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Theorem lncvrat 39287
Description: A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)
Hypotheses
Ref Expression
lncvrat.b 𝐡 = (Baseβ€˜πΎ)
lncvrat.l ≀ = (leβ€˜πΎ)
lncvrat.c 𝐢 = ( β‹– β€˜πΎ)
lncvrat.a 𝐴 = (Atomsβ€˜πΎ)
lncvrat.n 𝑁 = (Linesβ€˜πΎ)
lncvrat.m 𝑀 = (pmapβ€˜πΎ)
Assertion
Ref Expression
lncvrat (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ 𝑃𝐢𝑋)
Proof of Theorem lncvrat
Dummy variables π‘Ÿ π‘ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 769 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ (π‘€β€˜π‘‹) ∈ 𝑁)
2 simpl1 1188 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
3 simpl2 1189 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ 𝑋 ∈ 𝐡)
4 lncvrat.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
5 eqid 2728 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
6 lncvrat.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
7 lncvrat.n . . . . 5 𝑁 = (Linesβ€˜πΎ)
8 lncvrat.m . . . . 5 𝑀 = (pmapβ€˜πΎ)
94, 5, 6, 7, 8isline3 39281 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ ((π‘€β€˜π‘‹) ∈ 𝑁 ↔ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))))
102, 3, 9syl2anc 582 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ ((π‘€β€˜π‘‹) ∈ 𝑁 ↔ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))))
111, 10mpbid 231 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ)))
12 simp1l1 1263 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝐾 ∈ HL)
13 simp1l3 1265 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝑃 ∈ 𝐴)
14 simp2l 1196 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ π‘ž ∈ 𝐴)
15 simp2r 1197 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ π‘Ÿ ∈ 𝐴)
16 simp3l 1198 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ π‘ž β‰  π‘Ÿ)
17 simp1rr 1236 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝑃 ≀ 𝑋)
18 simp3r 1199 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))
1917, 18breqtrd 5178 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝑃 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ))
20 lncvrat.l . . . . . . 7 ≀ = (leβ€˜πΎ)
21 lncvrat.c . . . . . . 7 𝐢 = ( β‹– β€˜πΎ)
2220, 5, 21, 6atcvrj2 38938 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑃 ≀ (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝑃𝐢(π‘ž(joinβ€˜πΎ)π‘Ÿ))
2312, 13, 14, 15, 16, 19, 22syl132anc 1385 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝑃𝐢(π‘ž(joinβ€˜πΎ)π‘Ÿ))
2423, 18breqtrrd 5180 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) ∧ (π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) ∧ (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ))) β†’ 𝑃𝐢𝑋)
25243exp 1116 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ ((π‘ž ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) β†’ ((π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ 𝑃𝐢𝑋)))
2625rexlimdvv 3208 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ (βˆƒπ‘ž ∈ 𝐴 βˆƒπ‘Ÿ ∈ 𝐴 (π‘ž β‰  π‘Ÿ ∧ 𝑋 = (π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ 𝑃𝐢𝑋))
2711, 26mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ 𝑃𝐢𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆƒwrex 3067   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  joincjn 18310   β‹– ccvr 38766  Atomscatm 38767  HLchlt 38854  Linesclines 38999  pmapcpmap 39002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-lines 39006  df-pmap 39009
This theorem is referenced by:  2lnat  39289
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