Theorem lncvrat 33253

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 762	. . 3 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> ( M ` X ) e. N )
2		simpl1 1008	. . . 4 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( ( M ` X ) e. N /\ P .<_ X ) ) -> K e. HL )
3		simpl2 1009	. . . 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 2422	. . . . 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 33247	. . . 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 665	. . 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 213	. 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 1098	. . . . . 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 1100	. . . . . 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 1031	. . . . . 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 1032	. . . . . 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 1033	. . . . . 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 1071	. . . . . . 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 1034	. . . . . . 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 4384	. . . . . 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 32904	. . . . . 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 1282	. . . . 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 4386	. . . 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 1204	. . 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 2856	. 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 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   =/= wne 2593   E.wrex 2709   class class class wbr 4359   ` cfv 5537  (class class class)co 6242   Basecbs 15057   lecple 15133   joincjn 16125   <o ccvr 32734   Atomscatm 32735   HLchlt 32822   Linesclines 32965   pmapcpmap 32968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-reu 2715  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-riota 6204  df-ov 6245  df-oprab 6246  df-preset 16109  df-poset 16127  df-plt 16140  df-lub 16156  df-glb 16157  df-join 16158  df-meet 16159  df-p0 16221  df-lat 16228  df-clat 16290  df-oposet 32648  df-ol 32650  df-oml 32651  df-covers 32738  df-ats 32739  df-atl 32770  df-cvlat 32794  df-hlat 32823  df-lines 32972  df-pmap 32975

This theorem is referenced by:  2lnat  33255