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Theorem pmodlem2 35133
Description: Lemma for pmod1i 35134. (Contributed by NM, 9-Mar-2012.)
Hypotheses
Ref Expression
pmodlem.l |- .<_ = ( le ` K )
pmodlem.j |- .\/ = ( join ` K )
pmodlem.a |- A = ( Atoms ` K )
pmodlem.s |- S = ( PSubSp ` K )
pmodlem.p |- .+ = ( +P ` K )
Assertion
Ref Expression
pmodlem2 |- ( ( 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 ) ) )
Proof of Theorem pmodlem2
Dummy variables q p r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 477 . . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> X = (/) )
21oveq1d 6665 . . . . 5 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> ( X .+ Y ) = ( (/) .+ Y ) )
3 simpl1 1064 . . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> K e. HL )
4 simpl22 1140 . . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> Y C_ A )
5 pmodlem.a . . . . . . 7 |- A = ( Atoms ` K )
6 pmodlem.p . . . . . . 7 |- .+ = ( +P ` K )
75, 6padd02 35098 . . . . . 6 |- ( ( K e. HL /\ Y C_ A ) -> ( (/) .+ Y ) = Y )
83, 4, 7syl2anc 693 . . . . 5 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> ( (/) .+ Y ) = Y )
92, 8eqtrd 2656 . . . 4 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> ( X .+ Y ) = Y )
109ineq1d 3813 . . 3 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> ( ( X .+ Y ) i^i Z ) = ( Y i^i Z ) )
11 ssinss1 3841 . . . . 5 |- ( Y C_ A -> ( Y i^i Z ) C_ A )
124, 11syl 17 . . . 4 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> ( Y i^i Z ) C_ A )
13 simpl21 1139 . . . 4 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> X C_ A )
145, 6sspadd2 35102 . . . 4 |- ( ( K e. HL /\ ( Y i^i Z ) C_ A /\ X C_ A ) -> ( Y i^i Z ) C_ ( X .+ ( Y i^i Z ) ) )
153, 12, 13, 14syl3anc 1326 . . 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 ) ) )
1610, 15eqsstrd 3639 . 2 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ X = (/) ) -> ( ( X .+ Y ) i^i Z ) C_ ( X .+ ( Y i^i Z ) ) )
17 oveq2 6658 . . . . 5 |- ( Y = (/) -> ( X .+ Y ) = ( X .+ (/) ) )
18 simp1 1061 . . . . . 6 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) -> K e. HL )
19 simp21 1094 . . . . . 6 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) -> X C_ A )
205, 6padd01 35097 . . . . . 6 |- ( ( K e. HL /\ X C_ A ) -> ( X .+ (/) ) = X )
2118, 19, 20syl2anc 693 . . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) -> ( X .+ (/) ) = X )
2217, 21sylan9eqr 2678 . . . 4 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ Y = (/) ) -> ( X .+ Y ) = X )
2322ineq1d 3813 . . 3 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ Y = (/) ) -> ( ( X .+ Y ) i^i Z ) = ( X i^i Z ) )
24 inss1 3833 . . . 4 |- ( X i^i Z ) C_ X
25 simpl1 1064 . . . . 5 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ Y = (/) ) -> K e. HL )
26 simpl21 1139 . . . . 5 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ Y = (/) ) -> X C_ A )
27 simpl22 1140 . . . . . 6 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ Y = (/) ) -> Y C_ A )
2827, 11syl 17 . . . . 5 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ Y = (/) ) -> ( Y i^i Z ) C_ A )
295, 6sspadd1 35101 . . . . 5 |- ( ( K e. HL /\ X C_ A /\ ( Y i^i Z ) C_ A ) -> X C_ ( X .+ ( Y i^i Z ) ) )
3025, 26, 28, 29syl3anc 1326 . . . 4 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ Y = (/) ) -> X C_ ( X .+ ( Y i^i Z ) ) )
3124, 30syl5ss 3614 . . 3 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ Y = (/) ) -> ( X i^i Z ) C_ ( X .+ ( Y i^i Z ) ) )
3223, 31eqsstrd 3639 . 2 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ Y = (/) ) -> ( ( X .+ Y ) i^i Z ) C_ ( X .+ ( Y i^i Z ) ) )
33 elin 3796 . . . 4 |- ( p e. ( ( X .+ Y ) i^i Z ) <-> ( p e. ( X .+ Y ) /\ p e. Z ) )
34 simpl1 1064 . . . . . . . . . 10 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( ( X =/= (/) /\ Y =/= (/) ) /\ p e. Z ) ) -> K e. HL )
35 hllat 34650 . . . . . . . . . 10 |- ( K e. HL -> K e. Lat )
3634, 35syl 17 . . . . . . . . 9 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( ( X =/= (/) /\ Y =/= (/) ) /\ p e. Z ) ) -> K e. Lat )
37 simpl21 1139 . . . . . . . . 9 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( ( X =/= (/) /\ Y =/= (/) ) /\ p e. Z ) ) -> X C_ A )
