Theorem cdleme51finvN 35844

Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 14-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemef50.b	|- B = ( Base ` K )
cdlemef50.l	|- .<_ = ( le ` K )
cdlemef50.j	|- .\/ = ( join ` K )
cdlemef50.m	|- ./\ = ( meet ` K )
cdlemef50.a	|- A = ( Atoms ` K )
cdlemef50.h	|- H = ( LHyp ` K )
cdlemef50.u	|- U = ( ( P .\/ Q ) ./\ W )
cdlemef50.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs50.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemef50.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
cdlemef51.v	|- V = ( ( Q .\/ P ) ./\ W )
cdlemef51.n	|- N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) )
cdlemefs51.o	|- O = ( ( Q .\/ P ) ./\ ( N .\/ ( ( u .\/ v ) ./\ W ) ) )
cdlemef51.g	|- G = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = O ) ) , [_ u / v ]_ N ) .\/ ( a ./\ W ) ) ) ) , a ) )

Assertion

Ref	Expression
cdleme51finvN	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> `' F = G )

Distinct variable groups:   a, b, c, s, t, u, v, x, y, z, ./\   .\/ , a, b, c, s, t, u, v, x, y, z   .<_ , a, b, c, s, t, u, v, x, y, z   A, a, b, c, s, t, u, v, x, y, z   B, a, b, c, s, t, u, v, x, y, z   D, a, b, c, s, v, x, y, z   E, a, b, c, x, y, z   F, a, b, c, u, v   H, a, b, c, s, t, u, v, x, y, z   K, a, b, c, s, t, u, v, x, y, z   P, a, b, c, s, t, u, v, x, y, z   Q, a, b, c, s, t, u, v, x, y, z   U, a, b, c, s, t, v, x, y, z   W, a, b, c, s, t, u, v, x, y, z   G, s, t, x, y, z   N, a, b, c, t, u, x, y, z   O, a, b, c, x, y, z   V, a, b, c, t, u, v, x, y, z

Allowed substitution hints:   D( u, t)   U( u)   E( v, u, t, s)   F( x, y, z, t, s)   G( v, u, a, b, c)   N( v, s)   O( v, u, t, s)   V( s)

Proof of Theorem cdleme51finvN

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemef50.b	. . . . 5 |- B = ( Base ` K )
2		cdlemef50.l	. . . . 5 |- .<_ = ( le ` K )
3		cdlemef50.j	. . . . 5 |- .\/ = ( join ` K )
4		cdlemef50.m	. . . . 5 |- ./\ = ( meet ` K )
5		cdlemef50.a	. . . . 5 |- A = ( Atoms ` K )
6		cdlemef50.h	. . . . 5 |- H = ( LHyp ` K )
7		cdlemef50.u	. . . . 5 |- U = ( ( P .\/ Q ) ./\ W )
8		cdlemef50.d	. . . . 5 |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9		cdlemefs50.e	. . . . 5 |- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10		cdlemef50.f	. . . . 5 |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme50f1o 35834	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F : B -1-1-onto-> B )
12		dff1o4 6145	. . . 4 |- ( F : B -1-1-onto-> B <-> ( F Fn B /\ `' F Fn B ) )
13	11, 12	sylib 208	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( F Fn B /\ `' F Fn B ) )
14	13	simprd 479	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> `' F Fn B )
15		cdlemef51.v	. . . . 5 |- V = ( ( Q .\/ P ) ./\ W )
16		cdlemef51.n	. . . . 5 |- N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) )
17		cdlemefs51.o	. . . . 5 |- O = ( ( Q .\/ P ) ./\ ( N .\/ ( ( u .\/ v ) ./\ W ) ) )
18		cdlemef51.g	. . . . 5 |- G = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = O ) ) , [_ u / v ]_ N ) .\/ ( a ./\ W ) ) ) ) , a ) )
19	1, 2, 3, 4, 5, 6, 15, 16, 17, 18	cdleme50f1o 35834	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> G : B -1-1-onto-> B )
20	19	3com23 1271	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> G : B -1-1-onto-> B )
21		f1ofn 6138	. . 3 |- ( G : B -1-1-onto-> B -> G Fn B )
22	20, 21	syl 17	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> G Fn B )
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18	cdleme51finvfvN 35843	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ e e. B ) -> ( `' F ` e ) = ( G ` e ) )
24	14, 22, 23	eqfnfvd 6314	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> `' F = G )

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   `'ccnv 5113   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270

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-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

This theorem is referenced by:  cdleme51finvtrN  35846