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Theorem dihjatcclem2 35370
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 ) )
dihjatcclem2.c  |-  ( ph  ->  ( I `  V
)  C_  ( (
I `  P )  .(+)  ( I `  Q
) ) )
Assertion
Ref Expression
dihjatcclem2  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  Q ) ) )

Proof of Theorem dihjatcclem2
StepHypRef Expression
1 dihjatcclem.b . . 3  |-  B  =  ( Base `  K
)
2 dihjatcclem.l . . 3  |-  .<_  =  ( le `  K )
3 dihjatcclem.h . . 3  |-  H  =  ( LHyp `  K
)
4 dihjatcclem.j . . 3  |-  .\/  =  ( join `  K )
5 dihjatcclem.m . . 3  |-  ./\  =  ( meet `  K )
6 dihjatcclem.a . . 3  |-  A  =  ( Atoms `  K )
7 dihjatcclem.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
8 dihjatcclem.s . . 3  |-  .(+)  =  (
LSSum `  U )
9 dihjatcclem.i . . 3  |-  I  =  ( ( DIsoH `  K
) `  W )
10 dihjatcclem.v . . 3  |-  V  =  ( ( P  .\/  Q )  ./\  W )
11 dihjatcclem.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
12 dihjatcclem.p . . 3  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
13 dihjatcclem.q . . 3  |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dihjatcclem1 35369 . 2  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( ( I `  P ) 
.(+)  ( I `  Q ) )  .(+)  ( I `  V ) ) )
153, 7, 11dvhlmod 35061 . . . . 5  |-  ( ph  ->  U  e.  LMod )
16 eqid 2451 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
1716lsssssubg 17145 . . . . 5  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
1815, 17syl 16 . . . 4  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
1912simpld 459 . . . . . . 7  |-  ( ph  ->  P  e.  A )
201, 6atbase 33240 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
2119, 20syl 16 . . . . . 6  |-  ( ph  ->  P  e.  B )
221, 3, 9, 7, 16dihlss 35201 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  B
)  ->  ( I `  P )  e.  (
LSubSp `  U ) )
2311, 21, 22syl2anc 661 . . . . 5  |-  ( ph  ->  ( I `  P
)  e.  ( LSubSp `  U ) )
2413simpld 459 . . . . . . 7  |-  ( ph  ->  Q  e.  A )
251, 6atbase 33240 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
2624, 25syl 16 . . . . . 6  |-  ( ph  ->  Q  e.  B )
271, 3, 9, 7, 16dihlss 35201 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  B
)  ->  ( I `  Q )  e.  (
LSubSp `  U ) )
2811, 26, 27syl2anc 661 . . . . 5  |-  ( ph  ->  ( I `  Q
)  e.  ( LSubSp `  U ) )
2916, 8lsmcl 17270 . . . . 5  |-  ( ( U  e.  LMod  /\  (
I `  P )  e.  ( LSubSp `  U )  /\  ( I `  Q
)  e.  ( LSubSp `  U ) )  -> 
( ( I `  P )  .(+)  ( I `
 Q ) )  e.  ( LSubSp `  U
) )
3015, 23, 28, 29syl3anc 1219 . . . 4  |-  ( ph  ->  ( ( I `  P )  .(+)  ( I `
 Q ) )  e.  ( LSubSp `  U
) )
3118, 30sseldd 3455 . . 3  |-  ( ph  ->  ( ( I `  P )  .(+)  ( I `
 Q ) )  e.  (SubGrp `  U
) )
3210fveq2i 5792 . . . . 5  |-  ( I `
 V )  =  ( I `  (
( P  .\/  Q
)  ./\  W )
)
3311simpld 459 . . . . . . . 8  |-  ( ph  ->  K  e.  HL )
34 hllat 33314 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
3533, 34syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
361, 4, 6hlatjcl 33317 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
3733, 19, 24, 36syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( P  .\/  Q
)  e.  B )
3811simprd 463 . . . . . . . 8  |-  ( ph  ->  W  e.  H )
391, 3lhpbase 33948 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
4038, 39syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  B )
411, 5latmcl 15324 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  Q
)  ./\  W )  e.  B )
4235, 37, 40, 41syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  e.  B )
431, 3, 9, 7, 16dihlss 35201 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P 
.\/  Q )  ./\  W )  e.  B )  ->  ( I `  ( ( P  .\/  Q )  ./\  W )
)  e.  ( LSubSp `  U ) )
4411, 42, 43syl2anc 661 . . . . 5  |-  ( ph  ->  ( I `  (
( P  .\/  Q
)  ./\  W )
)  e.  ( LSubSp `  U ) )
4532, 44syl5eqel 2543 . . . 4  |-  ( ph  ->  ( I `  V
)  e.  ( LSubSp `  U ) )
4618, 45sseldd 3455 . . 3  |-  ( ph  ->  ( I `  V
)  e.  (SubGrp `  U ) )
47 dihjatcclem2.c . . 3  |-  ( ph  ->  ( I `  V
)  C_  ( (
I `  P )  .(+)  ( I `  Q
) ) )
488lsmss2 16269 . . 3  |-  ( ( ( ( I `  P )  .(+)  ( I `
 Q ) )  e.  (SubGrp `  U
)  /\  ( I `  V )  e.  (SubGrp `  U )  /\  (
I `  V )  C_  ( ( I `  P )  .(+)  ( I `
 Q ) ) )  ->  ( (
( I `  P
)  .(+)  ( I `  Q ) )  .(+)  ( I `  V ) )  =  ( ( I `  P ) 
.(+)  ( I `  Q ) ) )
4931, 46, 47, 48syl3anc 1219 . 2  |-  ( ph  ->  ( ( ( I `
 P )  .(+)  ( I `  Q ) )  .(+)  ( I `  V ) )  =  ( ( I `  P )  .(+)  ( I `
 Q ) ) )
5014, 49eqtrd 2492 1  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3426   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   meetcmee 15217   Latclat 15317  SubGrpcsubg 15777   LSSumclsm 16237   LModclmod 17054   LSubSpclss 17119   Atomscatm 33214   HLchlt 33301   LHypclh 33934   DVecHcdvh 35029   DIsoHcdih 35179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-riotaBAD 32910
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-tpos 6845  df-undef 6892  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-0g 14482  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-p1 15312  df-lat 15318  df-clat 15380  df-mnd 15517  df-submnd 15567  df-grp 15647  df-minusg 15648  df-sbg 15649  df-subg 15780  df-cntz 15937  df-lsm 16239  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-oppr 16821  df-dvdsr 16839  df-unit 16840  df-invr 16870  df-dvr 16881  df-drng 16940  df-lmod 17056  df-lss 17120  df-lsp 17159  df-lvec 17290  df-oposet 33127  df-ol 33129  df-oml 33130  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302  df-llines 33448  df-lplanes 33449  df-lvols 33450  df-lines 33451  df-psubsp 33453  df-pmap 33454  df-padd 33746  df-lhyp 33938  df-laut 33939  df-ldil 34054  df-ltrn 34055  df-trl 34109  df-tendo 34705  df-edring 34707  df-disoa 34980  df-dvech 35030  df-dib 35090  df-dic 35124  df-dih 35180
This theorem is referenced by:  dihjatcc  35373
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