3835, 37sylib 196 . . . . . 6  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( C  vH  D )  C_  (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A ) )
39 ssrin 3719 . . . . . 6  |-  ( ( C  vH  D ) 
C_  ( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A
)  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B ) )
4038, 39syl 16 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B ) )
41 simpl 457 . . . . . 6  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  A  MH  B )
42 ssrin 3719 . . . . . . . 8  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( C  i^i  B ) )
4342, 20syl6ss 3511 . . . . . . 7  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
4443adantr 465 . . . . . 6  |-  ( ( A  C_  C  /\  A  C_  D )  -> 
( A  i^i  B
)  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
45 inss2 3715 . . . . . . . 8  |-  ( C  i^i  B )  C_  B
46 inss2 3715 . . . . . . . 8  |-  ( D  i^i  B )  C_  B
4718, 19, 6chlubi 26516 . . . . . . . . 9  |-  ( ( ( C  i^i  B
)  C_  B  /\  ( D  i^i  B ) 
C_  B )  <->  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B )
4847bicomi 202 . . . . . . . 8  |-  ( ( ( C  i^i  B
)  vH  ( D  i^i  B ) )  C_  B 
<->  ( ( C  i^i  B )  C_  B  /\  ( D  i^i  B ) 
C_  B ) )
4945, 46, 48mpbir2an 920 . . . . . . 7  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
5049a1i 11 . . . . . 6  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  (
( C  i^i  B
)  vH  ( D  i^i  B ) )  C_  B )
515, 6, 213pm3.2i 1174 . . . . . . 7  |-  ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH )
52 mdsl3 27362 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  e.  CH )  /\  ( A  MH  B  /\  ( A  i^i  B
)  C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
) )  ->  (
( ( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i  B )  =  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
5351, 52mpan 670 . . . . . 6  |-  ( ( A  MH  B  /\  ( A  i^i  B ) 
C_  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  /\  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  B
)  ->  ( (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5441, 44, 50, 53syl3an 1270 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( (
( ( C  i^i  B )  vH  ( D  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5540, 54sseqtrd 3535 . . . 4  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
56553expb 1197 . . 3  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
5710, 56sylan2b 475 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  C_  (
( C  i^i  B
)  vH  ( D  i^i  B ) ) )
583, 4, 6lediri 26582 . . 3  |-  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  (
( C  vH  D
)  i^i  B )
5958a1i 11 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  i^i  B )  vH  ( D  i^i  B ) )  C_  ( ( C  vH  D )  i^i 
B ) )
6057, 59eqssd 3516 1  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
B )  =  ( ( C  i^i  B
)  vH  ( D  i^i  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471   class class class wbr 4456  (class class class)co 6296   CHcch 25973    vH chj 25977    MH cmd 26010    MH* cdmd 26011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589  ax-hilex 26043  ax-hfvadd 26044  ax-hvcom 26045  ax-hvass 26046  ax-hv0cl 26047  ax-hvaddid 26048  ax-hfvmul 26049  ax-hvmulid 26050  ax-hvmulass 26051  ax-hvdistr1 26052  ax-hvdistr2 26053  ax-hvmul0 26054  ax-hfi 26123  ax-his1 26126  ax-his2 26127  ax-his3 26128  ax-his4 26129  ax-hcompl 26246
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-cn 19855  df-cnp 19856  df-lm 19857  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cfil 21820  df-cau 21821  df-cmet 21822  df-grpo 25320  df-gid 25321  df-ginv 25322  df-gdiv 25323  df-ablo 25411  df-subgo 25431  df-vc 25566  df-nv 25612  df-va 25615  df-ba 25616  df-sm 25617  df-0v 25618  df-vs 25619  df-nmcv 25620  df-ims 25621  df-dip 25738  df-ssp 25762  df-ph 25855  df-cbn 25906  df-hnorm 26012  df-hba 26013  df-hvsub 26015  df-hlim 26016  df-hcau 26017  df-sh 26251  df-ch 26266  df-oc 26297  df-ch0 26298  df-shs 26353  df-chj 26355  df-md 27326  df-dmd 27327
This theorem is referenced by:  mdslmd1lem1  27371  mdslmd1lem2  27372
  Copyright terms: Public domain W3C validator