Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lncvrat Structured version   Visualization version   GIF version

Theorem lncvrat 36360
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 ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → 𝑃𝐶𝑋)

Proof of Theorem lncvrat
Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 758 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → (𝑀𝑋) ∈ 𝑁)
2 simpl1 1171 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → 𝐾 ∈ HL)
3 simpl2 1172 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → 𝑋𝐵)
4 lncvrat.b . . . . 5 𝐵 = (Base‘𝐾)
5 eqid 2779 . . . . 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)
94, 5, 6, 7, 8isline3 36354 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))))
102, 3, 9syl2anc 576 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))))
111, 10mpbid 224 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟)))
12 simp1l1 1246 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝐾 ∈ HL)
13 simp1l3 1248 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃𝐴)
14 simp2l 1179 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑞𝐴)
15 simp2r 1180 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑟𝐴)
16 simp3l 1181 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑞𝑟)
17 simp1rr 1219 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃 𝑋)
18 simp3r 1182 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑋 = (𝑞(join‘𝐾)𝑟))
1917, 18breqtrd 4955 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃 (𝑞(join‘𝐾)𝑟))
20 lncvrat.l . . . . . . 7 = (le‘𝐾)
21 lncvrat.c . . . . . . 7 𝐶 = ( ⋖ ‘𝐾)
2220, 5, 21, 6atcvrj2 36011 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑃 (𝑞(join‘𝐾)𝑟))) → 𝑃𝐶(𝑞(join‘𝐾)𝑟))
2312, 13, 14, 15, 16, 19, 22syl132anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃𝐶(𝑞(join‘𝐾)𝑟))
2423, 18breqtrrd 4957 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) ∧ (𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟))) → 𝑃𝐶𝑋)
25243exp 1099 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → ((𝑞𝐴𝑟𝐴) → ((𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟)) → 𝑃𝐶𝑋)))
2625rexlimdvv 3239 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = (𝑞(join‘𝐾)𝑟)) → 𝑃𝐶𝑋))
2711, 26mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → 𝑃𝐶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2968  wrex 3090   class class class wbr 4929  cfv 6188  (class class class)co 6976  Basecbs 16339  lecple 16428  joincjn 17412  ccvr 35840  Atomscatm 35841  HLchlt 35928  Linesclines 36072  pmapcpmap 36075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-proset 17396  df-poset 17414  df-plt 17426  df-lub 17442  df-glb 17443  df-join 17444  df-meet 17445  df-p0 17507  df-lat 17514  df-clat 17576  df-oposet 35754  df-ol 35756  df-oml 35757  df-covers 35844  df-ats 35845  df-atl 35876  df-cvlat 35900  df-hlat 35929  df-lines 36079  df-pmap 36082
This theorem is referenced by:  2lnat  36362
  Copyright terms: Public domain W3C validator