Theorem lplnllnneN 34842

Description: Two lattice lines defined by atoms defining a lattice plane are not equal. (Contributed by NM, 9-Oct-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lplnri1.j	|- .\/ = ( join ` K )
lplnri1.a	|- A = ( Atoms ` K )
lplnri1.p	|- P = ( LPlanes ` K )
lplnri1.y	|- Y = ( ( Q .\/ R ) .\/ S )

Assertion

Ref	Expression
lplnllnneN	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> ( Q .\/ S ) =/= ( R .\/ S ) )

Proof of Theorem lplnllnneN

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( le ` K ) = ( le ` K )
2		lplnri1.j	. . 3 |- .\/ = ( join ` K )
3		lplnri1.a	. . 3 |- A = ( Atoms ` K )
4		lplnri1.p	. . 3 |- P = ( LPlanes ` K )
5		lplnri1.y	. . 3 |- Y = ( ( Q .\/ R ) .\/ S )
6	1, 2, 3, 4, 5	lplnriaN 34836	. 2 |- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> -. Q ( le ` K ) ( R .\/ S ) )
7		simpl1 1064	. . . . . 6 |- ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> K e. HL )
8		simpl21 1139	. . . . . 6 |- ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> Q e. A )
9		simpl23 1141	. . . . . 6 |- ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> S e. A )
10	1, 2, 3	hlatlej1 34661	. . . . . 6 |- ( ( K e. HL /\ Q e. A /\ S e. A ) -> Q ( le ` K ) ( Q .\/ S ) )
11	7, 8, 9, 10	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> Q ( le ` K ) ( Q .\/ S ) )
12		simpr 477	. . . . 5 |- ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> ( Q .\/ S ) = ( R .\/ S ) )
13	11, 12	breqtrd 4679	. . . 4 |- ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) /\ ( Q .\/ S ) = ( R .\/ S ) ) -> Q ( le ` K ) ( R .\/ S ) )
14	13	ex 450	. . 3 |- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> ( ( Q .\/ S ) = ( R .\/ S ) -> Q ( le ` K ) ( R .\/ S ) ) )
15	14	necon3bd 2808	. 2 |- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> ( -. Q ( le ` K ) ( R .\/ S ) -> ( Q .\/ S ) =/= ( R .\/ S ) ) )
16	6, 15	mpd 15	1 |- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> ( Q .\/ S ) =/= ( R .\/ S ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Atomscatm 34550   HLchlt 34637   LPlanesclpl 34778

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-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

This theorem is referenced by:  cdleme16aN  35546