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Theorem dihjat 34790
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 2441 . . 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 462 . . 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 462 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  P  e.  A
)
12 simprl 750 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  P ( le
`  K ) W )
1311, 12jca 529 . . 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 462 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  Q  e.  A
)
16 simprr 751 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  Q ( le
`  K ) W )
1715, 16jca 529 . . 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 34783 . 2  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  Q
( le `  K
) W ) )  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `  P ) 
.(+)  ( I `  Q ) ) )
19 eqid 2441 . . 3  |-  ( Base `  K )  =  (
Base `  K )
208adantr 462 . . 3  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2119, 4atbase 32656 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2210, 21syl 16 . . . . 5  |-  ( ph  ->  P  e.  ( Base `  K ) )
2322adantr 462 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  P  e.  ( Base `  K )
)
24 simprl 750 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  P ( le `  K ) W )
2523, 24jca 529 . . 3  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  ( P  e.  ( Base `  K
)  /\  P ( le `  K ) W ) )
2614adantr 462 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  Q  e.  A )
27 simprr 751 . . . 4  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  -.  Q
( le `  K
) W )
2826, 27jca 529 . . 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 34784 . 2  |-  ( (
ph  /\  ( P
( le `  K
) W  /\  -.  Q ( le `  K ) W ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( ( I `  P
)  .(+)  ( I `  Q ) ) )
308adantr 462 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3119, 4atbase 32656 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3214, 31syl 16 . . . . . 6  |-  ( ph  ->  Q  e.  ( Base `  K ) )
3332adantr 462 . . . . 5  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  Q  e.  ( Base `  K )
)
34 simprr 751 . . . . 5  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  Q ( le `  K ) W )
3533, 34jca 529 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( Q  e.  ( Base `  K
)  /\  Q ( le `  K ) W ) )
3610adantr 462 . . . . 5  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  P  e.  A )
37 simprl 750 . . . . 5  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  -.  P
( le `  K
) W )
3836, 37jca 529 . . . 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 34784 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( I `  ( Q  .\/  P
) )  =  ( ( I `  Q
)  .(+)  ( I `  P ) ) )
408simpld 456 . . . . . 6  |-  ( ph  ->  K  e.  HL )
413, 4hlatjcom 32734 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4240, 10, 14, 41syl3anc 1213 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4342fveq2d 5692 . . . 4  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( I `  ( Q  .\/  P ) ) )
4443adantr 462 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( I `  ( Q 
.\/  P ) ) )
452, 5, 8dvhlmod 34477 . . . . . 6  |-  ( ph  ->  U  e.  LMod )
46 lmodabl 16972 . . . . . 6  |-  ( U  e.  LMod  ->  U  e. 
Abel )
4745, 46syl 16 . . . . 5  |-  ( ph  ->  U  e.  Abel )
48 eqid 2441 . . . . . . . 8  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
4948lsssssubg 17017 . . . . . . 7  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
5045, 49syl 16 . . . . . 6  |-  ( ph  ->  ( LSubSp `  U )  C_  (SubGrp `  U )
)
5119, 2, 7, 5, 48dihlss 34617 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  (
Base `  K )
)  ->  ( I `  P )  e.  (
LSubSp `  U ) )
528, 22, 51syl2anc 656 . . . . . 6  |-  ( ph  ->  ( I `  P
)  e.  ( LSubSp `  U ) )
5350, 52sseldd 3354 . . . . 5  |-  ( ph  ->  ( I `  P
)  e.  (SubGrp `  U ) )
5419, 2, 7, 5, 48dihlss 34617 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  (
Base `  K )
)  ->  ( I `  Q )  e.  (
LSubSp `  U ) )
558, 32, 54syl2anc 656 . . . . . 6  |-  ( ph  ->  ( I `  Q
)  e.  ( LSubSp `  U ) )
5650, 55sseldd 3354 . . . . 5  |-  ( ph  ->  ( I `  Q
)  e.  (SubGrp `  U ) )
576lsmcom 16333 . . . . 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 1213 . . . 4  |-  ( ph  ->  ( ( I `  P )  .(+)  ( I `
 Q ) )  =  ( ( I `
 Q )  .(+)  ( I `  P ) ) )
5958adantr 462 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( (
I `  P )  .(+)  ( I `  Q
) )  =  ( ( I `  Q
)  .(+)  ( I `  P ) ) )
6039, 44, 593eqtr4d 2483 . 2  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  Q ( le `  K ) W ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( ( I `  P
)  .(+)  ( I `  Q ) ) )
618adantr 462 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6210adantr 462 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  P  e.  A )
63 simprl 750 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  -.  P
( le `  K
) W )
6462, 63jca 529 . . 3  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  ( P  e.  A  /\  -.  P
( le `  K
) W ) )
6514adantr 462 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  Q  e.  A )
66 simprr 751 . . . 4  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  -.  Q
( le `  K
) W )
6765, 66jca 529 . . 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 34789 . 2  |-  ( (
ph  /\  ( -.  P ( le `  K ) W  /\  -.  Q ( le `  K ) W ) )  ->  ( I `  ( P  .\/  Q
) )  =  ( ( I `  P
)  .(+)  ( I `  Q ) ) )
6918, 29, 60, 684casesdan 936 1  |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `
 P )  .(+)  ( I `  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    C_ wss 3325   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   lecple 14241   joincjn 15110  SubGrpcsubg 15668   LSSumclsm 16126   Abelcabel 16271   LModclmod 16928   LSubSpclss 16991   Atomscatm 32630   HLchlt 32717   LHypclh 33350   DVecHcdvh 34445   DIsoHcdih 34595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-riotaBAD 32326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-undef 6788  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-0g 14376  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-mnd 15411  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-cntz 15828  df-lsm 16128  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-lmod 16930  df-lss 16992  df-lsp 17031  df-lvec 17162  df-lsatoms 32343  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864  df-lplanes 32865  df-lvols 32866  df-lines 32867  df-psubsp 32869  df-pmap 32870  df-padd 33162  df-lhyp 33354  df-laut 33355  df-ldil 33470  df-ltrn 33471  df-trl 33525  df-tgrp 34109  df-tendo 34121  df-edring 34123  df-dveca 34369  df-disoa 34396  df-dvech 34446  df-dib 34506  df-dic 34540  df-dih 34596  df-doch 34715  df-djh 34762
This theorem is referenced by:  dihprrnlem2  34792
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