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Theorem cdlemg14f 36795
Description: TODO: FIX COMMENT (Contributed by NM, 6-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg14f  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )

Proof of Theorem cdlemg14f
StepHypRef Expression
1 simp1 994 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp32 1031 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  ->  G  e.  T )
3 simp2l 1020 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
4 simp2r 1021 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
5 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
6 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
7 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
8 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
10 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
115, 6, 7, 8, 9, 10ltrnu 36261 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( Q 
.\/  ( G `  Q ) )  ./\  W ) )
121, 2, 3, 4, 11syl211anc 1232 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( Q 
.\/  ( G `  Q ) )  ./\  W ) )
13 simp31 1030 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  ->  F  e.  T )
145, 8, 9, 10ltrnel 36279 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
151, 2, 3, 14syl3anc 1226 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
16 simp33 1032 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( F `  P
)  =  P )
175, 8, 9, 10ltrnateq 36322 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  ( G `  P ) )  =  ( G `  P
) )
181, 13, 3, 15, 16, 17syl131anc 1239 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( F `  ( G `  P )
)  =  ( G `
 P ) )
1918oveq2d 6286 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  =  ( P  .\/  ( G `  P ) ) )
2019oveq1d 6285 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( P 
.\/  ( G `  P ) )  ./\  W ) )
215, 8, 9, 10ltrnel 36279 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( G `  Q )  e.  A  /\  -.  ( G `  Q )  .<_  W ) )
221, 2, 4, 21syl3anc 1226 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( ( G `  Q )  e.  A  /\  -.  ( G `  Q )  .<_  W ) )
235, 8, 9, 10ltrnateq 36322 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( G `  Q )  e.  A  /\  -.  ( G `  Q )  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  ( G `  Q ) )  =  ( G `  Q
) )
241, 13, 3, 22, 16, 23syl131anc 1239 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( F `  ( G `  Q )
)  =  ( G `
 Q ) )
2524oveq2d 6286 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  =  ( Q  .\/  ( G `  Q ) ) )
2625oveq1d 6285 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( ( Q  .\/  ( F `  ( G `
 Q ) ) )  ./\  W )  =  ( ( Q 
.\/  ( G `  Q ) )  ./\  W ) )
2712, 20, 263eqtr4d 2505 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14794   joincjn 15775   meetcmee 15776   Atomscatm 35404   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   trLctrl 36299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300
This theorem is referenced by:  cdlemg15a  36797  cdlemg22  36829  cdlemg29  36847  cdlemg39  36858
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