Theorem cdlemk14-2N 35551

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 19 on p. 119. Q, C are k₂, f₂. (Contributed by NM, 1-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
cdlemk14-2N	|- ( ( ( 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 ) ) ) -> ( N ` P ) .<_ ( ( Q ` P ) .\/ ( R ` ( F o. `' C ) ) ) )

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 cdlemk14-2N

Step	Hyp	Ref	Expression
1		simp11 1021	. . 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 1022	. . 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 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 ) ) ) -> F e. T )
5		simp22 1025	. 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 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 ) ) ) -> N e. T )
7		simp33 1029	. 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 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 ) ) ) -> ( R ` F ) = ( R ` N ) )
9		simp32l 1116	. 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 1117	. 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 1027	. 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	cdlemk14 35527	. 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 ) ) ) -> ( N ` P ) .<_ ( ( Q ` P ) .\/ ( R ` ( F o. `' C ) ) ) )
23	3, 4, 5, 6, 7, 8, 9, 10, 11, 22	syl333anc 1255	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 ) ) ) -> ( N ` P ) .<_ ( ( Q ` P ) .\/ ( R ` ( F o. `' C ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   =/= wne 2657   class class class wbr 4442   |-> cmpt 4500   _I cid 4785   `'ccnv 4993   |` cres 4996   o. ccom 4998   ` cfv 5581   iota_crio 6237  (class class class)co 6277   Basecbs 14481   lecple 14553   joincjn 15422   meetcmee 15423   Atomscatm 33937   HLchlt 34024   LHypclh 34657   LTrncltrn 34774   trLctrl 34831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-riotaBAD 33633

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-undef 6994  df-map 7414  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-llines 34171  df-lplanes 34172  df-lvols 34173  df-lines 34174  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832

This theorem is referenced by: (None)