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Theorem pmod1i 30089
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 2358 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2358 . . . . 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 30088 . . . 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 3465 . . . . 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 30059 . . . . . 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 29995 . . . . . . . 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 3473 . . . . . . . 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 30058 . . . . . . 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 3466 . . . . . . . 8  |-  ( Y  i^i  Z )  C_  Z
273, 5paddss2 30059 . . . . . . . 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 30082 . . . . . . 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 3290 . . . . 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 3265 . . . 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 3469 . . 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 3272 . 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 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   ` cfv 5334  (class class class)co 5942   lecple 13306   joincjn 14171   Atomscatm 29505   HLchlt 29592   PSubSpcpsubsp 29737   + Pcpadd 30036
This theorem is referenced by:  pmod2iN  30090  pmodN  30091  pmodl42N  30092  hlmod1i  30097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-poset 14173  df-plt 14185  df-lub 14201  df-join 14203  df-lat 14245  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593  df-psubsp 29744  df-padd 30037
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