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Theorem dihmeetlem10N 34980
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem9.b  |-  B  =  ( Base `  K
)
dihmeetlem9.l  |-  .<_  =  ( le `  K )
dihmeetlem9.h  |-  H  =  ( LHyp `  K
)
dihmeetlem9.j  |-  .\/  =  ( join `  K )
dihmeetlem9.m  |-  ./\  =  ( meet `  K )
dihmeetlem9.a  |-  A  =  ( Atoms `  K )
dihmeetlem9.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihmeetlem9.s  |-  .(+)  =  (
LSSum `  U )
dihmeetlem9.i  |-  I  =  ( ( DIsoH `  K
) `  W )
Assertion
Ref Expression
dihmeetlem10N  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( I `  ( ( X  ./\  Y )  .\/  p ) )  =  ( ( I `  X )  i^i  ( I `  ( Y  .\/  p ) ) ) )

Proof of Theorem dihmeetlem10N
StepHypRef Expression
1 simpl1l 1039 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  K  e.  HL )
2 simpl2 992 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  X  e.  B )
3 simpl3 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  Y  e.  B )
4 simprll 761 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  p  e.  A )
5 simprr 756 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  p  .<_  X )
6 dihmeetlem9.b . . . . 5  |-  B  =  ( Base `  K
)
7 dihmeetlem9.l . . . . 5  |-  .<_  =  ( le `  K )
8 dihmeetlem9.j . . . . 5  |-  .\/  =  ( join `  K )
9 dihmeetlem9.m . . . . 5  |-  ./\  =  ( meet `  K )
10 dihmeetlem9.a . . . . 5  |-  A  =  ( Atoms `  K )
116, 7, 8, 9, 10dihmeetlem5 34972 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  p  .<_  X ) )  ->  ( X  ./\  ( Y  .\/  p
) )  =  ( ( X  ./\  Y
)  .\/  p )
)
121, 2, 3, 4, 5, 11syl32anc 1226 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( X  ./\  ( Y  .\/  p
) )  =  ( ( X  ./\  Y
)  .\/  p )
)
1312fveq2d 5710 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( I `  ( X  ./\  ( Y  .\/  p ) ) )  =  ( I `
 ( ( X 
./\  Y )  .\/  p ) ) )
14 simpl1 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
15 hllat 33027 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
161, 15syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  K  e.  Lat )
176, 10atbase 32953 . . . . 5  |-  ( p  e.  A  ->  p  e.  B )
184, 17syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  p  e.  B )
196, 8latjcl 15236 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  p  e.  B )  ->  ( Y  .\/  p
)  e.  B )
2016, 3, 18, 19syl3anc 1218 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( Y  .\/  p )  e.  B
)
21 dihmeetlem9.h . . . 4  |-  H  =  ( LHyp `  K
)
226, 7, 21, 8, 9, 10dihmeetlem6 34973 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  -.  ( X  ./\  ( Y  .\/  p ) )  .<_  W )
23 dihmeetlem9.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
246, 7, 9, 21, 23dihmeetcN 34966 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  ( Y 
.\/  p )  e.  B )  /\  -.  ( X  ./\  ( Y 
.\/  p ) ) 
.<_  W )  ->  (
I `  ( X  ./\  ( Y  .\/  p
) ) )  =  ( ( I `  X )  i^i  (
I `  ( Y  .\/  p ) ) ) )
2514, 2, 20, 22, 24syl121anc 1223 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( I `  ( X  ./\  ( Y  .\/  p ) ) )  =  ( ( I `  X )  i^i  ( I `  ( Y  .\/  p ) ) ) )
2613, 25eqtr3d 2477 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) )  ->  ( I `  ( ( X  ./\  Y )  .\/  p ) )  =  ( ( I `  X )  i^i  ( I `  ( Y  .\/  p ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3342   class class class wbr 4307   ` cfv 5433  (class class class)co 6106   Basecbs 14189   lecple 14260   joincjn 15129   meetcmee 15130   Latclat 15230   LSSumclsm 16148   Atomscatm 32927   HLchlt 33014   LHypclh 33647   DVecHcdvh 34742   DIsoHcdih 34892
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 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-riotaBAD 32623
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 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-tpos 6760  df-undef 6807  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-n0 10595  df-z 10662  df-uz 10877  df-fz 11453  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-sca 14269  df-vsca 14270  df-0g 14395  df-poset 15131  df-plt 15143  df-lub 15159  df-glb 15160  df-join 15161  df-meet 15162  df-p0 15224  df-p1 15225  df-lat 15231  df-clat 15293  df-mnd 15430  df-submnd 15480  df-grp 15560  df-minusg 15561  df-sbg 15562  df-subg 15693  df-cntz 15850  df-lsm 16150  df-cmn 16294  df-abl 16295  df-mgp 16607  df-ur 16619  df-rng 16662  df-oppr 16730  df-dvdsr 16748  df-unit 16749  df-invr 16779  df-dvr 16790  df-drng 16849  df-lmod 16965  df-lss 17029  df-lsp 17068  df-lvec 17199  df-oposet 32840  df-ol 32842  df-oml 32843  df-covers 32930  df-ats 32931  df-atl 32962  df-cvlat 32986  df-hlat 33015  df-llines 33161  df-lplanes 33162  df-lvols 33163  df-lines 33164  df-psubsp 33166  df-pmap 33167  df-padd 33459  df-lhyp 33651  df-laut 33652  df-ldil 33767  df-ltrn 33768  df-trl 33822  df-tendo 34418  df-edring 34420  df-disoa 34693  df-dvech 34743  df-dib 34803  df-dic 34837  df-dih 34893
This theorem is referenced by:  dihmeetlem11N  34981
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