Theorem pexmidlem8N 35263

Description: Lemma for pexmidN 35255. The contradiction of pexmidlem6N 35261 and pexmidlem7N 35262 shows that there can be no atom p that is not in X .+ ( ._|_ ` X ), which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pexmidALT.a	|- A = ( Atoms ` K )
pexmidALT.p	|- .+ = ( +P ` K )
pexmidALT.o	|- ._|_ = ( _|_P ` K )

Assertion

Ref	Expression
pexmidlem8N	|- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> ( X .+ ( ._|_ ` X ) ) = A )

Proof of Theorem pexmidlem8N

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nonconne 2806	. 2 |- -. ( X = X /\ X =/= X )
2		simpll 790	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> K e. HL )
3		simplr 792	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> X C_ A )
4		pexmidALT.a	. . . . . . 7 |- A = ( Atoms ` K )
5		pexmidALT.o	. . . . . . 7 |- ._|_ = ( _|_P ` K )
6	4, 5	polssatN 35194	. . . . . 6 |- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) C_ A )
7	6	adantr 481	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> ( ._|_ ` X ) C_ A )
8		pexmidALT.p	. . . . . 6 |- .+ = ( +P ` K )
9	4, 8	paddssat 35100	. . . . 5 |- ( ( K e. HL /\ X C_ A /\ ( ._|_ ` X ) C_ A ) -> ( X .+ ( ._|_ ` X ) ) C_ A )
10	2, 3, 7, 9	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> ( X .+ ( ._|_ ` X ) ) C_ A )
11		df-pss 3590	. . . . . . 7 |- ( ( X .+ ( ._|_ ` X ) ) C. A <-> ( ( X .+ ( ._|_ ` X ) ) C_ A /\ ( X .+ ( ._|_ ` X ) ) =/= A ) )
12		pssnel 4039	. . . . . . 7 |- ( ( X .+ ( ._|_ ` X ) ) C. A -> E. p ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) )
13	11, 12	sylbir 225	. . . . . 6 |- ( ( ( X .+ ( ._|_ ` X ) ) C_ A /\ ( X .+ ( ._|_ ` X ) ) =/= A ) -> E. p ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) )
14		df-rex 2918	. . . . . 6 |- ( E. p e. A -. p e. ( X .+ ( ._|_ ` X ) ) <-> E. p ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) )
15	13, 14	sylibr 224	. . . . 5 |- ( ( ( X .+ ( ._|_ ` X ) ) C_ A /\ ( X .+ ( ._|_ ` X ) ) =/= A ) -> E. p e. A -. p e. ( X .+ ( ._|_ ` X ) ) )
16		simplll 798	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) /\ ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> K e. HL )
17		simpllr 799	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) /\ ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> X C_ A )
18		simprl 794	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) /\ ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> p e. A )
19		simplrl 800	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) /\ ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )
20		simplrr 801	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) /\ ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> X =/= (/) )
21		simprr 796	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) /\ ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> -. p e. ( X .+ ( ._|_ ` X ) ) )
22		eqid 2622	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
23		eqid 2622	. . . . . . . . . 10 |- ( join ` K ) = ( join ` K )
24		eqid 2622	. . . . . . . . . 10 |- ( X .+ { p } ) = ( X .+ { p } )
25	22, 23, 4, 8, 5, 24	pexmidlem6N 35261	. . . . . . . . 9 |- ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( X .+ { p } ) = X )
26	22, 23, 4, 8, 5, 24	pexmidlem7N 35262	. . . . . . . . 9 |- ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( X .+ { p } ) =/= X )
27	25, 26	jca 554	. . . . . . . 8 |- ( ( ( K e. HL /\ X C_ A /\ p e. A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( ( X .+ { p } ) = X /\ ( X .+ { p } ) =/= X ) )
28	16, 17, 18, 19, 20, 21, 27	syl33anc 1341	. . . . . . 7 |- ( ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) /\ ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( ( X .+ { p } ) = X /\ ( X .+ { p } ) =/= X ) )
29		nonconne 2806	. . . . . . . 8 |- -. ( ( X .+ { p } ) = X /\ ( X .+ { p } ) =/= X )
30	29, 1	2false 365	. . . . . . 7 |- ( ( ( X .+ { p } ) = X /\ ( X .+ { p } ) =/= X ) <-> ( X = X /\ X =/= X ) )
31	28, 30	sylib 208	. . . . . 6 |- ( ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) /\ ( p e. A /\ -. p e. ( X .+ ( ._|_ ` X ) ) ) ) -> ( X = X /\ X =/= X ) )
32	31	rexlimdvaa 3032	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> ( E. p e. A -. p e. ( X .+ ( ._|_ ` X ) ) -> ( X = X /\ X =/= X ) ) )
33	15, 32	syl5 34	. . . 4 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> ( ( ( X .+ ( ._|_ ` X ) ) C_ A /\ ( X .+ ( ._|_ ` X ) ) =/= A ) -> ( X = X /\ X =/= X ) ) )
34	10, 33	mpand 711	. . 3 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> ( ( X .+ ( ._|_ ` X ) ) =/= A -> ( X = X /\ X =/= X ) ) )
35	34	necon1bd 2812	. 2 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> ( -. ( X = X /\ X =/= X ) -> ( X .+ ( ._|_ ` X ) ) = A ) )
36	1, 35	mpi 20	1 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ( ._|_ ` ( ._|_ ` X ) ) = X /\ X =/= (/) ) ) -> ( X .+ ( ._|_ ` X ) ) = A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   C. wpss 3575   (/)c0 3915   {csn 4177   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Atomscatm 34550   HLchlt 34637   +Pcpadd 35081   _|_PcpolN 35188

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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  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-iin 4523  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-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  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-psubsp 34789  df-pmap 34790  df-padd 35082  df-polarityN 35189  df-psubclN 35221

This theorem is referenced by:  pexmidALTN  35264