Theorem cdlemefrs32fva1 30883

Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT (Contributed by NM, 29-Mar-2013.)

Hypotheses

Ref	Expression
cdlemefrs27.b	|- B = ( Base ` K )
cdlemefrs27.l	|- .<_ = ( le ` K )
cdlemefrs27.j	|- .\/ = ( join ` K )
cdlemefrs27.m	|- ./\ = ( meet ` K )
cdlemefrs27.a	|- A = ( Atoms ` K )
cdlemefrs27.h	|- H = ( LHyp ` K )
cdlemefrs27.eq	|- ( s = R -> ( ph <-> ps ) )
cdlemefrs27.nb	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ ph ) ) ) -> N e. B )
cdlemefrs27.rnb	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> [_ R / s ]_ N e. B )
cdleme29frs.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
cdleme29frs.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )

Assertion

Ref	Expression
cdlemefrs32fva1	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> ( F ` R ) = [_ R / s ]_ N )

Distinct variable groups:   z, s, A   H, s   .\/ , s   K, s   .<_ , s   P, s   Q, s   R, s   W, s   ps, s   z, A   z, B   z, H   z, K   z, .<_   z, N   z, P   z, Q   z, R   z, W   ps, z   B, s   z, .\/   ./\ , s, z   ph, z   x, z, A   x, B   x, .\/   x, .<_   x, ./\   x, N   x, s, R   x, W   x, P   x, Q

Allowed substitution hints:   ph( x, s)   ps( x)   F( x, z, s)   H( x)   K( x)   N( s)   O( x, z, s)

Proof of Theorem cdlemefrs32fva1

Step	Hyp	Ref	Expression
1		simp2rl 1026	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> R e. A )
2		cdlemefrs27.b	. . . . 5 |- B = ( Base ` K )
3		cdlemefrs27.a	. . . . 5 |- A = ( Atoms ` K )
4	2, 3	atbase 29772	. . . 4 |- ( R e. A -> R e. B )
5	1, 4	syl 16	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> R e. B )
6		simp2l 983	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> P =/= Q )
7		simp2rr 1027	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> -. R .<_ W )
8		cdleme29frs.o	. . . 4 |- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
9		cdleme29frs.f	. . . 4 |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
10	8, 9	cdleme31fv1s 30874	. . 3 |- ( ( R e. B /\ ( P =/= Q /\ -. R .<_ W ) ) -> ( F ` R ) = [_ R / x ]_ O )
11	5, 6, 7, 10	syl12anc 1182	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> ( F ` R ) = [_ R / x ]_ O )
12		cdlemefrs27.l	. . 3 |- .<_ = ( le ` K )
13		cdlemefrs27.j	. . 3 |- .\/ = ( join ` K )
14		cdlemefrs27.m	. . 3 |- ./\ = ( meet ` K )
15		cdlemefrs27.h	. . 3 |- H = ( LHyp ` K )
16		cdlemefrs27.eq	. . 3 |- ( s = R -> ( ph <-> ps ) )
17		cdlemefrs27.nb	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ ph ) ) ) -> N e. B )
18		cdlemefrs27.rnb	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> [_ R / s ]_ N e. B )
19	2, 12, 13, 14, 3, 15, 16, 17, 18, 8	cdlemefrs32fva 30882	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> [_ R / x ]_ O = [_ R / s ]_ N )
20	11, 19	eqtrd 2436	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ ps ) -> ( F ` R ) = [_ R / s ]_ N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   [_csb 3211   ifcif 3699   class class class wbr 4172   e. cmpt 4226   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466

This theorem is referenced by:  cdlemefr32fva1  30892  cdlemefs32fva1  30905

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470