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Theorem pmod2iN 30038
Description: Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmod.a  |-  A  =  ( Atoms `  K )
pmod.s  |-  S  =  ( PSubSp `  K )
pmod.p  |-  .+  =  ( + P `  K
)
Assertion
Ref Expression
pmod2iN  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( Z  C_  X  ->  ( ( X  i^i  Y )  .+  Z )  =  ( X  i^i  ( Y 
.+  Z ) ) ) )

Proof of Theorem pmod2iN
StepHypRef Expression
1 incom 3361 . . . . . 6  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
21oveq1i 5868 . . . . 5  |-  ( ( X  i^i  Y ) 
.+  Z )  =  ( ( Y  i^i  X )  .+  Z )
3 hllat 29553 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 976 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  K  e.  Lat )
5 simp22 989 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  Y  C_  A )
6 ssinss1 3397 . . . . . . 7  |-  ( Y 
C_  A  ->  ( Y  i^i  X )  C_  A )
75, 6syl 15 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Y  i^i  X
)  C_  A )
8 simp23 990 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  Z  C_  A )
9 pmod.a . . . . . . 7  |-  A  =  ( Atoms `  K )
10 pmod.p . . . . . . 7  |-  .+  =  ( + P `  K
)
119, 10paddcom 30002 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Y  i^i  X ) 
C_  A  /\  Z  C_  A )  ->  (
( Y  i^i  X
)  .+  Z )  =  ( Z  .+  ( Y  i^i  X ) ) )
124, 7, 8, 11syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( Y  i^i  X )  .+  Z )  =  ( Z  .+  ( Y  i^i  X ) ) )
132, 12syl5eq 2327 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( X  i^i  Y )  .+  Z )  =  ( Z  .+  ( Y  i^i  X ) ) )
14 simp21 988 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  X  e.  S )
158, 5, 143jca 1132 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Z  C_  A  /\  Y  C_  A  /\  X  e.  S )
)
16 pmod.s . . . . . . 7  |-  S  =  ( PSubSp `  K )
179, 16, 10pmod1i 30037 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Z  C_  A  /\  Y  C_  A  /\  X  e.  S ) )  -> 
( Z  C_  X  ->  ( ( Z  .+  Y )  i^i  X
)  =  ( Z 
.+  ( Y  i^i  X ) ) ) )
18173impia 1148 . . . . 5  |-  ( ( K  e.  HL  /\  ( Z  C_  A  /\  Y  C_  A  /\  X  e.  S )  /\  Z  C_  X )  ->  (
( Z  .+  Y
)  i^i  X )  =  ( Z  .+  ( Y  i^i  X ) ) )
1915, 18syld3an2 1229 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( Z  .+  Y )  i^i  X
)  =  ( Z 
.+  ( Y  i^i  X ) ) )
209, 10paddcom 30002 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  C_  A  /\  Y  C_  A )  ->  ( Z  .+  Y )  =  ( Y  .+  Z
) )
214, 8, 5, 20syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Z  .+  Y
)  =  ( Y 
.+  Z ) )
2221ineq1d 3369 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( Z  .+  Y )  i^i  X
)  =  ( ( Y  .+  Z )  i^i  X ) )
2313, 19, 223eqtr2d 2321 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( X  i^i  Y )  .+  Z )  =  ( ( Y 
.+  Z )  i^i 
X ) )
24 incom 3361 . . 3  |-  ( ( Y  .+  Z )  i^i  X )  =  ( X  i^i  ( Y  .+  Z ) )
2523, 24syl6eq 2331 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( ( X  i^i  Y )  .+  Z )  =  ( X  i^i  ( Y  .+  Z ) ) )
26253expia 1153 1  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A ) )  ->  ( Z  C_  X  ->  ( ( X  i^i  Y )  .+  Z )  =  ( X  i^i  ( Y 
.+  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   Latclat 14151   Atomscatm 29453   HLchlt 29540   PSubSpcpsubsp 29685   + Pcpadd 29984
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
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