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Theorem dihjatcclem1 36508
Description: Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
Hypotheses
Ref Expression
dihjatcclem.b  |-  B  =  ( Base `  K
)
dihjatcclem.l  |-  .<_  =  ( le `  K )
dihjatcclem.h  |-  H  =  ( LHyp `  K
)
dihjatcclem.j  |-  .\/  =  ( join `  K )
dihjatcclem.m  |-  ./\  =  ( meet `  K )
dihjatcclem.a  |-  A  =  ( Atoms `  K )
dihjatcclem.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihjatcclem.s  |-  .(+)  =  (
LSSum `  U )
dihjatcclem.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihjatcclem.v  |-  V  =  ( ( P  .\/  Q )  ./\  W )
dihjatcclem.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dihjatcclem.p  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
dihjatcclem.q  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
Assertion
Ref Expression
dihjatcclem1  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( ( I `  P ) 
.(+)  ( I `  Q ) )  .(+)  ( I `  V ) ) )

Proof of Theorem dihjatcclem1
StepHypRef Expression
1 dihjatcclem.h . . . . 5  |-  H  =  ( LHyp `  K
)
2 dihjatcclem.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
3 dihjatcclem.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3dvhlmod 36200 . . . 4  |-  ( ph  ->  U  e.  LMod )
5 lmodabl 17405 . . . 4  |-  ( U  e.  LMod  ->  U  e. 
Abel )
64, 5syl 16 . . 3  |-  ( ph  ->  U  e.  Abel )
7 eqid 2467 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
87lsssssubg 17452 . . . . 5  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
94, 8syl 16 . . . 4  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
10 dihjatcclem.p . . . . . . 7  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
1110simpld 459 . . . . . 6  |-  ( ph  ->  P  e.  A )
12 dihjatcclem.b . . . . . . 7  |-  B  =  ( Base `  K
)
13 dihjatcclem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
1412, 13atbase 34379 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
1511, 14syl 16 . . . . 5  |-  ( ph  ->  P  e.  B )
16 dihjatcclem.i . . . . . 6  |-  I  =  ( ( DIsoH `  K
) `  W )
1712, 1, 16, 2, 7dihlss 36340 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  B
)  ->  ( I `  P )  e.  (
LSubSp `  U ) )
183, 15, 17syl2anc 661 . . . 4  |-  ( ph  ->  ( I `  P
)  e.  ( LSubSp `  U ) )
199, 18sseldd 3510 . . 3  |-  ( ph  ->  ( I `  P
)  e.  (SubGrp `  U ) )
20 dihjatcclem.v . . . . . 6  |-  V  =  ( ( P  .\/  Q )  ./\  W )
213simpld 459 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
22 hllat 34453 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
2321, 22syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
24 dihjatcclem.q . . . . . . . . 9  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
2524simpld 459 . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
26 dihjatcclem.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
2712, 26, 13hlatjcl 34456 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
2821, 11, 25, 27syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( P  .\/  Q
)  e.  B )
293simprd 463 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
3012, 1lhpbase 35087 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
3129, 30syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  B )
32 dihjatcclem.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
3312, 32latmcl 15551 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  Q
)  ./\  W )  e.  B )
