Theorem ltrniotafvawN 35866

Description: Version of cdleme46fvaw 35789 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ltrniotaval.l	|- .<_ = ( le ` K )
ltrniotaval.a	|- A = ( Atoms ` K )
ltrniotaval.h	|- H = ( LHyp ` K )
ltrniotaval.t	|- T = ( ( LTrn ` K ) ` W )
ltrniotaval.f	|- F = ( iota_ f e. T ( f ` P ) = Q )

Assertion

Ref	Expression
ltrniotafvawN	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( ( F ` R ) e. A /\ -. ( F ` R ) .<_ W ) )

Distinct variable groups:   A, f   f, H   f, K   .<_ , f   P, f   Q, f   T, f   f, W

Allowed substitution hints:   R( f)   F( f)

Proof of Theorem ltrniotafvawN

Dummy variables s t x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( Base ` K ) = ( Base ` K )
2		ltrniotaval.l	. 2 |- .<_ = ( le ` K )
3		eqid 2622	. 2 |- ( join ` K ) = ( join ` K )
4		eqid 2622	. 2 |- ( meet ` K ) = ( meet ` K )
5		ltrniotaval.a	. 2 |- A = ( Atoms ` K )
6		ltrniotaval.h	. 2 |- H = ( LHyp ` K )
7		eqid 2622	. 2 |- ( ( P ( join ` K ) Q ) ( meet ` K ) W ) = ( ( P ( join ` K ) Q ) ( meet ` K ) W )
8		eqid 2622	. 2 |- ( ( t ( join ` K ) ( ( P ( join ` K ) Q ) ( meet ` K ) W ) ) ( meet ` K ) ( Q ( join ` K ) ( ( P ( join ` K ) t ) ( meet ` K ) W ) ) ) = ( ( t ( join ` K ) ( ( P ( join ` K ) Q ) ( meet ` K ) W ) ) ( meet ` K ) ( Q ( join ` K ) ( ( P ( join ` K ) t ) ( meet ` K ) W ) ) )
9		eqid 2622	. 2 |- ( ( P ( join ` K ) Q ) ( meet ` K ) ( ( ( t ( join ` K ) ( ( P ( join ` K ) Q ) ( meet ` K ) W ) ) ( meet ` K ) ( Q ( join ` K ) ( ( P ( join ` K ) t ) ( meet ` K ) W ) ) ) ( join ` K ) ( ( s ( join ` K ) t ) ( meet ` K ) W ) ) ) = ( ( P ( join ` K ) Q ) ( meet ` K ) ( ( ( t ( join ` K ) ( ( P ( join ` K ) Q ) ( meet ` K ) W ) ) ( meet ` K ) ( Q ( join ` K ) ( ( P ( join ` K ) t ) ( meet ` K ) W ) ) ) ( join ` K ) ( ( s ( join ` K ) t ) ( meet ` K ) W ) ) )
10		eqid 2622	. 2 |- ( x e. ( Base ` K ) |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. ( Base ` K ) A. s e. A ( ( -. s .<_ W /\ ( s ( join ` K ) ( x ( meet ` K ) W ) ) = x ) -> z = ( if ( s .<_ ( P ( join ` K ) Q ) , ( iota_ y e. ( Base ` K ) A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P ( join ` K ) Q ) ) -> y = ( ( P ( join ` K ) Q ) ( meet ` K ) ( ( ( t ( join ` K ) ( ( P ( join ` K ) Q ) ( meet ` K ) W ) ) ( meet ` K ) ( Q ( join ` K ) ( ( P ( join ` K ) t ) ( meet ` K ) W ) ) ) ( join ` K ) ( ( s ( join ` K ) t ) ( meet ` K ) W ) ) ) ) ) , [_ s / t ]_ ( ( t ( join ` K ) ( ( P ( join ` K ) Q ) ( meet ` K ) W ) ) ( meet ` K ) ( Q ( join ` K ) ( ( P ( join ` K ) t ) ( meet ` K ) W ) ) ) ) ( join ` K ) ( x ( meet ` K ) W ) ) ) ) , x ) ) = ( x e. ( Base ` K ) |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. ( Base ` K ) A. s e. A ( ( -. s .<_ W /\ ( s ( join ` K ) ( x ( meet ` K ) W ) ) = x ) -> z = ( if ( s .<_ ( P ( join ` K ) Q ) , ( iota_ y e. ( Base ` K ) A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P ( join ` K ) Q ) ) -> y = ( ( P ( join ` K ) Q ) ( meet ` K ) ( ( ( t ( join ` K ) ( ( P ( join ` K ) Q ) ( meet ` K ) W ) ) ( meet ` K ) ( Q ( join ` K ) ( ( P ( join ` K ) t ) ( meet ` K ) W ) ) ) ( join ` K ) ( ( s ( join ` K ) t ) ( meet ` K ) W ) ) ) ) ) , [_ s / t ]_ ( ( t ( join ` K ) ( ( P ( join ` K ) Q ) ( meet ` K ) W ) ) ( meet ` K ) ( Q ( join ` K ) ( ( P ( join ` K ) t ) ( meet ` K ) W ) ) ) ) ( join ` K ) ( x ( meet ` K ) W ) ) ) ) , x ) )
11		ltrniotaval.t	. 2 |- T = ( ( LTrn ` K ) ` W )
12		ltrniotaval.f	. 2 |- F = ( iota_ f e. T ( f ` P ) = Q )
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemg1fvawlemN 35861	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( ( F ` R ) e. A /\ -. ( F ` R ) .<_ W ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by: (None)