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Theorem dihmeetlem10N 34622
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 1056 . . . 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 1009 . . . 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 1010 . . . 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 770 . . . 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 764 . . . 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 34614 . . . 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 1272 . . 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 5876 . 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 1008 . . 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 32667 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
161, 15syl 17 . . . 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 32593 . . . . 5  |-  ( p  e.  A  ->  p  e.  B )
184, 17syl 17 . . . 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 16241 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  p  e.  B )  ->  ( Y  .\/  p
)  e.  B )
2016, 3, 18, 19syl3anc 1264 . . 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 34615 . . 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 34608 . . 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 1269 . 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 2463 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867    i^i cin 3432   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Basecbs 15073   lecple 15149   joincjn 16133   meetcmee 16134   Latclat 16235   LSSumclsm 17214   Atomscatm 32567   HLchlt 32654   LHypclh 33287   DVecHcdvh 34384   DIsoHcdih 34534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-riotaBAD 32263
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6972  df-undef 7019  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-sca 15158  df-vsca 15159  df-0g 15292  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-p0 16229  df-p1 16230  df-lat 16236  df-clat 16298  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-submnd 16527  df-grp 16617  df-minusg 16618  df-sbg 16619  df-subg 16758  df-cntz 16915  df-lsm 17216  df-cmn 17360  df-abl 17361  df-mgp 17652  df-ur 17664  df-ring 17710  df-oppr 17779  df-dvdsr 17797  df-unit 17798  df-invr 17828  df-dvr 17839  df-drng 17905  df-lmod 18021  df-lss 18084  df-lsp 18123  df-lvec 18254  df-oposet 32480  df-ol 32482  df-oml 32483  df-covers 32570  df-ats 32571  df-atl 32602  df-cvlat 32626  df-hlat 32655  df-llines 32801  df-lplanes 32802  df-lvols 32803  df-lines 32804  df-psubsp 32806  df-pmap 32807  df-padd 33099  df-lhyp 33291  df-laut 33292  df-ldil 33407  df-ltrn 33408  df-trl 33463  df-tendo 34060  df-edring 34062  df-disoa 34335  df-dvech 34385  df-dib 34445  df-dic 34479  df-dih 34535
This theorem is referenced by:  dihmeetlem11N  34623
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