38 simpl22 1140 . . . . . . . . 9 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( ( X =/= (/) /\ Y =/= (/) ) /\ p e. Z ) ) -> Y C_ A )
39 simprl 794 . . . . . . . . 9 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( ( X =/= (/) /\ Y =/= (/) ) /\ p e. Z ) ) -> ( X =/= (/) /\ Y =/= (/) ) )
40 pmodlem.l . . . . . . . . . 10 |- .<_ = ( le ` K )
41 pmodlem.j . . . . . . . . . 10 |- .\/ = ( join ` K )
4240, 41, 5, 6elpaddn0 35086 . . . . . . . . 9 |- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( p e. ( X .+ Y ) <-> ( p e. A /\ E. q e. X E. r e. Y p .<_ ( q .\/ r ) ) ) )
4336, 37, 38, 39, 42syl31anc 1329 . . . . . . . 8 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( ( X =/= (/) /\ Y =/= (/) ) /\ p e. Z ) ) -> ( p e. ( X .+ Y ) <-> ( p e. A /\ E. q e. X E. r e. Y p .<_ ( q .\/ r ) ) ) )
44 simpl1 1064 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> K e. HL )
45 simpl21 1139 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> X C_ A )
46 simpl22 1140 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> Y C_ A )
47 simpl23 1141 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> Z e. S )
48 simpl3 1066 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> X C_ Z )
49 simpr1 1067 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> p e. Z )
50 simpr2l 1120 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> q e. X )
51 simpr2r 1121 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> r e. Y )
52 simpr3 1069 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> p .<_ ( q .\/ r ) )
53 pmodlem.s . . . . . . . . . . . . . . 15 |- S = ( PSubSp ` K )
5440, 41, 5, 53, 6pmodlem1 35132 . . . . . . . . . . . . . 14 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( Z e. S /\ X C_ Z /\ p e. Z ) /\ ( q e. X /\ r e. Y /\ p .<_ ( q .\/ r ) ) ) -> p e. ( X .+ ( Y i^i Z ) ) )
5544, 45, 46, 47, 48, 49, 50, 51, 52, 54syl333anc 1358 . . . . . . . . . . . . 13 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( p e. Z /\ ( q e. X /\ r e. Y ) /\ p .<_ ( q .\/ r ) ) ) -> p e. ( X .+ ( Y i^i Z ) ) )
56553exp2 1285 . . . . . . . . . . . 12 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) -> ( p e. Z -> ( ( q e. X /\ r e. Y ) -> ( p .<_ ( q .\/ r ) -> p e. ( X .+ ( Y i^i Z ) ) ) ) ) )
5756imp 445 . . . . . . . . . . 11 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ p e. Z ) -> ( ( q e. X /\ r e. Y ) -> ( p .<_ ( q .\/ r ) -> p e. ( X .+ ( Y i^i Z ) ) ) ) )
5857rexlimdvv 3037 . . . . . . . . . 10 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ p e. Z ) -> ( E. q e. X E. r e. Y p .<_ ( q .\/ r ) -> p e. ( X .+ ( Y i^i Z ) ) ) )
5958adantld 483 . . . . . . . . 9 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ p e. Z ) -> ( ( p e. A /\ E. q e. X E. r e. Y p .<_ ( q .\/ r ) ) -> p e. ( X .+ ( Y i^i Z ) ) ) )
6059adantrl 752 . . . . . . . 8 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( ( X =/= (/) /\ Y =/= (/) ) /\ p e. Z ) ) -> ( ( p e. A /\ E. q e. X E. r e. Y p .<_ ( q .\/ r ) ) -> p e. ( X .+ ( Y i^i Z ) ) ) )
6143, 60sylbid 230 . . . . . . 7 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( ( X =/= (/) /\ Y =/= (/) ) /\ p e. Z ) ) -> ( p e. ( X .+ Y ) -> p e. ( X .+ ( Y i^i Z ) ) ) )
6261exp32 631 . . . . . 6 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) -> ( ( X =/= (/) /\ Y =/= (/) ) -> ( p e. Z -> ( p e. ( X .+ Y ) -> p e. ( X .+ ( Y i^i Z ) ) ) ) ) )
6362com34 91 . . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) -> ( ( X =/= (/) /\ Y =/= (/) ) -> ( p e. ( X .+ Y ) -> ( p e. Z -> p e. ( X .+ ( Y i^i Z ) ) ) ) ) )
6463imp4b 613 . . . 4 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( ( p e. ( X .+ Y ) /\ p e. Z ) -> p e. ( X .+ ( Y i^i Z ) ) ) )
6533, 64syl5bi 232 . . 3 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( p e. ( ( X .+ Y ) i^i Z ) -> p e. ( X .+ ( Y i^i Z ) ) ) )
6665ssrdv 3609 . 2 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z e. S ) /\ X C_ Z ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( ( X .+ Y ) i^i Z ) C_ ( X .+ ( Y i^i Z ) ) )
6716, 32, 66pm2.61da2ne 2882 1 |- ( ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   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:  pmod1i  35134
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