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Theorem pmod1i 33588
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 2443 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2443 . . . . 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 33587 . . . 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 1187 . . 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 3591 . . . . 5  |-  ( Y  i^i  Z )  C_  Y
9 simpll 753 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  K  e.  HL )
10 simplr2 1031 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Y  C_  A
)
11 simplr1 1030 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  X  C_  A
)
123, 5paddss2 33558 . . . . . 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 1218 . . . . 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 17 . . . 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 457 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  K  e.  HL )
163, 4psubssat 33494 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Z  e.  S )  ->  Z  C_  A )
17163ad2antr3 1155 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Z  C_  A )
18 simpr2 995 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Y  C_  A )
19 ssinss1 3599 . . . . . . . 8  |-  ( Y 
C_  A  ->  ( Y  i^i  Z )  C_  A )
2018, 19syl 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  -> 
( Y  i^i  Z
)  C_  A )
213, 5paddss1 33557 . . . . . . 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 1218 . . . . . 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 429 . . . . 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 1032 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Z  e.  S )
259, 24, 16syl2anc 661 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Z  C_  A
)
26 inss2 3592 . . . . . . . 8  |-  ( Y  i^i  Z )  C_  Z
273, 5paddss2 33558 . . . . . . . 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 17 . . . . . . 7  |-  ( ( K  e.  HL  /\  Z  C_  A  /\  Z  C_  A )  ->  ( Z  .+  ( Y  i^i  Z ) )  C_  ( Z  .+  Z ) )
299, 25, 25, 28syl3anc 1218 . . . . . 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 33581 . . . . . . 7  |-  ( ( K  e.  HL  /\  Z  e.  S )  ->  ( Z  .+  Z
)  =  Z )
319, 24, 30syl2anc 661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( Z  .+  Z )  =  Z )
3229, 31sseqtrd 3413 . . . . 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 3387 . . . 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 3595 . . 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 3394 . 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 434 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349   ` cfv 5439  (class class class)co 6112   lecple 14266   joincjn 15135   Atomscatm 33004   HLchlt 33091   PSubSpcpsubsp 33236   +Pcpadd 33535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-lat 15237  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-psubsp 33243  df-padd 33536
This theorem is referenced by:  pmod2iN  33589  pmodN  33590  pmodl42N  33591  hlmod1i  33596
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