Theorem cdleme43aN 35777

Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 penultimate line: g(f(r)) = (p v q) ^ (g(s) v v₁). (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme43.b	|- B = ( Base ` K )
cdleme43.l	|- .<_ = ( le ` K )
cdleme43.j	|- .\/ = ( join ` K )
cdleme43.m	|- ./\ = ( meet ` K )
cdleme43.a	|- A = ( Atoms ` K )
cdleme43.h	|- H = ( LHyp ` K )
cdleme43.u	|- U = ( ( P .\/ Q ) ./\ W )
cdleme43.x	|- X = ( ( Q .\/ P ) ./\ W )
cdleme43.c	|- C = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme43.f	|- Z = ( ( P .\/ Q ) ./\ ( C .\/ ( ( R .\/ S ) ./\ W ) ) )
cdleme43.d	|- D = ( ( S .\/ X ) ./\ ( P .\/ ( ( Q .\/ S ) ./\ W ) ) )
cdleme43.g	|- G = ( ( Q .\/ P ) ./\ ( D .\/ ( ( Z .\/ S ) ./\ W ) ) )
cdleme43.e	|- E = ( ( D .\/ U ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) )
cdleme43.v	|- V = ( ( Z .\/ S ) ./\ W )
cdleme43.y	|- Y = ( ( R .\/ D ) ./\ W )

Assertion

Ref	Expression
cdleme43aN	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> G = ( ( P .\/ Q ) ./\ ( D .\/ V ) ) )

Proof of Theorem cdleme43aN

Step	Hyp	Ref	Expression
1		cdleme43.j	. . . 4 |- .\/ = ( join ` K )
2		cdleme43.a	. . . 4 |- A = ( Atoms ` K )
3	1, 2	hlatjcom 34654	. . 3 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
4		cdleme43.v	. . . . 5 |- V = ( ( Z .\/ S ) ./\ W )
5	4	oveq2i 6661	. . . 4 |- ( D .\/ V ) = ( D .\/ ( ( Z .\/ S ) ./\ W ) )
6	5	a1i 11	. . 3 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( D .\/ V ) = ( D .\/ ( ( Z .\/ S ) ./\ W ) ) )
7	3, 6	oveq12d 6668	. 2 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( ( P .\/ Q ) ./\ ( D .\/ V ) ) = ( ( Q .\/ P ) ./\ ( D .\/ ( ( Z .\/ S ) ./\ W ) ) ) )
8		cdleme43.g	. 2 |- G = ( ( Q .\/ P ) ./\ ( D .\/ ( ( Z .\/ S ) ./\ W ) ) )
9	7, 8	syl6reqr 2675	1 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> G = ( ( P .\/ Q ) ./\ ( D .\/ V ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (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

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-reu 2919  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-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-lub 16974  df-join 16976  df-lat 17046  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by: (None)