Theorem lncvrat 33789

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 991	. . . 4 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> K e. HL )
3		simpl2 992	. . . 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 2454	. . . . 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 33783	. . . 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 661	. . 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 1081	. . . . . 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 1083	. . . . . 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 1014	. . . . . 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 1015	. . . . . 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 1016	. . . . . 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 1054	. . . . . . 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 1017	. . . . . . 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 4427	. . . . . 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 33440	. . . . . 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 1237	. . . . 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 4429	. . . 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 1187	. . 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 2953	. 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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2648   E.wrex 2800   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   <o ccvr 33270   Atomscatm 33271   HLchlt 33358   Linesclines 33501   pmapcpmap 33504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-lines 33508  df-pmap 33511

This theorem is referenced by:  2lnat  33791