Theorem cdlemksat 34829

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 1018	. 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 34828	. 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 1107	. 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 34123	. 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 1219	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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2648   class class class wbr 4401   |-> cmpt 4459   _I cid 4740   `'ccnv 4948   |` cres 4951   o. ccom 4953   ` cfv 5527   iota_crio 6161  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   Atomscatm 33247   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   trLctrl 34141

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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-riotaBAD 32943

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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-undef 6903  df-map 7327  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482  df-lvols 33483  df-lines 33484  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142

This theorem is referenced by:  cdlemk7  34831  cdlemk11  34832  cdlemk12  34833  cdlemk14  34837  cdlemk15  34838