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Theorem dihjatcclem1 35068
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 34760 . . . 4  |-  ( ph  ->  U  e.  LMod )
5 lmodabl 16997 . . . 4  |-  ( U  e.  LMod  ->  U  e. 
Abel )
64, 5syl 16 . . 3  |-  ( ph  ->  U  e.  Abel )
7 eqid 2443 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
87lsssssubg 17044 . . . . 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 32939 . . . . . 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 34900 . . . . 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 3362 . . 3  |-  ( ph  ->  ( I `  P
)  e.  (SubGrp `  U ) )
20 dihjatcclem.v . . . . . 6  |-  V  =  ( ( P  .\/  Q )  ./\  W )
213simpld 459 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
22 hllat 33013 . . . . . . . 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 33016 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
2821, 11, 25, 27syl3anc 1218 . . . . . . 7  |-  ( ph  ->  ( P  .\/  Q
)  e.  B )
293simprd 463 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
3012, 1lhpbase 33647 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
3129, 30syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  B )
32 dihjatcclem.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
3312, 32latmcl 15227 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  Q
)  ./\  W )  e.  B )
3423, 28, 31, 33syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  e.  B )
3520, 34syl5eqel 2527 . . . . 5  |-  ( ph  ->  V  e.  B )
3612, 1, 16, 2, 7dihlss 34900 . . . . 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 3362 . . 3  |-  ( ph  ->  ( I `  V
)  e.  (SubGrp `  U ) )
3912, 13atbase 32939 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
4025, 39syl 16 . . . . 5  |-  ( ph  ->  Q  e.  B )
4112, 1, 16, 2, 7dihlss 34900 . . . . 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 3362 . . 3  |-  ( ph  ->  ( I `  Q
)  e.  (SubGrp `  U ) )
44 dihjatcclem.s . . . 4  |-  .(+)  =  (
LSSum `  U )
4544lsm4 16347 . . 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 1227 . 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 907 . . . . . . 7  |-  ( ph  ->  -.  ( P  .<_  W  /\  Q  .<_  W ) )
49 dihjatcclem.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
5012, 49, 26latjle12 15237 . . . . . . . 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 1220 . . . . . . 7  |-  ( ph  ->  ( ( P  .<_  W  /\  Q  .<_  W )  <-> 
( P  .\/  Q
)  .<_  W ) )
5248, 51mtbid 300 . . . . . 6  |-  ( ph  ->  -.  ( P  .\/  Q )  .<_  W )
5349, 26, 13hlatlej1 33024 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
5421, 11, 25, 53syl3anc 1218 . . . . . 6  |-  ( ph  ->  P  .<_  ( P  .\/  Q ) )
5512, 49, 26, 32, 13, 1, 16, 2, 44dihvalcq2 34897 . . . . . 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 1227 . . . . 5  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  ( ( P  .\/  Q ) 
./\  W ) ) ) )
5720fveq2i 5699 . . . . . 6  |-  ( I `
 V )  =  ( I `  (
( P  .\/  Q
)  ./\  W )
)
5857oveq2i 6107 . . . . 5  |-  ( ( I `  P ) 
.(+)  ( I `  V ) )  =  ( ( I `  P )  .(+)  ( I `
 ( ( P 
.\/  Q )  ./\  W ) ) )
5956, 58syl6eqr 2493 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  V ) ) )
6049, 26, 13hlatlej2 33025 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
6121, 11, 25, 60syl3anc 1218 . . . . . 6  |-  ( ph  ->  Q  .<_  ( P  .\/  Q ) )
6212, 49, 26, 32, 13, 1, 16, 2, 44dihvalcq2 34897 . . . . . 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 1227 . . . . 5  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 Q )  .(+)  ( I `  ( ( P  .\/  Q ) 
./\  W ) ) ) )
6457oveq2i 6107 . . . . 5  |-  ( ( I `  Q ) 
.(+)  ( I `  V ) )  =  ( ( I `  Q )  .(+)  ( I `
 ( ( P 
.\/  Q )  ./\  W ) ) )
6563, 64syl6eqr 2493 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 Q )  .(+)  ( I `  V ) ) )
6659, 65oveq12d 6114 . . 3  |-  ( ph  ->  ( ( I `  ( P  .\/  Q ) )  .(+)  ( I `  ( P  .\/  Q
) ) )  =  ( ( ( I `
 P )  .(+)  ( I `  V ) )  .(+)  ( (
I `  Q )  .(+)  ( I `  V
) ) ) )
6712, 1, 16, 2, 7dihlss 34900 . . . . . 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 3362 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  e.  (SubGrp `  U
) )
7044lsmidm 16166 . . . 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 2477 . 2  |-  ( ph  ->  ( ( ( I `
 P )  .(+)  ( I `  V ) )  .(+)  ( (
I `  Q )  .(+)  ( I `  V
) ) )  =  ( I `  ( P  .\/  Q ) ) )
7344lsmidm 16166 . . . 4  |-  ( ( I `  V )  e.  (SubGrp `  U
)  ->  ( (
I `  V )  .(+)  ( I `  V
) )  =  ( I `  V ) )
7438, 73syl 16 . . 3  |-  ( ph  ->  ( ( I `  V )  .(+)  ( I `
 V ) )  =  ( I `  V ) )
7574oveq2d 6112 . 2  |-  ( ph  ->  ( ( ( I `
 P )  .(+)  ( I `  Q ) )  .(+)  ( (
I `  V )  .(+)  ( I `  V
) ) )  =  ( ( ( I `
 P )  .(+)  ( I `  Q ) )  .(+)  ( I `  V ) ) )
7646, 72, 753eqtr3d 2483 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 1369    e. wcel 1756    C_ wss 3333   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   joincjn 15119   meetcmee 15120   Latclat 15220  SubGrpcsubg 15680   LSSumclsm 16138   Abelcabel 16283   LModclmod 16953   LSubSpclss 17018   Atomscatm 32913   HLchlt 33000   LHypclh 33633   DVecHcdvh 34728   DIsoHcdih 34878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-riotaBAD 32609
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-undef 6797  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-0g 14385  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-subg 15683  df-cntz 15840  df-lsm 16140  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-dvr 16780  df-drng 16839  df-lmod 16955  df-lss 17019  df-lsp 17058  df-lvec 17189  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-llines 33147  df-lplanes 33148  df-lvols 33149  df-lines 33150  df-psubsp 33152  df-pmap 33153  df-padd 33445  df-lhyp 33637  df-laut 33638  df-ldil 33753  df-ltrn 33754  df-trl 33808  df-tendo 34404  df-edring 34406  df-disoa 34679  df-dvech 34729  df-dib 34789  df-dic 34823  df-dih 34879
This theorem is referenced by:  dihjatcclem2  35069
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