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Theorem dihjatcclem1 31585
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 31277 . . . 4  |-  ( ph  ->  U  e.  LMod )
5 lmodabl 15912 . . . 4  |-  ( U  e.  LMod  ->  U  e. 
Abel )
64, 5syl 16 . . 3  |-  ( ph  ->  U  e.  Abel )
7 eqid 2381 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
87lsssssubg 15955 . . . . 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 446 . . . . . 6  |-  ( ph  ->  P  e.  A )
12 dihjatcclem.b . . . . . . 7  |-  B  =  ( Base `  K
)
13 dihjatcclem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
1412, 13atbase 29456 . . . . . 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 31417 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  B
)  ->  ( I `  P )  e.  (
LSubSp `  U ) )
183, 15, 17syl2anc 643 . . . 4  |-  ( ph  ->  ( I `  P
)  e.  ( LSubSp `  U ) )
199, 18sseldd 3286 . . 3  |-  ( ph  ->  ( I `  P
)  e.  (SubGrp `  U ) )
20 dihjatcclem.v . . . . . 6  |-  V  =  ( ( P  .\/  Q )  ./\  W )
213simpld 446 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
22 hllat 29530 . . . . . . . 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 446 . . . . . . . 8  |-  ( ph  ->  Q  e.  A )
26 dihjatcclem.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
2712, 26, 13hlatjcl 29533 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
2821, 11, 25, 27syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( P  .\/  Q
)  e.  B )
293simprd 450 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
3012, 1lhpbase 30164 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
3129, 30syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  B )
32 dihjatcclem.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
3312, 32latmcl 14401 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  Q
)  ./\  W )  e.  B )
3423, 28, 31, 33syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  e.  B )
3520, 34syl5eqel 2465 . . . . 5  |-  ( ph  ->  V  e.  B )
3612, 1, 16, 2, 7dihlss 31417 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  B
)  ->  ( I `  V )  e.  (
LSubSp `  U ) )
373, 35, 36syl2anc 643 . . . 4  |-  ( ph  ->  ( I `  V
)  e.  ( LSubSp `  U ) )
389, 37sseldd 3286 . . 3  |-  ( ph  ->  ( I `  V
)  e.  (SubGrp `  U ) )
3912, 13atbase 29456 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
4025, 39syl 16 . . . . 5  |-  ( ph  ->  Q  e.  B )
4112, 1, 16, 2, 7dihlss 31417 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( I `  Q )  e.  (
LSubSp `  U ) )
423, 40, 41syl2anc 643 . . . 4  |-  ( ph  ->  ( I `  Q
)  e.  ( LSubSp `  U ) )
439, 42sseldd 3286 . . 3  |-  ( ph  ->  ( I `  Q
)  e.  (SubGrp `  U ) )
44 dihjatcclem.s . . . 4  |-  .(+)  =  (
LSSum `  U )
4544lsm4 15396 . . 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 1193 . 2  |-  ( ph  ->  ( ( ( I `
 P )  .(+)  ( I `  V ) )  .(+)  ( (
I `  Q )  .(+)  ( I `  V
) ) )  =  ( ( ( I `
 P )  .(+)  ( I `  Q ) )  .(+)  ( (
I `  V )  .(+)  ( I `  V
) ) ) )
4724simprd 450 . . . . . . . 8  |-  ( ph  ->  -.  Q  .<_  W )
4847intnand 883 . . . . . . 7  |-  ( ph  ->  -.  ( P  .<_  W  /\  Q  .<_  W ) )
49 dihjatcclem.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
5012, 49, 26latjle12 14412 . . . . . . . 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 1186 . . . . . . 7  |-  ( ph  ->  ( ( P  .<_  W  /\  Q  .<_  W )  <-> 
( P  .\/  Q
)  .<_  W ) )
5248, 51mtbid 292 . . . . . 6  |-  ( ph  ->  -.  ( P  .\/  Q )  .<_  W )
5349, 26, 13hlatlej1 29541 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
5421, 11, 25, 53syl3anc 1184 . . . . . 6  |-  ( ph  ->  P  .<_  ( P  .\/  Q ) )
5512, 49, 26, 32, 13, 1, 16, 2, 44dihvalcq2 31414 . . . . . 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 1193 . . . . 5  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  ( ( P  .\/  Q ) 
./\  W ) ) ) )
5720fveq2i 5665 . . . . . 6  |-  ( I `
 V )  =  ( I `  (
( P  .\/  Q
)  ./\  W )
)
5857oveq2i 6025 . . . . 5  |-  ( ( I `  P ) 
.(+)  ( I `  V ) )  =  ( ( I `  P )  .(+)  ( I `
