Theorem lncvrat 32763

Description: A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)

Hypotheses

Ref	Expression
lncvrat.b	|- B = ( Base ` K )
lncvrat.l	|- .<_ = ( le ` K )
lncvrat.c	|- C = ( <o ` K )
lncvrat.a	|- A = ( Atoms ` K )
lncvrat.n	|- N = ( Lines ` K )
lncvrat.m	|- M = ( pmap ` K )

Assertion

Ref	Expression
lncvrat	|- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> P C X )

Proof of Theorem lncvrat

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

Step	Hyp	Ref	Expression
1		simprl 755	. . 3 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> ( M ` X ) e. N )
2		simpl1 998	. . . 4 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> K e. HL )
3		simpl2 999	. . . 4 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> X e. B )
4		lncvrat.b	. . . . 5 |- B = ( Base ` K )
5		eqid 2400	. . . . 5 |- ( join ` K ) = ( join ` K )
6		lncvrat.a	. . . . 5 |- A = ( Atoms ` K )
7		lncvrat.n	. . . . 5 |- N = ( Lines ` K )
8		lncvrat.m	. . . . 5 |- M = ( pmap ` K )
9	4, 5, 6, 7, 8	isline3 32757	. . . 4 |- ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) e. N <-> E. q e. A E. r e. A ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) )
10	2, 3, 9	syl2anc 659	. . 3 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> ( ( M ` X ) e. N <-> E. q e. A E. r e. A ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) )
11	1, 10	mpbid 210	. 2 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> E. q e. A E. r e. A ( q =/= r /\ X = ( q ( join ` K ) r ) ) )
12		simp1l1 1088	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> K e. HL )
13		simp1l3 1090	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> P e. A )
14		simp2l 1021	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> q e. A )
15		simp2r 1022	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> r e. A )
16		simp3l 1023	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> q =/= r )
17		simp1rr 1061	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> P .<_ X )
18		simp3r 1024	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> X = ( q ( join ` K ) r ) )
19	17, 18	breqtrd 4416	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> P .<_ ( q ( join ` K ) r ) )
20		lncvrat.l	. . . . . . 7 |- .<_ = ( le ` K )
21		lncvrat.c	. . . . . . 7 |- C = ( <o ` K )
22	20, 5, 21, 6	atcvrj2 32414	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ q e. A /\ r e. A ) /\ ( q =/= r /\ P .<_ ( q ( join ` K ) r ) ) ) -> P C ( q ( join ` K ) r ) )
23	12, 13, 14, 15, 16, 19, 22	syl132anc 1246	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> P C ( q ( join ` K ) r ) )
24	23, 18	breqtrrd 4418	. . . 4 |- ( ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) /\ ( q e. A /\ r e. A ) /\ ( q =/= r /\ X = ( q ( join ` K ) r ) ) ) -> P C X )
25	24	3exp 1194	. . 3 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> ( ( q e. A /\ r e. A ) -> ( ( q =/= r /\ X = ( q ( join ` K ) r ) ) -> P C X ) ) )
26	25	rexlimdvv 2899	. 2 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> ( E. q e. A E. r e. A ( q =/= r /\ X = ( q ( join ` K ) r ) ) -> P C X ) )
27	11, 26	mpd 15	1 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> P C X )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 972   = wceq 1403   e. wcel 1840   =/= wne 2596   E.wrex 2752   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   <o ccvr 32244   Atomscatm 32245   HLchlt 32332   Linesclines 32475   pmapcpmap 32478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-lines 32482  df-pmap 32485

This theorem is referenced by:  2lnat  32765