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Theorem dihord2cN 31411
Description: Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihjust.b  |-  B  =  ( Base `  K
)
dihjust.l  |-  .<_  =  ( le `  K )
dihjust.j  |-  .\/  =  ( join `  K )
dihjust.m  |-  ./\  =  ( meet `  K )
dihjust.a  |-  A  =  ( Atoms `  K )
dihjust.h  |-  H  =  ( LHyp `  K
)
dihjust.i  |-  I  =  ( ( DIsoB `  K
) `  W )
dihjust.J  |-  J  =  ( ( DIsoC `  K
) `  W )
dihjust.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihjust.s  |-  .(+)  =  (
LSSum `  U )
dihord2c.t  |-  T  =  ( ( LTrn `  K
) `  W )
dihord2c.r  |-  R  =  ( ( trL `  K
) `  W )
dihord2c.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
dihord2cN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  <. f ,  O >.  e.  ( I `  ( X  ./\  W ) ) )
Distinct variable groups:    .\/ , f    ./\ , f    .(+) ,
f    f, h, A    f, I    f, J    R, f    B, f, h    f, H, h    f, K, h    .<_ , f, h    T, f, h    f, W, h   
f, X
Allowed substitution hints:    .(+) ( h)    R( h)    U( f, h)    I( h)    J( h)    .\/ ( h)    ./\ ( h)    O( f, h)    X( h)

Proof of Theorem dihord2cN
StepHypRef Expression
1 simp3 957 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )
2 eqidd 2284 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  O  =  O )
3 simp1 955 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4 simp1l 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  K  e.  HL )
5 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  K  e.  Lat )
7 simp2 956 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  X  e.  B )
8 simp1r 980 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  W  e.  H )
9 dihjust.b . . . . . 6  |-  B  =  ( Base `  K
)
10 dihjust.h . . . . . 6  |-  H  =  ( LHyp `  K
)
119, 10lhpbase 30187 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
128, 11syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  W  e.  B )
13 dihjust.m . . . . 5  |-  ./\  =  ( meet `  K )
149, 13latmcl 14157 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
156, 7, 12, 14syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( X  ./\  W
)  e.  B )
16 dihjust.l . . . . 5  |-  .<_  =  ( le `  K )
179, 16, 13latmle2 14183 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
186, 7, 12, 17syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( X  ./\  W
)  .<_  W )
19 dihord2c.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
20 dihord2c.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
21 dihord2c.o . . . 4  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
22 dihjust.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
239, 16, 10, 19, 20, 21, 22dibopelval3 31338 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( X 
./\  W )  e.  B  /\  ( X 
./\  W )  .<_  W ) )  -> 
( <. f ,  O >.  e.  ( I `  ( X  ./\  W ) )  <->  ( ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W
) )  /\  O  =  O ) ) )
243, 15, 18, 23syl12anc 1180 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  -> 
( <. f ,  O >.  e.  ( I `  ( X  ./\  W ) )  <->  ( ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W
) )  /\  O  =  O ) ) )
251, 2, 24mpbir2and 888 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f
)  .<_  ( X  ./\  W ) ) )  ->  <. f ,  O >.  e.  ( I `  ( X  ./\  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023    e. cmpt 4077    _I cid 4304    |` cres 4691   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   LSSumclsm 14945   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   DVecHcdvh 31268   DIsoBcdib 31328   DIsoCcdic 31362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-glb 14109  df-meet 14111  df-lat 14152  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-disoa 31219  df-dib 31329
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