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Theorem dihjat 31684
Description: Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)
Hypotheses
Ref Expression
dihjat.h  |-  H  =  ( LHyp `  K
)
dihjat.j  |-  .\/  =  ( join `  K )
dihjat.a  |-  A  =  ( Atoms `  K )
dihjat.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihjat.s  |-  .(+)  =  (
LSSum `  U )
dihjat.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihjat.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dihjat.p  |-  ( ph  ->  P  e.  A )
dihjat.q  |-  ( ph  ->  Q  e.  A )
Assertion
Ref Expression
dihjat  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  Q ) ) )

Proof of Theorem dihjat
StepHypRef Expression
1 eqid 2366 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 dihjat.h . . 3  |-  H  =  ( LHyp `  K
)
3 dihjat.j . . 3  |-  .\/  =  ( join `  K )
4 dihjat.a . . 3  |-  A  =  ( Atoms `  K )
5 dihjat.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
6 dihjat.s . . 3  |-  .(+)  =  (
LSSum `  U )
7 dihjat.i . . 3  |-  I  =  ( ( DIsoH `  K
) `  W )
8 dihjat.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
98adantr 451 . . 3  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 dihjat.p . . . . 5  |-  ( ph  ->  P  e.  A )
1110adantr 451 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  P  e.  A
)
12 simprl 732 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  P ( le
`  K ) W )
1311, 12jca 518 . . 3  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  ( P  e.  A  /\  P ( le `  K ) W ) )
14 dihjat.q . . . . 5  |-  ( ph  ->  Q  e.  A )
1514adantr 451 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  Q  e.  A
)
16 simprr 733 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  Q ( le
`  K ) W )
1715, 16jca 518 . . 3  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  ( Q  e.  A  /\  Q ( le `  K ) W ) )
181, 2, 3, 4, 5, 6, 7, 9, 13, 17dihjatb 31677 . 2  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `  P ) 
.(+)  ( I `  Q ) ) )
19 eqid 2366 . . 3  |-  ( Base `  K )  =  (
Base `  K )
208adantr 451 . . 3  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2119, 4atbase 29550 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2210, 21syl 15 . . . . 5  |-  ( ph  ->  P  e.  ( Base `  K ) )
2322adantr 451 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  P  e.  ( Base `  K )
)
24 simprl 732 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  P ( le `  K ) W )
2523, 24jca 518 . . 3  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  ( P  e.  ( Base `  K
)  /\  P ( le `  K ) W ) )
2614adantr 451 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  Q  e.  A )
27 simprr 733 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  -.  Q
( le `  K
) W )
2826, 27jca 518 . . 3  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  ( Q  e.  A  /\  -.  Q
( le `  K
) W ) )
2919, 1, 2, 3, 4, 5, 6, 7, 20, 25, 28dihjatc 31678 . 2  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( ( I `  P
)  .(+)  ( I `  Q ) ) )
308adantr 451 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3119, 4atbase 29550 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3214, 31syl 15 . . . . . 6  |-  ( ph  ->  Q  e.  ( Base `  K ) )
3332adantr 451 . . . . 5  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  Q  e.  ( Base `  K )
)
34 simprr 733 . . . . 5  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  Q ( le `  K ) W )
3533, 34jca 518 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( Q  e.  ( Base `  K
)  /\  Q ( le `  K ) W ) )
3610adantr 451 . . . . 5  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  P  e.  A )
37 simprl 732 . . . . 5  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  -.  P
( le `  K
) W )
3836, 37jca 518 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( P  e.  A  /\  -.  P
( le `  K
) W ) )
3919, 1, 2, 3, 4, 5, 6, 7, 30, 35, 38dihjatc 31678 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( I `  ( Q  .\/  P
) )  =  ( ( I `  Q
)  .(+)  ( I `  P ) ) )
408simpld 445 . . . . . 6  |-  ( ph  ->  K  e.  HL )
413, 4hlatjcom 29628 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4240, 10, 14, 41syl3anc 1183 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4342fveq2d 5636 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( I `  ( Q  .\/  P ) ) )
4443adantr 451 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( I `  ( Q 
.\/  P ) ) )
452, 5, 8dvhlmod 31371 . . . . . 6  |-  ( ph  ->  U  e.  LMod )
46 lmodabl 15882 . . . . . 6  |-  ( U  e.  LMod  ->  U  e. 
