Theorem cdlemksat 37024

Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)

Hypotheses

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

Assertion

Ref	Expression
cdlemksat	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) e. A )

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

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

Proof of Theorem cdlemksat

Step	Hyp	Ref	Expression
1		simp11 1024	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) )
2		cdlemk.b	. . 3 |- B = ( Base ` K )
3		cdlemk.l	. . 3 |- .<_ = ( le ` K )
4		cdlemk.j	. . 3 |- .\/ = ( join ` K )
5		cdlemk.a	. . 3 |- A = ( Atoms ` K )
6		cdlemk.h	. . 3 |- H = ( LHyp ` K )
7		cdlemk.t	. . 3 |- T = ( ( LTrn ` K ) ` W )
8		cdlemk.r	. . 3 |- R = ( ( trL ` K ) ` W )
9		cdlemk.m	. . 3 |- ./\ = ( meet ` K )
10		cdlemk.s	. . 3 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
11	2, 3, 4, 5, 6, 7, 8, 9, 10	cdlemksel 37023	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( S ` G ) e. T )
12		simp22l 1113	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> P e. A )
13	3, 5, 6, 7	ltrnat 36316	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( S ` G ) e. T /\ P e. A ) -> ( ( S ` G ) ` P ) e. A )
14	1, 11, 12, 13	syl3anc 1226	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1399   e. wcel 1836   =/= wne 2591   class class class wbr 4384   |-> cmpt 4442   _I cid 4721   `'ccnv 4929   |` cres 4932   o. ccom 4934   ` cfv 5513   iota_crio 6179  (class class class)co 6218   Basecbs 14657   lecple 14732   joincjn 15713   meetcmee 15714   Atomscatm 35440   HLchlt 35527   LHypclh 36160   LTrncltrn 36277   trLctrl 36335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-riotaBAD 35136

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-iin 4263  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-undef 6942  df-map 7362  df-preset 15697  df-poset 15715  df-plt 15728  df-lub 15744  df-glb 15745  df-join 15746  df-meet 15747  df-p0 15809  df-p1 15810  df-lat 15816  df-clat 15878  df-oposet 35353  df-ol 35355  df-oml 35356  df-covers 35443  df-ats 35444  df-atl 35475  df-cvlat 35499  df-hlat 35528  df-llines 35674  df-lplanes 35675  df-lvols 35676  df-lines 35677  df-psubsp 35679  df-pmap 35680  df-padd 35972  df-lhyp 36164  df-laut 36165  df-ldil 36280  df-ltrn 36281  df-trl 36336

This theorem is referenced by:  cdlemk7  37026  cdlemk11  37027  cdlemk12  37028  cdlemk14  37032  cdlemk15  37033