Theorem cdleme17d4 35785

Description: TODO: FIX COMMENT. (Contributed by NM, 11-Apr-2013.)

Hypotheses

Ref	Expression
cdlemef46.b	|- B = ( Base ` K )
cdlemef46.l	|- .<_ = ( le ` K )
cdlemef46.j	|- .\/ = ( join ` K )
cdlemef46.m	|- ./\ = ( meet ` K )
cdlemef46.a	|- A = ( Atoms ` K )
cdlemef46.h	|- H = ( LHyp ` K )
cdlemef46.u	|- U = ( ( P .\/ Q ) ./\ W )
cdlemef46.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs46.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemef46.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 ) )

Assertion

Ref	Expression
cdleme17d4	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) -> ( F ` P ) = Q )

Distinct variable groups:   t, s, x, y, z, A   B, s, t, x, y, z   D, s, x, y, z   x, E, y, z   H, s, t, x, y, z   .\/ , s, t, x, y, z   K, s, t, x, y, z   .<_ , s, t, x, y, z   ./\ , s, t, x, y, z   P, s, t, x, y, z   Q, s, t, x, y, z   U, s, t, x, y, z   W, s, t, x, y, z

Allowed substitution hints:   D( t)   E( t, s)   F( x, y, z, t, s)

Proof of Theorem cdleme17d4

Step	Hyp	Ref	Expression
1		simp2l 1087	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> P e. A )
2		cdlemef46.b	. . . . 5 |- B = ( Base ` K )
3		cdlemef46.a	. . . . 5 |- A = ( Atoms ` K )
4	2, 3	atbase 34576	. . . 4 |- ( P e. A -> P 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 e. B )
6		cdlemef46.f	. . . 4 |- 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 ) )
7	6	cdleme31id 35682	. . 3 |- ( ( P e. B /\ P = Q ) -> ( F ` P ) = P )
8	5, 7	sylan 488	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) -> ( F ` P ) = P )
9		simpr 477	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) -> P = Q )
10	8, 9	eqtrd 2656	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) -> ( F ` P ) = Q )

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

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-ats 34554

This theorem is referenced by:  cdleme17d  35786