 ( ( P 
.\/  Q )  ./\  W ) ) )
5956, 58syl6eqr 2431 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  V ) ) )
6049, 26, 13hlatlej2 29542 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
6121, 11, 25, 60syl3anc 1184 . . . . . 6  |-  ( ph  ->  Q  .<_  ( P  .\/  Q ) )
6212, 49, 26, 32, 13, 1, 16, 2, 44dihvalcq2 31414 . . . . . 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 1193 . . . . 5  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 Q )  .(+)  ( I `  ( ( P  .\/  Q ) 
./\  W ) ) ) )
6457oveq2i 6025 . . . . 5  |-  ( ( I `  Q ) 
.(+)  ( I `  V ) )  =  ( ( I `  Q )  .(+)  ( I `
 ( ( P 
.\/  Q )  ./\  W ) ) )
6563, 64syl6eqr 2431 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 Q )  .(+)  ( I `  V ) ) )
6659, 65oveq12d 6032 . . 3  |-  ( ph  ->  ( ( I `  ( P  .\/  Q ) )  .(+)  ( I `  ( P  .\/  Q
) ) )  =  ( ( ( I `
 P )  .(+)  ( I `  V ) )  .(+)  ( (
I `  Q )  .(+)  ( I `  V
) ) ) )
6712, 1, 16, 2, 7dihlss 31417 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  .\/  Q )  e.  B )  ->  ( I `  ( P  .\/  Q ) )  e.  ( LSubSp `  U ) )
683, 28, 67syl2anc 643 . . . . 5  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  e.  ( LSubSp `  U
) )
699, 68sseldd 3286 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  e.  (SubGrp `  U
) )
7044lsmidm 15217 . . . 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 2415 . 2  |-  ( ph  ->  ( ( ( I `
 P )  .(+)  ( I `  V ) )  .(+)  ( (
I `  Q )  .(+)  ( I `  V
) ) )  =  ( I `  ( P  .\/  Q ) ) )
7344lsmidm 15217 . . . 4  |-  ( ( I `  V )  e.  (SubGrp `  U
)  ->  ( (
I `  V )  .(+)  ( I `  V
) )  =  ( I `  V ) )
7438, 73syl 16 . . 3  |-  ( ph  ->  ( ( I `  V )  .(+)  ( I `
 V ) )  =  ( I `  V ) )
7574oveq2d 6030 . 2  |-  ( ph  ->  ( ( ( I `
 P )  .(+)  ( I `  Q ) )  .(+)  ( (
I `  V )  .(+)  ( I `  V
) ) )  =  ( ( ( I `
 P )  .(+)  ( I `  Q ) )  .(+)  ( I `  V ) ) )
7646, 72, 753eqtr3d 2421 1  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( ( I `  P ) 
.(+)  ( I `  Q ) )  .(+)  ( I `  V ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3257   class class class wbr 4147   ` cfv 5388  (class class class)co 6014   Basecbs 13390   lecple 13457   joincjn 14322   meetcmee 14323   Latclat 14395  SubGrpcsubg 14859   LSSumclsm 15189   Abelcabel 15334   LModclmod 15871   LSubSpclss 15929   Atomscatm 29430   HLchlt 29517   LHypclh 30150   DVecHcdvh 31245   DIsoHcdih 31395
This theorem is referenced by:  dihjatcclem2  31586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-iin 4032  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-tpos 6409  df-undef 6473  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-oadd 6658  df-er 6835  df-map 6950  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-nn 9927  df-2 9984  df-3 9985  df-4 9986  df-5 9987  df-6 9988  df-n0 10148  df-z 10209  df-uz 10415  df-fz 10970  df-struct 13392  df-ndx 13393  df-slot 13394  df-base 13395  df-sets 13396  df-ress 13397  df-plusg 13463  df-mulr 13464  df-sca 13466  df-vsca 13467  df-0g 13648  df-poset 14324  df-plt 14336  df-lub 14352  df-glb 14353  df-join 14354  df-meet 14355  df-p0 14389  df-p1 14390  df-lat 14396  df-clat 14458  df-mnd 14611  df-submnd 14660  df-grp 14733  df-minusg 14734  df-sbg 14735  df-subg 14862  df-cntz 15037  df-lsm 15191  df-cmn 15335  df-abl 15336  df-mgp 15570  df-rng 15584  df-ur 15586  df-oppr 15649  df-dvdsr 15667  df-unit 15668  df-invr 15698  df-dvr 15709  df-drng 15758  df-lmod 15873  df-lss 15930  df-lsp 15969  df-lvec 16096  df-oposet 29343  df-ol 29345  df-oml 29346  df-covers 29433  df-ats 29434  df-atl 29465  df-cvlat 29489  df-hlat 29518  df-llines 29664  df-lplanes 29665  df-lvols 29666  df-lines 29667  df-psubsp 29669  df-pmap 29670  df-padd 29962  df-lhyp 30154  df-laut 30155  df-ldil 30270  df-ltrn 30271  df-trl 30325  df-tendo 30921  df-edring 30923  df-disoa 31196  df-dvech 31246  df-dib 31306  df-dic 31340  df-dih 31396
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