Theorem pmod2iN 35135

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

Step	Hyp	Ref	Expression
1		incom 3805	. . . . . 6 |- ( X i^i Y ) = ( Y i^i X )
2	1	oveq1i 6660	. . . . 5 |- ( ( X i^i Y ) .+ Z ) = ( ( Y i^i X ) .+ Z )
3		hllat 34650	. . . . . . 7 |- ( K e. HL -> K e. Lat )
4	3	3ad2ant1 1082	. . . . . 6 |- ( ( K e. HL /\ ( X e. S /\ Y C_ A /\ Z C_ A ) /\ Z C_ X ) -> K e. Lat )
5		simp22 1095	. . . . . . 7 |- ( ( K e. HL /\ ( X e. S /\ Y C_ A /\ Z C_ A ) /\ Z C_ X ) -> Y C_ A )
6		ssinss1 3841	. . . . . . 7 |- ( Y C_ A -> ( Y i^i X ) C_ A )
7	5, 6	syl 17	. . . . . 6 |- ( ( K e. HL /\ ( X e. S /\ Y C_ A /\ Z C_ A ) /\ Z C_ X ) -> ( Y i^i X ) C_ A )
8		simp23 1096	. . . . . 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 )
11	9, 10	paddcom 35099	. . . . . 6 |- ( ( K e. Lat /\ ( Y i^i X ) C_ A /\ Z C_ A ) -> ( ( Y i^i X ) .+ Z ) = ( Z .+ ( Y i^i X ) ) )
12	4, 7, 8, 11	syl3anc 1326	. . . . 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 ) ) )
13	2, 12	syl5eq 2668	. . . 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 1094	. . . . . 6 |- ( ( K e. HL /\ ( X e. S /\ Y C_ A /\ Z C_ A ) /\ Z C_ X ) -> X e. S )
15	8, 5, 14	3jca 1242	. . . . 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 )
17	9, 16, 10	pmod1i 35134	. . . . . 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 ) ) ) )
18	17	3impia 1261	. . . . 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 ) ) )
19	15, 18	syld3an2 1373	. . . 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 ) ) )
20	9, 10	paddcom 35099	. . . . . 6 |- ( ( K e. Lat /\ Z C_ A /\ Y C_ A ) -> ( Z .+ Y ) = ( Y .+ Z ) )
21	4, 8, 5, 20	syl3anc 1326	. . . . 5 |- ( ( K e. HL /\ ( X e. S /\ Y C_ A /\ Z C_ A ) /\ Z C_ X ) -> ( Z .+ Y ) = ( Y .+ Z ) )
22	21	ineq1d 3813	. . . 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 ) )
23	13, 19, 22	3eqtr2d 2662	. . 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 3805	. . 3 |- ( ( Y .+ Z ) i^i X ) = ( X i^i ( Y .+ Z ) )
25	23, 24	syl6eq 2672	. 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 ) ) )
26	25	3expia 1267	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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Latclat 17045   Atomscatm 34550   HLchlt 34637   PSubSpcpsubsp 34782   +Pcpadd 35081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-padd 35082

This theorem is referenced by: (None)