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Theorem cdlemkfid3N 31114
Description: TODO: is this useful or should it be deleted? (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
Assertion
Ref Expression
cdlemkfid3N  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  [_ G  /  g ]_ Y  =  ( G `  P )
)
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b    g, G
Allowed substitution hints:    A( g, b)    B( b)    P( b)    R( b)    T( b)    F( g, b)    G( b)    H( g, b)    .\/ ( b)    K( g,
b)    .<_ ( g, b)    ./\ ( b)    N( g, b)    W( g, b)    Y( g, b)    Z( b)

Proof of Theorem cdlemkfid3N
StepHypRef Expression
1 simp22 989 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  G  e.  T
)
2 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
32cdlemk41 31109 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
41, 3syl 15 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  [_ G  /  g ]_ Y  =  (
( P  .\/  ( R `  G )
)  ./\  ( Z  .\/  ( R `  ( G  o.  `' b
) ) ) ) )
5 simp1 955 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N ) )
6 simp21l 1072 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T
)
7 simp21r 1073 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
8 simp23l 1076 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  b  e.  T
)
9 simp31 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  b )  =/=  ( R `  F )
)
10 simp33 993 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
11 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
12 cdlemk5.l . . . . . 6  |-  .<_  =  ( le `  K )
13 cdlemk5.j . . . . . 6  |-  .\/  =  ( join `  K )
14 cdlemk5.m . . . . . 6  |-  ./\  =  ( meet `  K )
15 cdlemk5.a . . . . . 6  |-  A  =  ( Atoms `  K )
16 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
17 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
18 cdlemk5.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
19 cdlemk5.z . . . . . 6  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
2011, 12, 13, 14, 15, 16, 17, 18, 19cdlemkfid2N 31112 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  b  e.  T
)  /\  ( ( R `  b )  =/=  ( R `  F
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  Z  =  ( b `  P ) )
215, 6, 7, 8, 9, 10, 20syl132anc 1200 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  Z  =  ( b `  P ) )
2221oveq1d 5873 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( Z  .\/  ( R `  ( G  o.  `' b ) ) )  =  ( ( b `  P
)  .\/  ( R `  ( G  o.  `' b ) ) ) )
2322oveq2d 5874 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
b `  P )  .\/  ( R `  ( G  o.  `' b
) ) ) ) )
24 simp1l 979 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
25 simp23r 1077 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  b  =/=  (  _I  |`  B ) )
26 simp32 992 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  b )  =/=  ( R `  G )
)
2726necomd 2529 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  G )  =/=  ( R `  b )
)
2811, 12, 13, 14, 15, 16, 17, 18cdlemkfid1N 31110 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B )  /\  G  e.  T
)  /\  ( ( R `  G )  =/=  ( R `  b
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( b `  P )  .\/  ( R `  ( G  o.  `' b ) ) ) )  =  ( G `  P ) )
2924, 8, 25, 1, 27, 10, 28syl132anc 1200 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( b `  P )  .\/  ( R `  ( G  o.  `' b ) ) ) )  =  ( G `  P ) )
304, 23, 293eqtrd 2319 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  =  N )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  G  e.  T  /\  ( b  e.  T  /\  b  =/=  (  _I  |`  B ) ) )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  [_ G  /  g ]_ Y  =  ( G `  P )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   [_csb 3081   class class class wbr 4023    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
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-3or 935  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-rmo 2551  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-iin 3908  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-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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