Theorem llni2 34798

Description: The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.)

Hypotheses

Ref	Expression
llni2.j	|- .\/ = ( join ` K )
llni2.a	|- A = ( Atoms ` K )
llni2.n	|- N = ( LLines ` K )

Assertion

Ref	Expression
llni2	|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. N )

Proof of Theorem llni2

Dummy variables s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1065	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> P e. A )
2		simpl3 1066	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> Q e. A )
3		simpr 477	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> P =/= Q )
4		eqidd 2623	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) = ( P .\/ Q ) )
5		neeq1 2856	. . . . 5 |- ( r = P -> ( r =/= s <-> P =/= s ) )
6		oveq1 6657	. . . . . 6 |- ( r = P -> ( r .\/ s ) = ( P .\/ s ) )
7	6	eqeq2d 2632	. . . . 5 |- ( r = P -> ( ( P .\/ Q ) = ( r .\/ s ) <-> ( P .\/ Q ) = ( P .\/ s ) ) )
8	5, 7	anbi12d 747	. . . 4 |- ( r = P -> ( ( r =/= s /\ ( P .\/ Q ) = ( r .\/ s ) ) <-> ( P =/= s /\ ( P .\/ Q ) = ( P .\/ s ) ) ) )
9		neeq2 2857	. . . . 5 |- ( s = Q -> ( P =/= s <-> P =/= Q ) )
10		oveq2 6658	. . . . . 6 |- ( s = Q -> ( P .\/ s ) = ( P .\/ Q ) )
11	10	eqeq2d 2632	. . . . 5 |- ( s = Q -> ( ( P .\/ Q ) = ( P .\/ s ) <-> ( P .\/ Q ) = ( P .\/ Q ) ) )
12	9, 11	anbi12d 747	. . . 4 |- ( s = Q -> ( ( P =/= s /\ ( P .\/ Q ) = ( P .\/ s ) ) <-> ( P =/= Q /\ ( P .\/ Q ) = ( P .\/ Q ) ) ) )
13	8, 12	rspc2ev 3324	. . 3 |- ( ( P e. A /\ Q e. A /\ ( P =/= Q /\ ( P .\/ Q ) = ( P .\/ Q ) ) ) -> E. r e. A E. s e. A ( r =/= s /\ ( P .\/ Q ) = ( r .\/ s ) ) )
14	1, 2, 3, 4, 13	syl112anc 1330	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> E. r e. A E. s e. A ( r =/= s /\ ( P .\/ Q ) = ( r .\/ s ) ) )
15		simpl1 1064	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> K e. HL )
16		eqid 2622	. . . . 5 |- ( Base ` K ) = ( Base ` K )
17		llni2.j	. . . . 5 |- .\/ = ( join ` K )
18		llni2.a	. . . . 5 |- A = ( Atoms ` K )
19	16, 17, 18	hlatjcl 34653	. . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
20	19	adantr 481	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( Base ` K ) )
21		llni2.n	. . . 4 |- N = ( LLines ` K )
22	16, 17, 18, 21	islln3 34796	. . 3 |- ( ( K e. HL /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) e. N <-> E. r e. A E. s e. A ( r =/= s /\ ( P .\/ Q ) = ( r .\/ s ) ) ) )
23	15, 20, 22	syl2anc 693	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( ( P .\/ Q ) e. N <-> E. r e. A E. s e. A ( r =/= s /\ ( P .\/ Q ) = ( r .\/ s ) ) ) )
24	14, 23	mpbird 247	1 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. N )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   ` cfv 5888  (class class class)co 6650   Basecbs 15857   joincjn 16944   Atomscatm 34550   HLchlt 34637   LLinesclln 34777

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

This theorem is referenced by:  2atneat  34801  islln2a  34803  2at0mat0  34811  ps-2c  34814  lplnnle2at  34827  2atmat  34847  lplnexllnN  34850  dalempjsen  34939  dalemcea  34946  dalem2  34947  dalemdea  34948  dalem16  34965  dalemcjden  34978  dalem23  34982  dalem54  35012  dalem60  35018  llnexchb2  35155  arglem1N  35477  cdlemc5  35482  cdleme20l1  35608  cdleme20l2  35609  cdleme20l  35610  cdleme22b  35629  cdlemeg46req  35817  cdlemh  36105