Abel )
4745, 46syl 15 . . . . 5  |-  ( ph  ->  U  e.  Abel )
48 eqid 2366 . . . . . . . 8  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
4948lsssssubg 15925 . . . . . . 7  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
5045, 49syl 15 . . . . . 6  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
5119, 2, 7, 5, 48dihlss 31511 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  (
Base `  K )
)  ->  ( I `  P )  e.  (
LSubSp `  U ) )
528, 22, 51syl2anc 642 . . . . . 6  |-  ( ph  ->  ( I `  P
)  e.  ( LSubSp `  U ) )
5350, 52sseldd 3267 . . . . 5  |-  ( ph  ->  ( I `  P
)  e.  (SubGrp `  U ) )
5419, 2, 7, 5, 48dihlss 31511 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  (
Base `  K )
)  ->  ( I `  Q )  e.  (
LSubSp `  U ) )
558, 32, 54syl2anc 642 . . . . . 6  |-  ( ph  ->  ( I `  Q
)  e.  ( LSubSp `  U ) )
5650, 55sseldd 3267 . . . . 5  |-  ( ph  ->  ( I `  Q
)  e.  (SubGrp `  U ) )
576lsmcom 15360 . . . . 5  |-  ( ( U  e.  Abel  /\  (
I `  P )  e.  (SubGrp `  U )  /\  ( I `  Q
)  e.  (SubGrp `  U ) )  -> 
( ( I `  P )  .(+)  ( I `
 Q ) )  =  ( ( I `
 Q )  .(+)  ( I `  P ) ) )
5847, 53, 56, 57syl3anc 1183 . . . 4  |-  ( ph  ->  ( ( I `  P )  .(+)  ( I `
 Q ) )  =  ( ( I `
 Q )  .(+)  ( I `  P ) ) )
5958adantr 451 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( (
I `  P )  .(+)  ( I `  Q
) )  =  ( ( I `  Q
)  .(+)  ( I `  P ) ) )
6039, 44, 593eqtr4d 2408 . 2  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( ( I `  P
)  .(+)  ( I `  Q ) ) )
618adantr 451 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6210adantr 451 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  P  e.  A )
63 simprl 732 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  -.  P
( le `  K
) W )
6462, 63jca 518 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  ( P  e.  A  /\  -.  P
( le `  K
) W ) )
6514adantr 451 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  Q  e.  A )
66 simprr 733 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  -.  Q
( le `  K
) W )
6765, 66jca 518 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  ( Q  e.  A  /\  -.  Q
( le `  K
) W ) )
681, 2, 3, 4, 5, 6, 7, 61, 64, 67dihjatcc 31683 . 2  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( ( I `  P
)  .(+)  ( I `  Q ) ) )
6918, 29, 60, 684casesdan 916 1  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    C_ wss 3238   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288  SubGrpcsubg 14825   LSSumclsm 15155   Abelcabel 15300   LModclmod 15837   LSubSpclss 15899   Atomscatm 29524   HLchlt 29611   LHypclh 30244   DVecHcdvh 31339   DIsoHcdih 31489
This theorem is referenced by:  dihprrnlem2  31686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-fal 1325  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-tpos 6376  df-undef 6440  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-0g 13614  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-mnd 14577  df-submnd 14626  df-grp 14699  df-minusg 14700  df-sbg 14701  df-subg 14828  df-cntz 15003  df-lsm 15157  df-cmn 15301  df-abl 15302  df-mgp 15536  df-rng 15550  df-ur 15552  df-oppr 15615  df-dvdsr 15633  df-unit 15634  df-invr 15664  df-dvr 15675  df-drng 15724  df-lmod 15839  df-lss 15900  df-lsp 15939  df-lvec 16066  df-lsatoms 29237  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758  df-lplanes 29759  df-lvols 29760  df-lines 29761  df-psubsp 29763  df-pmap 29764  df-padd 30056  df-lhyp 30248  df-laut 30249  df-ldil 30364  df-ltrn 30365  df-trl 30419  df-tgrp 31003  df-tendo 31015  df-edring 31017  df-dveca 31263  df-disoa 31290  df-dvech 31340  df-dib 31400  df-dic 31434  df-dih 31490  df-doch 31609  df-djh 31656
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