Theorem cdlemefrs32fva1 34039

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 1099	. . . 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 32926	. . . 4 |- ( R e. A -> R e. B )
5	1, 4	syl 17	. . 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 1056	. . 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 1100	. . 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 34030	. . 3 |- ( ( R e. B /\ ( P =/= Q /\ -. R .<_ W ) ) -> ( F ` R ) = [_ R / x ]_ O )
11	5, 6, 7, 10	syl12anc 1290	. 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 34038	. 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 2505	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   [_csb 3349   ifcif 3872   class class class wbr 4395   |-> cmpt 4454   ` cfv 5589   iota_crio 6269  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   Atomscatm 32900   HLchlt 32987   LHypclh 33620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-lhyp 33624

This theorem is referenced by:  cdlemefr32fva1  34048  cdlemefs32fva1  34061