3423, 28, 31, 33syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  e.  B )
3520, 34syl5eqel 2559 . . . . 5  |-  ( ph  ->  V  e.  B )
3612, 1, 16, 2, 7dihlss 36340 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  B
)  ->  ( I `  V )  e.  (
LSubSp `  U ) )
373, 35, 36syl2anc 661 . . . 4  |-  ( ph  ->  ( I `  V
)  e.  ( LSubSp `  U ) )
389, 37sseldd 3510 . . 3  |-  ( ph  ->  ( I `  V
)  e.  (SubGrp `  U ) )
3912, 13atbase 34379 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
4025, 39syl 16 . . . . 5  |-  ( ph  ->  Q  e.  B )
4112, 1, 16, 2, 7dihlss 36340 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( I `  Q )  e.  (
LSubSp `  U ) )
423, 40, 41syl2anc 661 . . . 4  |-  ( ph  ->  ( I `  Q
)  e.  ( LSubSp `  U ) )
439, 42sseldd 3510 . . 3  |-  ( ph  ->  ( I `  Q
)  e.  (SubGrp `  U ) )
44 dihjatcclem.s . . . 4  |-  .(+)  =  (
LSSum `  U )
4544lsm4 16716 . . 3  |-  ( ( U  e.  Abel  /\  (
( I `  P
)  e.  (SubGrp `  U )  /\  (
I `  V )  e.  (SubGrp `  U )
)  /\  ( (
I `  Q )  e.  (SubGrp `  U )  /\  ( I `  V
)  e.  (SubGrp `  U ) ) )  ->  ( ( ( I `  P ) 
.(+)  ( I `  V ) )  .(+)  ( ( I `  Q
)  .(+)  ( I `  V ) ) )  =  ( ( ( I `  P ) 
.(+)  ( I `  Q ) )  .(+)  ( ( I `  V
)  .(+)  ( I `  V ) ) ) )
466, 19, 38, 43, 38, 45syl122anc 1237 . 2  |-  ( ph  ->  ( ( ( I `
 P )  .(+)  ( I `  V ) )  .(+)  ( (
I `  Q )  .(+)  ( I `  V
) ) )  =  ( ( ( I `
 P )  .(+)  ( I `  Q ) )  .(+)  ( (
I `  V )  .(+)  ( I `  V
) ) ) )
4724simprd 463 . . . . . . . 8  |-  ( ph  ->  -.  Q  .<_  W )
4847intnand 914 . . . . . . 7  |-  ( ph  ->  -.  ( P  .<_  W  /\  Q  .<_  W ) )
49 dihjatcclem.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
5012, 49, 26latjle12 15561 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  W  e.  B
) )  ->  (
( P  .<_  W  /\  Q  .<_  W )  <->  ( P  .\/  Q )  .<_  W ) )
5123, 15, 40, 31, 50syl13anc 1230 . . . . . . 7  |-  ( ph  ->  ( ( P  .<_  W  /\  Q  .<_  W )  <-> 
( P  .\/  Q
)  .<_  W ) )
5248, 51mtbid 300 . . . . . 6  |-  ( ph  ->  -.  ( P  .\/  Q )  .<_  W )
5349, 26, 13hlatlej1 34464 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
5421, 11, 25, 53syl3anc 1228 . . . . . 6  |-  ( ph  ->  P  .<_  ( P  .\/  Q ) )
5512, 49, 26, 32, 13, 1, 16, 2, 44dihvalcq2 36337 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P 
.\/  Q )  e.  B  /\  -.  ( P  .\/  Q )  .<_  W )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  P  .<_  ( P 
.\/  Q ) ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( ( I `  P
)  .(+)  ( I `  ( ( P  .\/  Q )  ./\  W )
) ) )
563, 28, 52, 10, 54, 55syl122anc 1237 . . . . 5  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  ( ( P  .\/  Q ) 
./\  W ) ) ) )
5720fveq2i 5874 . . . . . 6  |-  ( I `
 V )  =  ( I `  (
( P  .\/  Q
)  ./\  W )
)
5857oveq2i 6305 . . . . 5  |-  ( ( I `  P ) 
.(+)  ( I `  V ) )  =  ( ( I `  P )  .(+)  ( I `
 ( ( P 
.\/  Q )  ./\  W ) ) )
5956, 58syl6eqr 2526 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  V ) ) )
6049, 26, 13hlatlej2 34465 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
6121, 11, 25, 60syl3anc 1228 . . . . . 6  |-  ( ph  ->  Q  .<_  ( P  .\/  Q ) )
6212, 49, 26, 32, 13, 1, 16, 2, 44dihvalcq2 36337 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P 
.\/  Q )  e.  B  /\  -.  ( P  .\/  Q )  .<_  W )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  ( P 
.\/  Q ) ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( ( I `  Q
)  .(+)  ( I `  ( ( P  .\/  Q )  ./\  W )
) ) )
