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Theorem cdlemkfid3N 34927
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 1022 . . 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 34922 . . 3  |-  ( G  e.  T  ->  [_ G  /  g ]_ Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
41, 3syl 16 . 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 988 . . . . 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 1105 . . . . 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 1106 . . . . 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 1109 . . . . 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 1024 . . . . 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 1026 . . . . 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 34925 . . . . 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 1237 . . . 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 6218 . . 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 6219 . 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 1012 . . 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 1110 . . 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 1025 . . . 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 2723 . . 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 34923 . . 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 1237 . 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 2499 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   [_csb 3398   class class class wbr 4403    _I cid 4742   `'ccnv 4950    |` cres 4953    o. ccom 4955   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   meetcmee 15237   Atomscatm 33266   HLchlt 33353   LHypclh 33986   LTrncltrn 34103   trLctrl 34160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-riotaBAD 32962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-undef 6905  df-map 7329  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-llines 33500  df-lplanes 33501  df-lvols 33502  df-lines 33503  df-psubsp 33505  df-pmap 33506  df-padd 33798  df-lhyp 33990  df-laut 33991  df-ldil 34106  df-ltrn 34107  df-trl 34161
This theorem is referenced by: (None)
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