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Theorem pmod2iN 35313
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 3676 . . . . . 6  |-  ( X  i^i  Y )  =  ( Y  i^i  X
)
21oveq1i 6291 . . . . 5  |-  ( ( X  i^i  Y ) 
.+  Z )  =  ( ( Y  i^i  X )  .+  Z )
3 hllat 34828 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 1018 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  K  e.  Lat )
5 simp22 1031 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  Y  C_  A )
6 ssinss1 3711 . . . . . . 7  |-  ( Y 
C_  A  ->  ( Y  i^i  X )  C_  A )
75, 6syl 16 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Y  i^i  X
)  C_  A )
8 simp23 1032 . . . . . 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 35277 . . . . . 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 1229 . . . . 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 2496 . . . 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 1030 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  ->  X  e.  S )
158, 5, 143jca 1177 . . . . 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 35312 . . . . . 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 1194 . . . . 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 1276 . . . 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 35277 . . . . . 6  |-  ( ( K  e.  Lat  /\  Z  C_  A  /\  Y  C_  A )  ->  ( Z  .+  Y )  =  ( Y  .+  Z
) )
214, 8, 5, 20syl3anc 1229 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  S  /\  Y  C_  A  /\  Z  C_  A )  /\  Z  C_  X )  -> 
( Z  .+  Y
)  =  ( Y 
.+  Z ) )
2221ineq1d 3684 . . . 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 2490 . . 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 3676 . . 3  |-  ( ( Y  .+  Z )  i^i  X )  =  ( X  i^i  ( Y  .+  Z ) )
2523, 24syl6eq 2500 . 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 1199 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    i^i cin 3460    C_ wss 3461   ` cfv 5578  (class class class)co 6281   Latclat 15549   Atomscatm 34728   HLchlt 34815   PSubSpcpsubsp 34960   +Pcpadd 35259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-psubsp 34967  df-padd 35260
This theorem is referenced by: (None)
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