Theorem lncvrat 37533

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 = ( ⋖ ‘𝐾)
lncvrat.a	⊢ 𝐴 = (Atoms‘𝐾)
lncvrat.n	⊢ 𝑁 = (Lines‘𝐾)
lncvrat.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
lncvrat	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → 𝑃𝐶𝑋)

Proof of Theorem lncvrat

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

Step	Hyp	Ref	Expression
1		simprl 771	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → (𝑀‘𝑋) ∈ 𝑁)
2		simpl1 1193	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → 𝐾 ∈ HL)
3		simpl2 1194	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
4		lncvrat.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
5		eqid 2737	. . . . 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 37527	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))))
10	2, 3, 9	syl2anc 587	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))))
11	1, 10	mpbid 235	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟)))
12		simp1l1 1268	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝐾 ∈ HL)
13		simp1l3 1270	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃 ∈ 𝐴)
14		simp2l 1201	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑞 ∈ 𝐴)
15		simp2r 1202	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑟 ∈ 𝐴)
16		simp3l 1203	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑞 ≠ 𝑟)
17		simp1rr 1241	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃 ≤ 𝑋)
18		simp3r 1204	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑋 = (𝑞(join‘𝐾)𝑟))
19	17, 18	breqtrd 5079	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃 ≤ (𝑞(join‘𝐾)𝑟))
20		lncvrat.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
21		lncvrat.c	. . . . . . 7 ⊢ 𝐶 = ( ⋖ ‘𝐾)
22	20, 5, 21, 6	atcvrj2 37184	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑃 ≤ (𝑞(join‘𝐾)𝑟))) → 𝑃𝐶(𝑞(join‘𝐾)𝑟))
23	12, 13, 14, 15, 16, 19, 22	syl132anc 1390	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃𝐶(𝑞(join‘𝐾)𝑟))
24	23, 18	breqtrrd 5081	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃𝐶𝑋)
25	24	3exp 1121	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → ((𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → ((𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟)) → 𝑃𝐶𝑋)))
26	25	rexlimdvv 3212	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = (𝑞(join‘𝐾)𝑟)) → 𝑃𝐶𝑋))
27	11, 26	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → 𝑃𝐶𝑋)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∃wrex 3062   class class class wbr 5053  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  joincjn 17818   ⋖ ccvr 37013  Atomscatm 37014  HLchlt 37101  Linesclines 37245  pmapcpmap 37248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-lat 17938  df-clat 18005  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-lines 37252  df-pmap 37255

This theorem is referenced by:  2lnat  37535