Theorem osumcllem1N 35242

Description: Lemma for osumclN 35253. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
osumcllem.l	|- .<_ = ( le ` K )
osumcllem.j	|- .\/ = ( join ` K )
osumcllem.a	|- A = ( Atoms ` K )
osumcllem.p	|- .+ = ( +P ` K )
osumcllem.o	|- ._|_ = ( _|_P ` K )
osumcllem.c	|- C = ( PSubCl ` K )
osumcllem.m	|- M = ( X .+ { p } )
osumcllem.u	|- U = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) )

Assertion

Ref	Expression
osumcllem1N	|- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( U i^i M ) = M )

Proof of Theorem osumcllem1N

Step	Hyp	Ref	Expression
1		osumcllem.m	. . 3 |- M = ( X .+ { p } )
2		osumcllem.a	. . . . . . 7 |- A = ( Atoms ` K )
3		osumcllem.p	. . . . . . 7 |- .+ = ( +P ` K )
4	2, 3	sspadd1 35101	. . . . . 6 |- ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> X C_ ( X .+ Y ) )
5	4	adantr 481	. . . . 5 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ ( X .+ Y ) )
6		simpl1 1064	. . . . . . 7 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> K e. HL )
7	2, 3	paddssat 35100	. . . . . . . 8 |- ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) C_ A )
8	7	adantr 481	. . . . . . 7 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( X .+ Y ) C_ A )
9		osumcllem.o	. . . . . . . 8 |- ._|_ = ( _|_P ` K )
10	2, 9	2polssN 35201	. . . . . . 7 |- ( ( K e. HL /\ ( X .+ Y ) C_ A ) -> ( X .+ Y ) C_ ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) )
11	6, 8, 10	syl2anc 693	. . . . . 6 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( X .+ Y ) C_ ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) )
12		osumcllem.u	. . . . . 6 |- U = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) )
13	11, 12	syl6sseqr 3652	. . . . 5 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( X .+ Y ) C_ U )
14	5, 13	sstrd 3613	. . . 4 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ U )
15		simpr 477	. . . . 5 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> p e. U )
16	15	snssd 4340	. . . 4 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> { p } C_ U )
17		simpl2 1065	. . . . 5 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ A )
18	2, 9	polssatN 35194	. . . . . . . . 9 |- ( ( K e. HL /\ ( X .+ Y ) C_ A ) -> ( ._|_ ` ( X .+ Y ) ) C_ A )
19	6, 8, 18	syl2anc 693	. . . . . . . 8 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ._|_ ` ( X .+ Y ) ) C_ A )
20	2, 9	polssatN 35194	. . . . . . . 8 |- ( ( K e. HL /\ ( ._|_ ` ( X .+ Y ) ) C_ A ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) C_ A )
21	6, 19, 20	syl2anc 693	. . . . . . 7 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) C_ A )
22	12, 21	syl5eqss 3649	. . . . . 6 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> U C_ A )
23	16, 22	sstrd 3613	. . . . 5 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> { p } C_ A )
24		eqid 2622	. . . . . . . 8 |- ( PSubSp ` K ) = ( PSubSp ` K )
25	2, 24, 9	polsubN 35193	. . . . . . 7 |- ( ( K e. HL /\ ( ._|_ ` ( X .+ Y ) ) C_ A ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) e. ( PSubSp ` K ) )
26	6, 19, 25	syl2anc 693	. . . . . 6 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) ) e. ( PSubSp ` K ) )
27	12, 26	syl5eqel 2705	. . . . 5 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> U e. ( PSubSp ` K ) )
28	2, 24, 3	paddss 35131	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ { p } C_ A /\ U e. ( PSubSp ` K ) ) ) -> ( ( X C_ U /\ { p } C_ U ) <-> ( X .+ { p } ) C_ U ) )
29	6, 17, 23, 27, 28	syl13anc 1328	. . . 4 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( ( X C_ U /\ { p } C_ U ) <-> ( X .+ { p } ) C_ U ) )
30	14, 16, 29	mpbi2and 956	. . 3 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( X .+ { p } ) C_ U )
31	1, 30	syl5eqss 3649	. 2 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> M C_ U )
32		sseqin2 3817	. 2 |- ( M C_ U <-> ( U i^i M ) = M )
33	31, 32	sylib 208	1 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> ( U i^i M ) = M )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Atomscatm 34550   HLchlt 34637   PSubSpcpsubsp 34782   +Pcpadd 35081   _|_PcpolN 35188   PSubClcpscN 35220

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-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-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

This theorem is referenced by:  osumcllem2N  35243  osumcllem9N  35250