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Theorem pmod1i 30342
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 2412 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2412 . . . . 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 30341 . . . 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 1153 . . 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 3529 . . . . 5  |-  ( Y  i^i  Z )  C_  Y
9 simpll 731 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  K  e.  HL )
10 simplr2 1000 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Y  C_  A
)
11 simplr1 999 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  X  C_  A
)
123, 5paddss2 30312 . . . . . 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 1184 . . . . 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 444 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  K  e.  HL )
163, 4psubssat 30248 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Z  e.  S )  ->  Z  C_  A )
17163ad2antr3 1124 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Z  C_  A )
18 simpr2 964 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S ) )  ->  Y  C_  A )
19 ssinss1 3537 . . . . . . . 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 30311 . . . . . . 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 1184 . . . . . 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 419 . . . . 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 1001 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Z  e.  S )
259, 24, 16syl2anc 643 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  Z  C_  A
)
26 inss2 3530 . . . . . . . 8  |-  ( Y  i^i  Z )  C_  Z
273, 5paddss2 30312 . . . . . . . 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 1184 . . . . . 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 30335 . . . . . . 7  |-  ( ( K  e.  HL  /\  Z  e.  S )  ->  ( Z  .+  Z
)  =  Z )
319, 24, 30syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )
)  /\  X  C_  Z
)  ->  ( Z  .+  Z )  =  Z )
3229, 31sseqtrd 3352 . . . . 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 3326 . . . 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 3533 . . 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 3333 . 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 424 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3287    C_ wss 3288   ` cfv 5421  (class class class)co 6048   lecple 13499   joincjn 14364   Atomscatm 29758   HLchlt 29845   PSubSpcpsubsp 29990   + Pcpadd 30289
This theorem is referenced by:  pmod2iN  30343  pmodN  30344  pmodl42N  30345  hlmod1i  30350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-join 14396  df-lat 14438  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-psubsp 29997  df-padd 30290
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