Theorem osumcllem7N 35248

Description: Lemma for osumclN 35253. (Contributed by NM, 24-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
osumcllem7N	|- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> p e. ( X .+ Y ) )

Distinct variable groups:   A, q   K, q   M, q   ._|_ , q   .+ , q   X, q   Y, q   q, p

Allowed substitution hints:   A( p)   C( q, p)   .+ ( p)   U( q, p)   .\/ ( q, p)   K( p)   .<_ ( q, p)   M( p)   ._|_ ( p)   X( p)   Y( p)

Proof of Theorem osumcllem7N

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp11 1091	. . . 4 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> K e. HL )
2		hllat 34650	. . . 4 |- ( K e. HL -> K e. Lat )
3	1, 2	syl 17	. . 3 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> K e. Lat )
4		simp12 1092	. . 3 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> X C_ A )
5		simp23 1096	. . 3 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> p e. A )
6		simp22 1095	. . 3 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> X =/= (/) )
7		inss2 3834	. . . . . 6 |- ( Y i^i M ) C_ M
8	7	sseli 3599	. . . . 5 |- ( q e. ( Y i^i M ) -> q e. M )
9	8	3ad2ant3 1084	. . . 4 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> q e. M )
10		osumcllem.m	. . . 4 |- M = ( X .+ { p } )
11	9, 10	syl6eleq 2711	. . 3 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> q e. ( X .+ { p } ) )
12		osumcllem.l	. . . 4 |- .<_ = ( le ` K )
13		osumcllem.j	. . . 4 |- .\/ = ( join ` K )
14		osumcllem.a	. . . 4 |- A = ( Atoms ` K )
15		osumcllem.p	. . . 4 |- .+ = ( +P ` K )
16	12, 13, 14, 15	elpaddatiN 35091	. . 3 |- ( ( ( K e. Lat /\ X C_ A /\ p e. A ) /\ ( X =/= (/) /\ q e. ( X .+ { p } ) ) ) -> E. r e. X q .<_ ( r .\/ p ) )
17	3, 4, 5, 6, 11, 16	syl32anc 1334	. 2 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> E. r e. X q .<_ ( r .\/ p ) )
18		simp11 1091	. . . 4 |- ( ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) /\ r e. X /\ q .<_ ( r .\/ p ) ) -> ( K e. HL /\ X C_ A /\ Y C_ A ) )
19		simp121 1193	. . . 4 |- ( ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) /\ r e. X /\ q .<_ ( r .\/ p ) ) -> X C_ ( ._|_ ` Y ) )
20		simp123 1195	. . . 4 |- ( ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) /\ r e. X /\ q .<_ ( r .\/ p ) ) -> p e. A )
21		simp2 1062	. . . 4 |- ( ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) /\ r e. X /\ q .<_ ( r .\/ p ) ) -> r e. X )
22		inss1 3833	. . . . 5 |- ( Y i^i M ) C_ Y
23		simp13 1093	. . . . 5 |- ( ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) /\ r e. X /\ q .<_ ( r .\/ p ) ) -> q e. ( Y i^i M ) )
24	22, 23	sseldi 3601	. . . 4 |- ( ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) /\ r e. X /\ q .<_ ( r .\/ p ) ) -> q e. Y )
25		simp3 1063	. . . 4 |- ( ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) /\ r e. X /\ q .<_ ( r .\/ p ) ) -> q .<_ ( r .\/ p ) )
26		osumcllem.o	. . . . 5 |- ._|_ = ( _|_P ` K )
27		osumcllem.c	. . . . 5 |- C = ( PSubCl ` K )
28		osumcllem.u	. . . . 5 |- U = ( ._|_ ` ( ._|_ ` ( X .+ Y ) ) )
29	12, 13, 14, 15, 26, 27, 10, 28	osumcllem6N 35247	. . . 4 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ p e. A ) /\ ( r e. X /\ q e. Y /\ q .<_ ( r .\/ p ) ) ) -> p e. ( X .+ Y ) )
30	18, 19, 20, 21, 24, 25, 29	syl123anc 1343	. . 3 |- ( ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) /\ r e. X /\ q .<_ ( r .\/ p ) ) -> p e. ( X .+ Y ) )
31	30	rexlimdv3a 3033	. 2 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> ( E. r e. X q .<_ ( r .\/ p ) -> p e. ( X .+ Y ) ) )
32	17, 31	mpd 15	1 |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ ( X C_ ( ._|_ ` Y ) /\ X =/= (/) /\ p e. A ) /\ q e. ( Y i^i M ) ) -> p e. ( X .+ Y ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   HLchlt 34637   +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-pmap 34790  df-padd 35082  df-polarityN 35189

This theorem is referenced by:  osumcllem8N  35249