Theorem 4atexlemex4 35359

Description: Lemma for 4atexlem7 35361. Show that when C = S, D satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)

Hypotheses

Ref	Expression
4thatlem.ph	|- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) )
4thatlem0.l	|- .<_ = ( le ` K )
4thatlem0.j	|- .\/ = ( join ` K )
4thatlem0.m	|- ./\ = ( meet ` K )
4thatlem0.a	|- A = ( Atoms ` K )
4thatlem0.h	|- H = ( LHyp ` K )
4thatlem0.u	|- U = ( ( P .\/ Q ) ./\ W )
4thatlem0.v	|- V = ( ( P .\/ S ) ./\ W )
4thatlem0.c	|- C = ( ( Q .\/ T ) ./\ ( P .\/ S ) )
4thatlem0.d	|- D = ( ( R .\/ T ) ./\ ( P .\/ S ) )

Assertion

Ref	Expression
4atexlemex4	|- ( ( ph /\ C = S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )

Distinct variable groups:   z, A   z, C   z, .\/   z, .<_   z, P   z, S   z, W   z, D

Allowed substitution hints:   ph( z)   Q( z)   R( z)   T( z)   U( z)   H( z)   K( z)   ./\ ( z)   V( z)

Proof of Theorem 4atexlemex4

Step	Hyp	Ref	Expression
1		4thatlem.ph	. . . 4 |- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) )
2		4thatlem0.l	. . . 4 |- .<_ = ( le ` K )
3		4thatlem0.j	. . . 4 |- .\/ = ( join ` K )
4		4thatlem0.a	. . . 4 |- A = ( Atoms ` K )
5		4thatlem0.u	. . . 4 |- U = ( ( P .\/ Q ) ./\ W )
6	1, 2, 3, 4, 5	4atexlemswapqr 35349	. . 3 |- ( ph -> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) )
7		4thatlem0.m	. . . . 5 |- ./\ = ( meet ` K )
8		4thatlem0.h	. . . . 5 |- H = ( LHyp ` K )
9		4thatlem0.v	. . . . 5 |- V = ( ( P .\/ S ) ./\ W )
10		4thatlem0.c	. . . . 5 |- C = ( ( Q .\/ T ) ./\ ( P .\/ S ) )
11		4thatlem0.d	. . . . 5 |- D = ( ( R .\/ T ) ./\ ( P .\/ S ) )
12	1, 2, 3, 7, 4, 8, 5, 9, 10, 11	4atexlemcnd 35358	. . . 4 |- ( ph -> C =/= D )
13		pm13.18 2875	. . . . . 6 |- ( ( C = S /\ C =/= D ) -> S =/= D )
14	13	necomd 2849	. . . . 5 |- ( ( C = S /\ C =/= D ) -> D =/= S )
15	14	expcom 451	. . . 4 |- ( C =/= D -> ( C = S -> D =/= S ) )
16	12, 15	syl 17	. . 3 |- ( ph -> ( C = S -> D =/= S ) )
17		biid 251	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) )
18		eqid 2622	. . . 4 |- ( ( P .\/ R ) ./\ W ) = ( ( P .\/ R ) ./\ W )
19	17, 2, 3, 7, 4, 8, 18, 9, 11	4atexlemex2 35357	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) /\ D =/= S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )
20	6, 16, 19	syl6an 568	. 2 |- ( ph -> ( C = S -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) )
21	20	imp 445	1 |- ( ( ph /\ C = S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   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-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-lhyp 35274

This theorem is referenced by:  4atexlemex6  35360