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Theorem cdlemg31d 35988
Description: Eliminate ( F ` P ) =/= P from cdlemg31c 35987. TODO: Prove directly. TODO: do we need to eliminate ( F ` P ) =/= P? It might be better to do this all at once at the end. See also cdlemg29 35993 vs. cdlemg28 35992. (Contributed by NM, 29-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l |- .<_ = ( le ` K )
cdlemg12.j |- .\/ = ( join ` K )
cdlemg12.m |- ./\ = ( meet ` K )
cdlemg12.a |- A = ( Atoms ` K )
cdlemg12.h |- H = ( LHyp ` K )
cdlemg12.t |- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r |- R = ( ( trL ` K ) ` W )
cdlemg31.n |- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
Assertion
Ref Expression
cdlemg31d |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) -> -. N .<_ W )
Proof of Theorem cdlemg31d
StepHypRef Expression
1 simp22r 1181 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) -> -. Q .<_ W )
21adantr 481 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> -. Q .<_ W )
3 simpl1 1064 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) )
4 simp21l 1178 . . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) -> P e. A )
54adantr 481 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> P e. A )
6 simp22l 1180 . . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) -> Q e. A )
76adantr 481 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> Q e. A )
8 simp23l 1182 . . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) -> v e. A )
98adantr 481 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> v e. A )
10 simpl31 1142 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> F e. T )
11 cdlemg12.l . . . . . . . 8 |- .<_ = ( le ` K )
12 cdlemg12.j . . . . . . . 8 |- .\/ = ( join ` K )
13 cdlemg12.m . . . . . . . 8 |- ./\ = ( meet ` K )
14 cdlemg12.a . . . . . . . 8 |- A = ( Atoms ` K )
15 cdlemg12.h . . . . . . . 8 |- H = ( LHyp ` K )
16 cdlemg12.t . . . . . . . 8 |- T = ( ( LTrn ` K ) ` W )
17 cdlemg12b.r . . . . . . . 8 |- R = ( ( trL ` K ) ` W )
18 cdlemg31.n . . . . . . . 8 |- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
1911, 12, 13, 14, 15, 16, 17, 18cdlemg31b 35986 . . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( v e. A /\ F e. T ) ) -> N .<_ ( Q .\/ ( R ` F ) ) )
203, 5, 7, 9, 10, 19syl122anc 1335 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> N .<_ ( Q .\/ ( R ` F ) ) )
21 simpl21 1139 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) )
22 simpr 477 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P )
23 eqid 2622 . . . . . . . . . 10 |- ( 0. ` K ) = ( 0. ` K )
2411, 23, 14, 15, 16, 17trl0 35457 . . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = ( 0. ` K ) )
253, 21, 10, 22, 24syl112anc 1330 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> ( R ` F ) = ( 0. ` K ) )
2625oveq2d 6666 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( R ` F ) ) = ( Q .\/ ( 0. ` K ) ) )
27 simp1l 1085 . . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) -> K e. HL )
28 hlol 34648 . . . . . . . . . 10 |- ( K e. HL -> K e. OL )
2927, 28syl 17 . . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) -> K e. OL )
3029adantr 481 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> K e. OL )
31 eqid 2622 . . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
3231, 14atbase 34576 . . . . . . . . 9 |- ( Q e. A -> Q e. ( Base ` K ) )
337, 32syl 17 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> Q e. ( Base ` K ) )
3431, 12, 23olj01 34512 . . . . . . . 8 |- ( ( K e. OL /\ Q e. ( Base ` K ) ) -> ( Q .\/ ( 0. ` K ) ) = Q )
3530, 33, 34syl2anc 693 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( 0. ` K ) ) = Q )
3626, 35eqtrd 2656 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( R ` F ) ) = Q )
3720, 36breqtrd 4679 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> N .<_ Q )
38 hlatl 34647 . . . . . . . 8 |- ( K e. HL -> K e. AtLat )
3927, 38syl 17 . . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) -> K e. AtLat )
4039adantr 481 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> K e. AtLat )
41 simpl33 1144 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> N e. A )
4211, 14atcmp 34598 . . . . . 6 |- ( ( K e. AtLat /\ N e. A /\ Q e. A ) -> ( N .<_ Q <-> N = Q ) )
4340, 41, 7, 42syl3anc 1326 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> ( N .<_ Q <-> N = Q ) )
4437, 43mpbid 222 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> N = Q )
4544breq1d 4663 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> ( N .<_ W <-> Q .<_ W ) )
462, 45mtbird 315 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) = P ) -> -. N .<_ W )
47 simpl1 1064 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) =/= P ) -> ( K e. HL /\ W e. H ) )
48 simpl21 1139 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) =/= P ) -> ( P e. A /\ -. P .<_ W ) )
49 simpl22 1140 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) =/= P ) -> ( Q e. A /\ -. Q .<_ W ) )
50 simpl23 1141 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) =/= P ) -> ( v e. A /\ v .<_ W ) )
51 simpl31 1142 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) =/= P ) -> F e. T )
52 simpl32 1143 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) =/= P ) -> v =/= ( R ` F ) )
53 simpr 477 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) =/= P ) -> ( F ` P ) =/= P )
54 simpl33 1144 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) =/= P ) -> N e. A )
5511, 12, 13, 14, 15, 16, 17, 18cdlemg31c 35987 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ F e. T ) /\ ( v =/= ( R ` F ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> -. N .<_ W )
5647, 48, 49, 50, 51, 52, 53, 54, 55syl323anc 1356 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) /\ ( F ` P ) =/= P ) -> -. N .<_ W )
5746, 56pm2.61dane 2881 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) /\ N e. A ) ) -> -. N .<_ W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   0.cp0 17037   OLcol 34461   Atomscatm 34550   AtLatcal 34551   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445
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-iin 4523  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-map 7859  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-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446
This theorem is referenced by:  cdlemg33b0  35989  cdlemg33a  35994
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