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Theorem cdlemk13 30208
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119.  O,  D are k1, f1. (Contributed by NM, 1-Jul-2013.)
Hypotheses
Ref Expression
cdlemk1.b  |-  B  =  ( Base `  K
)
cdlemk1.l  |-  .<_  =  ( le `  K )
cdlemk1.j  |-  .\/  =  ( join `  K )
cdlemk1.m  |-  ./\  =  ( meet `  K )
cdlemk1.a  |-  A  =  ( Atoms `  K )
cdlemk1.h  |-  H  =  ( LHyp `  K
)
cdlemk1.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk1.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk1.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk1.o  |-  O  =  ( S `  D
)
Assertion
Ref Expression
cdlemk13  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( O `  P )  =  ( ( P  .\/  ( R `  D )
)  ./\  ( ( N `  P )  .\/  ( R `  ( D  o.  `' F
) ) ) ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    D, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i
Allowed substitution hints:    A( f)    B( f, i)    S( f, i)    H( f)    K( f)    .<_ ( f)    O( f, i)

Proof of Theorem cdlemk13
StepHypRef Expression
1 cdlemk1.o . . 3  |-  O  =  ( S `  D
)
21fveq1i 5459 . 2  |-  ( O `
 P )  =  ( ( S `  D ) `  P
)
3 cdlemk1.b . . 3  |-  B  =  ( Base `  K
)
4 cdlemk1.l . . 3  |-  .<_  =  ( le `  K )
5 cdlemk1.j . . 3  |-  .\/  =  ( join `  K )
6 cdlemk1.a . . 3  |-  A  =  ( Atoms `  K )
7 cdlemk1.h . . 3  |-  H  =  ( LHyp `  K
)
8 cdlemk1.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemk1.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
10 cdlemk1.m . . 3  |-  ./\  =  ( meet `  K )
11 cdlemk1.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
123, 4, 5, 6, 7, 8, 9, 10, 11cdlemksv2 30203 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( ( S `
 D ) `  P )  =  ( ( P  .\/  ( R `  D )
)  ./\  ( ( N `  P )  .\/  ( R `  ( D  o.  `' F
) ) ) ) )
132, 12syl5eq 2302 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( O `  P )  =  ( ( P  .\/  ( R `  D )
)  ./\  ( ( N `  P )  .\/  ( R `  ( D  o.  `' F
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997    e. cmpt 4051    _I cid 4276   `'ccnv 4660    |` cres 4663    o. ccom 4665   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   Atomscatm 28620   HLchlt 28707   LHypclh 29340   LTrncltrn 29457   trLctrl 29514
This theorem is referenced by:  cdlemkole  30209  cdlemk14  30210  cdlemk13-2N  30232
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515
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