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Theorem cdlemd5 29192
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
Hypotheses
Ref Expression
cdlemd4.l  |-  .<_  =  ( le `  K )
cdlemd4.j  |-  .\/  =  ( join `  K )
cdlemd4.a  |-  A  =  ( Atoms `  K )
cdlemd4.h  |-  H  =  ( LHyp `  K
)
cdlemd4.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemd5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )

Proof of Theorem cdlemd5
StepHypRef Expression
1 fveq2 5377 . . . 4  |-  ( R  =  P  ->  ( F `  R )  =  ( F `  P ) )
2 fveq2 5377 . . . 4  |-  ( R  =  P  ->  ( G `  R )  =  ( G `  P ) )
31, 2eqeq12d 2267 . . 3  |-  ( R  =  P  ->  (
( F `  R
)  =  ( G `
 R )  <->  ( F `  P )  =  ( G `  P ) ) )
4 simpll1 999 . . . 4  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  /\  R  =/=  P
)  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
) )
5 simpl21 1038 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
65adantr 453 . . . 4  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  /\  R  =/=  P
)  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
7 simpl22 1039 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
87adantr 453 . . . 4  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  /\  R  =/=  P
)  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
9 simp23 995 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  =/=  Q
)
109ad2antrr 709 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  /\  R  =/=  P
)  ->  P  =/=  Q )
11 simplr 734 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  /\  R  =/=  P
)  ->  R  .<_  ( P  .\/  Q ) )
12 simpr 449 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  /\  R  =/=  P
)  ->  R  =/=  P )
1310, 11, 123jca 1137 . . . 4  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  /\  R  =/=  P
)  ->  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )
14 simpll3 1001 . . . 4  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  /\  R  =/=  P
)  ->  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )
15 cdlemd4.l . . . . 5  |-  .<_  =  ( le `  K )
16 cdlemd4.j . . . . 5  |-  .\/  =  ( join `  K )
17 cdlemd4.a . . . . 5  |-  A  =  ( Atoms `  K )
18 cdlemd4.h . . . . 5  |-  H  =  ( LHyp `  K
)
19 cdlemd4.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
2015, 16, 17, 18, 19cdlemd4 29191 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )
214, 6, 8, 13, 14, 20syl131anc 1200 . . 3  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  /\  R  =/=  P
)  ->  ( F `  R )  =  ( G `  R ) )
22 simpl3l 1015 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  ->  ( F `  P )  =  ( G `  P ) )
233, 21, 22pm2.61ne 2487 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  R  .<_  ( P 
.\/  Q ) )  ->  ( F `  R )  =  ( G `  R ) )
24 simpl1 963 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
) )
25 simpl21 1038 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
26 simpl22 1039 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
27 simpl23 1040 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  P  =/=  Q )
28 simpr 449 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  -.  R  .<_  ( P  .\/  Q
) )
2927, 28jca 520 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )
30 simpl3 965 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )
3115, 16, 17, 18, 19cdlemd2 29189 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )
3224, 25, 26, 29, 30, 31syl131anc 1200 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  ( F `  R )  =  ( G `  R ) )
3323, 32pm2.61dan 769 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   Atomscatm 28254   HLchlt 28341   LHypclh 28974   LTrncltrn 29091
This theorem is referenced by:  cdlemd7  29194
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-psubsp 28493  df-pmap 28494  df-padd 28786  df-lhyp 28978  df-laut 28979  df-ldil 29094  df-ltrn 29095
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