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Theorem lncvrat 38648
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 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ ((π‘€β€˜π‘‹) ∈ 𝑁 ∧ 𝑃 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
3 simpl2 1192 . . . 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 2732 . . . . 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 38642 . . . 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 584 . . 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 1266 . . . . . 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 1268 . . . . . 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 1199 . . . . . 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 1200 . . . . . 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 1201 . . . . . 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 1239 . . . . . . 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 1202 . . . . . . 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 5174 . . . . . 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 38299 . . . . . 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 1388 . . . . 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 5176 . . . 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 1119 . . 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 3210 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  lecple 17203  joincjn 18263   β‹– ccvr 38127  Atomscatm 38128  HLchlt 38215  Linesclines 38360  pmapcpmap 38363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-lines 38367  df-pmap 38370
This theorem is referenced by:  2lnat  38650
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