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Theorem cdlemn3 30517
Description: Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.)
Hypotheses
Ref Expression
cdlemn3.l  |-  .<_  =  ( le `  K )
cdlemn3.a  |-  A  =  ( Atoms `  K )
cdlemn3.p  |-  P  =  ( ( oc `  K ) `  W
)
cdlemn3.h  |-  H  =  ( LHyp `  K
)
cdlemn3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn3.f  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
cdlemn3.g  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  R )
cdlemn3.j  |-  J  =  ( iota_ h  e.  T
( h `  Q
)  =  R )
Assertion
Ref Expression
cdlemn3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( J  o.  F )  =  G )
Distinct variable groups:    .<_ , h    A, h    h, H    h, K    P, h    Q, h    R, h    T, h    h, W
Allowed substitution hints:    F( h)    G( h)    J( h)

Proof of Theorem cdlemn3
StepHypRef Expression
1 simp1 960 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 cdlemn3.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
3 cdlemn3.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
4 cdlemn3.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
5 cdlemn3.p . . . . . . . . . 10  |-  P  =  ( ( oc `  K ) `  W
)
62, 3, 4, 5lhpocnel2 29338 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
763ad2ant1 981 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp2 961 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
9 cdlemn3.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
10 cdlemn3.f . . . . . . . . 9  |-  F  =  ( iota_ h  e.  T
( h `  P
)  =  Q )
112, 3, 4, 9, 10ltrniotacl 29898 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
121, 7, 8, 11syl3anc 1187 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  F  e.  T )
13 eqid 2256 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1413, 4, 9ltrn1o 29443 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
151, 12, 14syl2anc 645 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
16 f1of 5375 . . . . . 6  |-  ( F : ( Base `  K
)
-1-1-onto-> ( Base `  K )  ->  F : ( Base `  K ) --> ( Base `  K ) )
1715, 16syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  F :
( Base `  K ) --> ( Base `  K )
)
187simpld 447 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  P  e.  A )
1913, 3atbase 28609 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2018, 19syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  P  e.  ( Base `  K )
)
21 fvco3 5495 . . . . 5  |-  ( ( F : ( Base `  K ) --> ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  (
( J  o.  F
) `  P )  =  ( J `  ( F `  P ) ) )
2217, 20, 21syl2anc 645 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( J  o.  F ) `  P )  =  ( J `  ( F `
 P ) ) )
232, 3, 4, 9, 10ltrniotaval 29900 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( F `  P )  =  Q )
241, 7, 8, 23syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( F `  P )  =  Q )
2524fveq2d 5427 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( J `  ( F `  P
) )  =  ( J `  Q ) )
26 cdlemn3.j . . . . 5  |-  J  =  ( iota_ h  e.  T
( h `  Q
)  =  R )
272, 3, 4, 9, 26ltrniotaval 29900 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( J `  Q )  =  R )
2822, 25, 273eqtrd 2292 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( J  o.  F ) `  P )  =  R )
29 cdlemn3.g . . . . 5  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  R )
302, 3, 4, 9, 29ltrniotaval 29900 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( G `  P )  =  R )
317, 30syld3an2 1234 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( G `  P )  =  R )
3228, 31eqtr4d 2291 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( J  o.  F ) `  P )  =  ( G `  P ) )
332, 3, 4, 9, 26ltrniotacl 29898 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  J  e.  T )
344, 9ltrnco 30038 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  J  e.  T  /\  F  e.  T
)  ->  ( J  o.  F )  e.  T
)
351, 33, 12, 34syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( J  o.  F )  e.  T
)
362, 3, 4, 9, 29ltrniotacl 29898 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  G  e.  T )
377, 36syld3an2 1234 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  G  e.  T )
382, 3, 4, 9ltrneq3 29527 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( J  o.  F )  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( J  o.  F
) `  P )  =  ( G `  P )  <->  ( J  o.  F )  =  G ) )
391, 35, 37, 7, 38syl121anc 1192 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( (
( J  o.  F
) `  P )  =  ( G `  P )  <->  ( J  o.  F )  =  G ) )
4032, 39mpbid 203 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( J  o.  F )  =  G )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3963    o. ccom 4630   -->wf 4634   -1-1-onto->wf1o 4637   ` cfv 4638   iota_crio 6228   Basecbs 13075   lecple 13142   occoc 13143   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420
This theorem is referenced by:  cdlemn4  30518
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478
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