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Theorem cdlemd5 33220
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 5849 . . . 4  |-  ( R  =  P  ->  ( F `  R )  =  ( F `  P ) )
2 fveq2 5849 . . . 4  |-  ( R  =  P  ->  ( G `  R )  =  ( G `  P ) )
31, 2eqeq12d 2424 . . 3  |-  ( R  =  P  ->  (
( F `  R
)  =  ( G `
 R )  <->  ( F `  P )  =  ( G `  P ) ) )
4 simpll1 1036 . . . 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 1075 . . . . 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 463 . . . 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 1076 . . . . 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 463 . . . 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 1032 . . . . . 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 724 . . . . 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 754 . . . . 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 459 . . . . 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 1177 . . . 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 1038 . . . 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 33219 . . . 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 1243 . . 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 1052 . . 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 2718 . 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 1000 . . 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 1075 . . 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 1076 . . 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 1077 . . . 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 459 . . . 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 530 . . 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 1002 . . 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 33217 . . 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 1243 . 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 792 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   lecple 14916   joincjn 15897   Atomscatm 32281   HLchlt 32368   LHypclh 33001   LTrncltrn 33118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-map 7459  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122
This theorem is referenced by:  cdlemd7  33222
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