Theorem cdlemksel 34414

Description: Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma(p) function to be a translation. TODO: combine cdlemki 34410? (Contributed by NM, 26-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
cdlemksel	|- ( ( ( ( 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 )

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 cdlemksel

Step	Hyp	Ref	Expression
1		simp13 1041	. . 3 |- ( ( ( ( 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 ) ) ) -> G e. T )
2		cdlemk.b	. . . 4 |- B = ( Base ` K )
3		cdlemk.l	. . . 4 |- .<_ = ( le ` K )
4		cdlemk.j	. . . 4 |- .\/ = ( join ` K )
5		cdlemk.a	. . . 4 |- A = ( Atoms ` K )
6		cdlemk.h	. . . 4 |- H = ( LHyp ` K )
7		cdlemk.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
8		cdlemk.r	. . . 4 |- R = ( ( trL ` K ) ` W )
9		cdlemk.m	. . . 4 |- ./\ = ( meet ` K )
10		cdlemk.s	. . . 4 |- 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	cdlemksv 34413	. . 3 |- ( G e. T -> ( S ` G ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )
12	1, 11	syl 17	. 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 ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )
13		eqid 2452	. . 3 |- ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
14	2, 3, 4, 5, 6, 7, 8, 9, 13	cdlemki 34410	. 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 ) ) ) -> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) e. T )
15	12, 14	eqeltrd 2530	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 ) e. T )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 375   /\ w3a 986   = wceq 1448   e. wcel 1891   =/= wne 2622   class class class wbr 4374   |-> cmpt 4433   _I cid 4722   `'ccnv 4811   |` cres 4814   o. ccom 4816   ` cfv 5561   iota_crio 6237  (class class class)co 6276   Basecbs 15132   lecple 15208   joincjn 16200   meetcmee 16201   Atomscatm 32831   HLchlt 32918   LHypclh 33551   LTrncltrn 33668   trLctrl 33726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-riotaBAD 32527

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-iun 4250  df-iin 4251  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-1st 6781  df-2nd 6782  df-undef 7007  df-map 7461  df-preset 16184  df-poset 16202  df-plt 16215  df-lub 16231  df-glb 16232  df-join 16233  df-meet 16234  df-p0 16296  df-p1 16297  df-lat 16303  df-clat 16365  df-oposet 32744  df-ol 32746  df-oml 32747  df-covers 32834  df-ats 32835  df-atl 32866  df-cvlat 32890  df-hlat 32919  df-llines 33065  df-lplanes 33066  df-lvols 33067  df-lines 33068  df-psubsp 33070  df-pmap 33071  df-padd 33363  df-lhyp 33555  df-laut 33556  df-ldil 33671  df-ltrn 33672  df-trl 33727

This theorem is referenced by:  cdlemksat  34415  cdlemksv2  34416  cdlemk12  34419  cdlemkoatnle  34420