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Theorem pmod1i 33484
Description: The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)
Hypotheses
Ref Expression
pmod.a  |-  A  =  ( Atoms `  K )
pmod.s  |-  S  =  ( PSubSp `  K )
pmod.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
pmod1i  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( X  C_  Z  ->  ( ( X  .+  Y )  i^i  Z
)  =  ( X 
.+  ( Y  i^i  Z ) ) ) )

Proof of Theorem pmod1i
StepHypRef Expression
1 eqid 2471 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2471 . . . . 5  |-  ( join `  K )  =  (
join `  K )
3 pmod.a . . . . 5  |-  A  =  ( Atoms `  K )
4 pmod.s . . . . 5  |-  S  =  ( PSubSp `  K )
5 pmod.p . . . . 5  |-  .+  =  ( +P `  K
)
61, 2, 3, 4, 5pmodlem2 33483 . . . 4  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  (
( X  .+  Y
)  i^i  Z )  C_  ( X  .+  ( Y  i^i  Z ) ) )
763expa 1231 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( ( X  .+  Y )  i^i 
Z )  C_  ( X  .+  ( Y  i^i  Z ) ) )
8 inss1 3643 . . . . 5  |-  ( Y  i^i  Z )  C_  Y
9 simpll 768 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  K  e.  HL )
10 simplr2 1073 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Y  C_  A
)
11 simplr1 1072 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  X  C_  A
)
123, 5paddss2 33454 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A  /\  X  C_  A )  ->  (
( Y  i^i  Z
)  C_  Y  ->  ( X  .+  ( Y  i^i  Z ) ) 
C_  ( X  .+  Y ) ) )
139, 10, 11, 12syl3anc 1292 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( ( Y  i^i  Z )  C_  Y  ->  ( X  .+  ( Y  i^i  Z ) )  C_  ( X  .+  Y ) ) )
148, 13mpi 20 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( X  .+  ( Y  i^i  Z
) )  C_  ( X  .+  Y ) )
15 simpl 464 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  K  e.  HL )
163, 4psubssat 33390 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Z  e.  S )  ->  Z  C_  A )
17163ad2antr3 1197 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Z  C_  A )
18 simpr2 1037 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Y  C_  A )
19 ssinss1 3651 . . . . . . . 8  |-  ( Y 
C_  A  ->  ( Y  i^i  Z )  C_  A )
2018, 19syl 17 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( Y  i^i  Z
)  C_  A )
213, 5paddss1 33453 . . . . . . 7  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  ( Y  i^i  Z )  C_  A )  ->  ( X  C_  Z  ->  ( X  .+  ( Y  i^i  Z ) )  C_  ( Z  .+  ( Y  i^i  Z ) ) ) )
2215, 17, 20, 21syl3anc 1292 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( X  C_  Z  ->  ( X  .+  ( Y  i^i  Z ) ) 
C_  ( Z  .+  ( Y  i^i  Z ) ) ) )
2322imp 436 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( X  .+  ( Y  i^i  Z
) )  C_  ( Z  .+  ( Y  i^i  Z ) ) )
24 simplr3 1074 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Z  e.  S )
259, 24, 16syl2anc 673 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Z  C_  A
)
26 inss2 3644 . . . . . . . 8  |-  ( Y  i^i  Z )  C_  Z
273, 5paddss2 33454 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  Z  C_  A )  ->  (
( Y  i^i  Z
)  C_  Z  ->  ( Z  .+  ( Y  i^i  Z ) ) 
C_  ( Z  .+  Z ) ) )
2826, 27mpi 20 . . . . . . 7  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  Z  C_  A )  ->  ( Z  .+  ( Y  i^i  Z ) )  C_  ( Z  .+  Z ) )
299, 25, 25, 28syl3anc 1292 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( Z  .+  ( Y  i^i  Z
) )  C_  ( Z  .+  Z ) )
304, 5paddidm 33477 . . . . . . 7  |-  ( ( K  e.  HL  /\  Z  e.  S )  ->  ( Z  .+  Z
)  =  Z )
319, 24, 30syl2anc 673 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( Z  .+  Z )  =  Z )
3229, 31sseqtrd 3454 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( Z  .+  ( Y  i^i  Z
) )  C_  Z
)
3323, 32sstrd 3428 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( X  .+  ( Y  i^i  Z
) )  C_  Z
)
3414, 33ssind 3647 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( X  .+  ( Y  i^i  Z
) )  C_  (
( X  .+  Y
)  i^i  Z )
)
357, 34eqssd 3435 . 2  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( ( X  .+  Y )  i^i 
Z )  =  ( X  .+  ( Y  i^i  Z ) ) )
3635ex 441 1  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( X  C_  Z  ->  ( ( X  .+  Y )  i^i  Z
)  =  ( X 
.+  ( Y  i^i  Z ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390   ` cfv 5589  (class class class)co 6308   lecple 15275   joincjn 16267   Atomscatm 32900   HLchlt 32987   PSubSpcpsubsp 33132   +Pcpadd 33431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-lat 16370  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-psubsp 33139  df-padd 33432
This theorem is referenced by:  pmod2iN  33485  pmodN  33486  pmodl42N  33487  hlmod1i  33492
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