Theorem cdlemkoatnle-2N 34828

Description: Utility lemma. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemk2.b	|- B = ( Base ` K )
cdlemk2.l	|- .<_ = ( le ` K )
cdlemk2.j	|- .\/ = ( join ` K )
cdlemk2.m	|- ./\ = ( meet ` K )
cdlemk2.a	|- A = ( Atoms ` K )
cdlemk2.h	|- H = ( LHyp ` K )
cdlemk2.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk2.r	|- R = ( ( trL ` K ) ` W )
cdlemk2.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk2.q	|- Q = ( S ` C )

Assertion

Ref	Expression
cdlemkoatnle-2N	|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( Q ` P ) e. A /\ -. ( Q ` P ) .<_ W ) )

Distinct variable groups:   f, i, ./\   .<_ , i   .\/ , f, i   A, i   C, f, i   f, F, i   i, H   i, K   f, N, i   P, f, i   R, f, i   T, f, i   f, W, i

Allowed substitution hints:   A( f)   B( f, i)   Q( f, i)   S( f, i)   H( f)   K( f)   .<_ ( f)

Proof of Theorem cdlemkoatnle-2N

Step	Hyp	Ref	Expression
1		simp11 1018	. . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> K e. HL )
2		simp12 1019	. . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> W e. H )
3	1, 2	jca 532	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
4		simp21 1021	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
5		simp22 1022	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C e. T )
6		simp23 1023	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> N e. T )
7		simp33 1026	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
8		simp13 1020	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) = ( R ` N ) )
9		simp32l 1113	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) )
10		simp32r 1114	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C =/= ( _I |` B ) )
11		simp31 1024	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` C ) =/= ( R ` F ) )
12		cdlemk2.b	. . 3 |- B = ( Base ` K )
13		cdlemk2.l	. . 3 |- .<_ = ( le ` K )
14		cdlemk2.j	. . 3 |- .\/ = ( join ` K )
15		cdlemk2.m	. . 3 |- ./\ = ( meet ` K )
16		cdlemk2.a	. . 3 |- A = ( Atoms ` K )
17		cdlemk2.h	. . 3 |- H = ( LHyp ` K )
18		cdlemk2.t	. . 3 |- T = ( ( LTrn ` K ) ` W )
19		cdlemk2.r	. . 3 |- R = ( ( trL ` K ) ` W )
20		cdlemk2.s	. . 3 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
21		cdlemk2.q	. . 3 |- Q = ( S ` C )
22	12, 13, 14, 15, 16, 17, 18, 19, 20, 21	cdlemkoatnle 34804	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ C e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ ( R ` C ) =/= ( R ` F ) ) ) -> ( ( Q ` P ) e. A /\ -. ( Q ` P ) .<_ W ) )
23	3, 4, 5, 6, 7, 8, 9, 10, 11, 22	syl333anc 1251	1 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( Q ` P ) e. A /\ -. ( Q ` P ) .<_ W ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   class class class wbr 4393   |-> cmpt 4451   _I cid 4732   `'ccnv 4940   |` cres 4943   o. ccom 4945   ` cfv 5519   iota_crio 6153  (class class class)co 6193   Basecbs 14285   lecple 14356   joincjn 15225   meetcmee 15226   Atomscatm 33217   HLchlt 33304   LHypclh 33937   LTrncltrn 34054   trLctrl 34111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-riotaBAD 32913

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-undef 6895  df-map 7319  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451  df-lplanes 33452  df-lvols 33453  df-lines 33454  df-psubsp 33456  df-pmap 33457  df-padd 33749  df-lhyp 33941  df-laut 33942  df-ldil 34057  df-ltrn 34058  df-trl 34112

This theorem is referenced by: (None)