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Theorem dihmeetbclemN 36502
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetc.b  |-  B  =  ( Base `  K
)
dihmeetc.l  |-  .<_  =  ( le `  K )
dihmeetc.m  |-  ./\  =  ( meet `  K )
dihmeetc.h  |-  H  =  ( LHyp `  K
)
dihmeetc.i  |-  I  =  ( ( DIsoH `  K
) `  W )
Assertion
Ref Expression
dihmeetbclemN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
I `  ( X  ./\ 
Y ) )  =  ( ( ( I `
 X )  i^i  ( I `  Y
) )  i^i  (
I `  W )
) )

Proof of Theorem dihmeetbclemN
StepHypRef Expression
1 simp3 998 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  ( X  ./\  Y )  .<_  W )
2 simp1l 1020 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  K  e.  HL )
3 hllat 34561 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  K  e.  Lat )
5 simp2l 1022 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  X  e.  B )
6 simp2r 1023 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  Y  e.  B )
7 dihmeetc.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
8 dihmeetc.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
97, 8latmcl 15556 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
104, 5, 6, 9syl3anc 1228 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  ( X  ./\  Y )  e.  B )
11 simp1r 1021 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  W  e.  H )
12 dihmeetc.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
137, 12lhpbase 35195 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
1411, 13syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  W  e.  B )
15 dihmeetc.l . . . . . . . 8  |-  .<_  =  ( le `  K )
167, 15, 8latleeqm1 15583 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  W  e.  B )  ->  (
( X  ./\  Y
)  .<_  W  <->  ( ( X  ./\  Y )  ./\  W )  =  ( X 
./\  Y ) ) )
174, 10, 14, 16syl3anc 1228 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
( X  ./\  Y
)  .<_  W  <->  ( ( X  ./\  Y )  ./\  W )  =  ( X 
./\  Y ) ) )
181, 17mpbid 210 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
( X  ./\  Y
)  ./\  W )  =  ( X  ./\  Y ) )
19 hlol 34559 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
202, 19syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  K  e.  OL )
217, 8latmassOLD 34427 . . . . . 6  |-  ( ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  W  e.  B
) )  ->  (
( X  ./\  Y
)  ./\  W )  =  ( X  ./\  ( Y  ./\  W ) ) )
2220, 5, 6, 14, 21syl13anc 1230 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
( X  ./\  Y
)  ./\  W )  =  ( X  ./\  ( Y  ./\  W ) ) )
2318, 22eqtr3d 2510 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  ( X  ./\  Y )  =  ( X  ./\  ( Y  ./\  W ) ) )
2423fveq2d 5876 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
I `  ( X  ./\ 
Y ) )  =  ( I `  ( X  ./\  ( Y  ./\  W ) ) ) )
25 simp1 996 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  ( K  e.  HL  /\  W  e.  H ) )
267, 8latmcl 15556 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  ./\  W
)  e.  B )
274, 6, 14, 26syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  ( Y  ./\  W )  e.  B )
287, 15, 8latmle2 15581 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  ./\  W
)  .<_  W )
294, 6, 14, 28syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  ( Y  ./\  W )  .<_  W )
30 dihmeetc.i . . . . 5  |-  I  =  ( ( DIsoH `  K
) `  W )
317, 15, 8, 12, 30dihmeetbN 36501 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( ( Y  ./\  W )  e.  B  /\  ( Y  ./\  W ) 
.<_  W ) )  -> 
( I `  ( X  ./\  ( Y  ./\  W ) ) )  =  ( ( I `  X )  i^i  (
I `  ( Y  ./\ 
W ) ) ) )
3225, 5, 27, 29, 31syl112anc 1232 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
I `  ( X  ./\  ( Y  ./\  W
) ) )  =  ( ( I `  X )  i^i  (
I `  ( Y  ./\ 
W ) ) ) )
337, 15latref 15557 . . . . . 6  |-  ( ( K  e.  Lat  /\  W  e.  B )  ->  W  .<_  W )
344, 14, 33syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  W  .<_  W )
357, 15, 8, 12, 30dihmeetbN 36501 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  B  /\  ( W  e.  B  /\  W  .<_  W ) )  ->  ( I `  ( Y  ./\  W
) )  =  ( ( I `  Y
)  i^i  ( I `  W ) ) )
3625, 6, 14, 34, 35syl112anc 1232 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
I `  ( Y  ./\ 
W ) )  =  ( ( I `  Y )  i^i  (
I `  W )
) )
3736ineq2d 3705 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
( I `  X
)  i^i  ( I `  ( Y  ./\  W
) ) )  =  ( ( I `  X )  i^i  (
( I `  Y
)  i^i  ( I `  W ) ) ) )
3824, 32, 373eqtrd 2512 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
I `  ( X  ./\ 
Y ) )  =  ( ( I `  X )  i^i  (
( I `  Y
)  i^i  ( I `  W ) ) ) )
39 inass 3713 . 2  |-  ( ( ( I `  X
)  i^i  ( I `  Y ) )  i^i  ( I `  W
) )  =  ( ( I `  X
)  i^i  ( (
I `  Y )  i^i  ( I `  W
) ) )
4038, 39syl6eqr 2526 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  .<_  W )  ->  (
I `  ( X  ./\ 
Y ) )  =  ( ( ( I `
 X )  i^i  ( I `  Y
) )  i^i  (
I `  W )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3480   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   meetcmee 15449   Latclat 15549   OLcol 34372   HLchlt 34548   LHypclh 35181   DIsoHcdih 36426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-undef 7014  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356  df-tendo 35952  df-edring 35954  df-disoa 36227  df-dvech 36277  df-dib 36337  df-dic 36371  df-dih 36427
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator