Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmod1i Unicode version

Theorem pmod1i 30037
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 2283 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2283 . . . . 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 30036 . . . 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 1151 . . 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 3389 . . . . 5  |-  ( Y  i^i  Z )  C_  Y
9 simpll 730 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  K  e.  HL )
10 simplr2 998 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Y  C_  A
)
11 simplr1 997 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  X  C_  A
)
123, 5paddss2 30007 . . . . . 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 1182 . . . . 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 16 . . . 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 443 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  K  e.  HL )
163, 4psubssat 29943 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Z  e.  S )  ->  Z  C_  A )
17163ad2antr3 1122 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Z  C_  A )
18 simpr2 962 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Y  C_  A )
19 ssinss1 3397 . . . . . . . 8  |-  ( Y 
C_  A  ->  ( Y  i^i  Z )  C_  A )
2018, 19syl 15 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( Y  i^i  Z
)  C_  A )
213, 5paddss1 30006 . . . . . . 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 1182 . . . . . 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 418 . . . . 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 999 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Z  e.  S )
259, 24, 16syl2anc 642 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Z  C_  A
)
26 inss2 3390 . . . . . . . 8  |-  ( Y  i^i  Z )  C_  Z
273, 5paddss2 30007 . . . . . . . 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 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  Z  C_  A )  ->  ( Z  .+  ( Y  i^i  Z ) )  C_  ( Z  .+  Z ) )
299, 25, 25, 28syl3anc 1182 . . . . . 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 30030 . . . . . . 7  |-  ( ( K  e.  HL  /\  Z  e.  S )  ->  ( Z  .+  Z
)  =  Z )
319, 24, 30syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( Z  .+  Z )  =  Z )
3229, 31sseqtrd 3214 . . . . 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 3189 . . . 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 3393 . . 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 3196 . 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 423 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   PSubSpcpsubsp 29685   + Pcpadd 29984
This theorem is referenced by:  pmod2iN  30038  pmodN  30039  pmodl42N  30040  hlmod1i  30045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-join 14110  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-padd 29985
  Copyright terms: Public domain W3C validator