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Theorem cdlemk20-2N 31150
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be fi. Our  D,  C,  O,  Q,  U,  V represent their f1, f2, k1, k2, sigma1, sigma2. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk2.b  |-  B  =  ( Base `  K
)
cdlemk2.l  |-  .<_  =  ( le `  K )
cdlemk2.j  |-  .\/  =  ( join `  K )
cdlemk2.m  |-  ./\  =  ( meet `  K )
cdlemk2.a  |-  A  =  ( Atoms `  K )
cdlemk2.h  |-  H  =  ( LHyp `  K
)
cdlemk2.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk2.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk2.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk2.q  |-  Q  =  ( S `  C
)
cdlemk2.v  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T
( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
cdlemk2a.o  |-  O  =  ( S `  D
)
Assertion
Ref Expression
cdlemk20-2N  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( V `  D ) `  P )  =  ( O `  P ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    C, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i    ./\ , d    .\/ , d    C, d, k    Q, d    P, d    R, d    T, d    W, d    ./\ , k    .<_ , k    .\/ , k    A, k    C, k   
k, F    k, H    k, K    k, N    Q, k    P, k    R, k    T, k    k, W    F, d    i, k, f, D, d
Allowed substitution hints:    A( f, d)    B( f, i, k, d)    Q( f, i)    S( f, i, k, d)    H( f, d)    K( f, d)    .<_ ( f, d)    N( d)    O( f, i, k, d)    V( f, i, k, d)

Proof of Theorem cdlemk20-2N
StepHypRef Expression
1 simp11 985 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
2 simp12 986 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H )
31, 2jca 518 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp211 1093 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T )
5 simp212 1094 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  e.  T )
6 simp213 1095 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  N  e.  T )
7 simp22l 1074 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  D  e.  T )
86, 7jca 518 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( N  e.  T  /\  D  e.  T ) )
9 simp33 993 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
10 simp13 987 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  F )  =  ( R `  N ) )
11 simp32l 1080 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
12 simp32r 1081 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  =/=  (  _I  |`  B ) )
13 simp22r 1075 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  D  =/=  (  _I  |`  B ) )
1411, 12, 133jca 1132 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
15 simp31 991 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( R `  C )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 C ) ) )
16 cdlemk2.b . . 3  |-  B  =  ( Base `  K
)
17 cdlemk2.l . . 3  |-  .<_  =  ( le `  K )
18 cdlemk2.j . . 3  |-  .\/  =  ( join `  K )
19 cdlemk2.m . . 3  |-  ./\  =  ( meet `  K )
20 cdlemk2.a . . 3  |-  A  =  ( Atoms `  K )
21 cdlemk2.h . . 3  |-  H  =  ( LHyp `  K
)
22 cdlemk2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
23 cdlemk2.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
24 cdlemk2.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
25 cdlemk2.q . . 3  |-  Q  =  ( S `  C
)
26 cdlemk2.v . . 3  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T
( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
27 cdlemk2a.o . . 3  |-  O  =  ( S `  D
)
2816, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27cdlemk20 31132 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  C  e.  T )  /\  (
( N  e.  T  /\  D  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  C )
) ) )  -> 
( ( V `  D ) `  P
)  =  ( O `
 P ) )
293, 4, 5, 8, 9, 10, 14, 15, 28syl332anc 1213 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( V `  D ) `  P )  =  ( O `  P ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4104    e. cmpt 4158    _I cid 4386   `'ccnv 4770    |` cres 4773    o. ccom 4775   ` cfv 5337  (class class class)co 5945   iota_crio 6384   Basecbs 13245   lecple 13312   joincjn 14177   meetcmee 14178   Atomscatm 29522   HLchlt 29609   LHypclh 30242   LTrncltrn 30359   trLctrl 30416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-map 6862  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-p1 14245  df-lat 14251  df-clat 14313  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-llines 29756  df-lplanes 29757  df-lvols 29758  df-lines 29759  df-psubsp 29761  df-pmap 29762  df-padd 30054  df-lhyp 30246  df-laut 30247  df-ldil 30362  df-ltrn 30363  df-trl 30417
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