633, 28, 52, 24, 61, 62syl122anc 1237 . . . . 5  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 Q )  .(+)  ( I `  ( ( P  .\/  Q ) 
./\  W ) ) ) )
6457oveq2i 6305 . . . . 5  |-  ( ( I `  Q ) 
.(+)  ( I `  V ) )  =  ( ( I `  Q )  .(+)  ( I `
 ( ( P 
.\/  Q )  ./\  W ) ) )
6563, 64syl6eqr 2526 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 Q )  .(+)  ( I `  V ) ) )
6659, 65oveq12d 6312 . . 3  |-  ( ph  ->  ( ( I `  ( P  .\/  Q ) )  .(+)  ( I `  ( P  .\/  Q
) ) )  =  ( ( ( I `
 P )  .(+)  ( I `  V ) )  .(+)  ( (
I `  Q )  .(+)  ( I `  V
) ) ) )
6712, 1, 16, 2, 7dihlss 36340 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  .\/  Q )  e.  B )  ->  ( I `  ( P  .\/  Q ) )  e.  ( LSubSp `  U ) )
683, 28, 67syl2anc 661 . . . . 5  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  e.  ( LSubSp `  U
) )
699, 68sseldd 3510 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  e.  (SubGrp `  U
) )
7044lsmidm 16532 . . . 4  |-  ( ( I `  ( P 
.\/  Q ) )  e.  (SubGrp `  U
)  ->  ( (
I `  ( P  .\/  Q ) )  .(+)  ( I `  ( P 
.\/  Q ) ) )  =  ( I `
 ( P  .\/  Q ) ) )
7169, 70syl 16 . . 3  |-  ( ph  ->  ( ( I `  ( P  .\/  Q ) )  .(+)  ( I `  ( P  .\/  Q
) ) )  =  ( I `  ( P  .\/  Q ) ) )
7266, 71eqtr3d 2510 . 2  |-  ( ph  ->  ( ( ( I `
 P )  .(+)  ( I `  V ) )  .(+)  ( (
I `  Q )  .(+)  ( I `  V
) ) )  =  ( I `  ( P  .\/  Q ) ) )
7344lsmidm 16532 . . . 4  |-  ( ( I `  V )  e.  (SubGrp `  U
)  ->  ( (
I `  V )  .(+)  ( I `  V
) )  =  ( I `  V ) )
7438, 73syl 16 . . 3  |-  ( ph  ->  ( ( I `  V )  .(+)  ( I `
 V ) )  =  ( I `  V ) )
7574oveq2d 6310 . 2  |-  ( ph  ->  ( ( ( I `
 P )  .(+)  ( I `  Q ) )  .(+)  ( (
I `  V )  .(+)  ( I `  V
) ) )  =  ( ( ( I `
 P )  .(+)  ( I `  Q ) )  .(+)  ( I `  V ) ) )
7646, 72, 753eqtr3d 2516 1  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( ( I `  P ) 
.(+)  ( I `  Q ) )  .(+)  ( I `  V ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   Basecbs 14502   lecple 14574   joincjn 15443   meetcmee 15444   Latclat 15544  SubGrpcsubg 16044   LSSumclsm 16504   Abelcabl 16649   LModclmod 17360   LSubSpclss 17426   Atomscatm 34353   HLchlt 34440   LHypclh 35073   DVecHcdvh 36168   DIsoHcdih 36318
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-riotaBAD 34049
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-tpos 6965  df-undef 7012  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-sca 14583  df-vsca 14584  df-0g 14709  df-poset 15445  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-p1 15539  df-lat 15545  df-clat 15607  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-submnd 15820  df-grp 15906  df-minusg 15907  df-sbg 15908  df-subg 16047  df-cntz 16204  df-lsm 16506  df-cmn 16650  df-abl 16651  df-mgp 16991  df-ur 17003  df-ring 17049  df-oppr 17121  df-dvdsr 17139  df-unit 17140  df-invr 17170  df-dvr 17181  df-drng 17246  df-lmod 17362  df-lss 17427  df-lsp 17466  df-lvec 17597  df-oposet 34266  df-ol 34268  df-oml 34269  df-covers 34356  df-ats 34357  df-atl 34388  df-cvlat 34412  df-hlat 34441  df-llines 34587  df-lplanes 34588  df-lvols 34589  df-lines 34590  df-psubsp 34592  df-pmap 34593  df-padd 34885  df-lhyp 35077  df-laut 35078  df-ldil 35193  df-ltrn 35194  df-trl 35248  df-tendo 35844  df-edring 35846  df-disoa 36119  df-dvech 36169  df-dib 36229  df-dic 36263  df-dih 36319
This theorem is referenced by:  dihjatcclem2  36509
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