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Theorem lncvrat 39165
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 768 . . 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 2726 . . . . 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 39159 . . . 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 583 . . 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 5167 . . . . . 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 38816 . . . . . 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 5169 . . . 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 3204 . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  lecple 17210  joincjn 18273   β‹– ccvr 38644  Atomscatm 38645  HLchlt 38732  Linesclines 38877  pmapcpmap 38880
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-proset 18257  df-poset 18275  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-lat 18394  df-clat 18461  df-oposet 38558  df-ol 38560  df-oml 38561  df-covers 38648  df-ats 38649  df-atl 38680  df-cvlat 38704  df-hlat 38733  df-lines 38884  df-pmap 38887
This theorem is referenced by:  2lnat  39167
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