Theorem cdlemk18-3N 34883

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. N, Y, O, D are k, sigma₂ (p), k₁, f₁. (Contributed by NM, 7-Jul-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemk3.b	|- B = ( Base ` K )
cdlemk3.l	|- .<_ = ( le ` K )
cdlemk3.j	|- .\/ = ( join ` K )
cdlemk3.m	|- ./\ = ( meet ` K )
cdlemk3.a	|- A = ( Atoms ` K )
cdlemk3.h	|- H = ( LHyp ` K )
cdlemk3.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk3.r	|- R = ( ( trL ` K ) ` W )
cdlemk3.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk3.u1	|- Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )

Assertion

Ref	Expression
cdlemk18-3N	|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( D Y F ) ` P ) = ( N ` P ) )

Distinct variable groups:   e, d, f, i, ./\   .<_ , i   .\/ , d, e, f, i   A, i   j, d, D, e, f, i   f, F, i   i, H   i, K   f, N, i   P, d, e, f, i   R, d, e, f, i   T, d, e, f, i   W, d, e, f, i   ./\ , j   .<_ , j   .\/ , j   A, j   j, F   j, H   j, K   j, N   P, j   R, j   S, d, e, j   T, j   j, W   F, d, e

Allowed substitution hints:   A( e, f, d)   B( e, f, i, j, d)   S( f, i)   H( e, f, d)   K( e, f, d)   .<_ ( e, f, d)   N( e, d)   Y( e, f, i, j, d)

Proof of Theorem cdlemk18-3N

Step	Hyp	Ref	Expression
1		simp22 1022	. . . 4 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> D e. T )
2		simp21 1021	. . . 4 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
3		cdlemk3.b	. . . . 5 |- B = ( Base ` K )
4		cdlemk3.l	. . . . 5 |- .<_ = ( le ` K )
5		cdlemk3.j	. . . . 5 |- .\/ = ( join ` K )
6		cdlemk3.m	. . . . 5 |- ./\ = ( meet ` K )
7		cdlemk3.a	. . . . 5 |- A = ( Atoms ` K )
8		cdlemk3.h	. . . . 5 |- H = ( LHyp ` K )
9		cdlemk3.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
10		cdlemk3.r	. . . . 5 |- R = ( ( trL ` K ) ` W )
11		cdlemk3.s	. . . . 5 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
12		cdlemk3.u1	. . . . 5 |- Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
13		eqid 2454	. . . . 5 |- ( S ` D ) = ( S ` D )
14		eqid 2454	. . . . 5 |- ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
15	3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cdlemkuu 34878	. . . 4 |- ( ( D e. T /\ F e. T ) -> ( D Y F ) = ( ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) ` F ) )
16	1, 2, 15	syl2anc 661	. . 3 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( D Y F ) = ( ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) ` F ) )
17	16	fveq1d 5802	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( D Y F ) ` P ) = ( ( ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) ` F ) ` P ) )
18	3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14	cdlemk18-2N 34869	. 2 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( N ` P ) = ( ( ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) ` F ) ` P ) )
19	17, 18	eqtr4d 2498	1 |- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( D Y F ) ` P ) = ( N ` P ) )

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   |-> cmpt2 6203   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: (None)