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Theorem lncvrelatN 36916
Description: A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lncvrelat.b 𝐵 = (Base‘𝐾)
lncvrelat.c 𝐶 = ( ⋖ ‘𝐾)
lncvrelat.a 𝐴 = (Atoms‘𝐾)
lncvrelat.n 𝑁 = (Lines‘𝐾)
lncvrelat.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
lncvrelatN (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ ((𝑀𝑋) ∈ 𝑁𝑃𝐶𝑋)) → 𝑃𝐴)

Proof of Theorem lncvrelatN
Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 36498 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1129 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → 𝐾 ∈ Lat)
3 eqid 2821 . . . . 5 (join‘𝐾) = (join‘𝐾)
4 lncvrelat.a . . . . 5 𝐴 = (Atoms‘𝐾)
5 lncvrelat.n . . . . 5 𝑁 = (Lines‘𝐾)
6 lncvrelat.m . . . . 5 𝑀 = (pmap‘𝐾)
73, 4, 5, 6isline2 36909 . . . 4 (𝐾 ∈ Lat → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)))))
82, 7syl 17 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)))))
9 simpll1 1208 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝐾 ∈ HL)
10 simpll2 1209 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑋𝐵)
119, 1syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝐾 ∈ Lat)
12 simplrl 775 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑞𝐴)
13 lncvrelat.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1413, 4atbase 36424 . . . . . . . . 9 (𝑞𝐴𝑞𝐵)
1512, 14syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑞𝐵)
16 simplrr 776 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐴)
1713, 4atbase 36424 . . . . . . . . 9 (𝑟𝐴𝑟𝐵)
1816, 17syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐵)
1913, 3latjcl 17660 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2011, 15, 18, 19syl3anc 1367 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2113, 6pmap11 36897 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑞(join‘𝐾)𝑟) ∈ 𝐵) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
229, 10, 20, 21syl3anc 1367 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
23 breq2 5069 . . . . . . . 8 (𝑋 = (𝑞(join‘𝐾)𝑟) → (𝑃𝐶𝑋𝑃𝐶(𝑞(join‘𝐾)𝑟)))
2423biimpd 231 . . . . . . 7 (𝑋 = (𝑞(join‘𝐾)𝑟) → (𝑃𝐶𝑋𝑃𝐶(𝑞(join‘𝐾)𝑟)))
259adantr 483 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝐾 ∈ HL)
26 simpll3 1210 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑃𝐵)
2726, 12, 163jca 1124 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑃𝐵𝑞𝐴𝑟𝐴))
2827adantr 483 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → (𝑃𝐵𝑞𝐴𝑟𝐴))
29 simplr 767 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑞𝑟)
30 simpr 487 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑃𝐶(𝑞(join‘𝐾)𝑟))
31 lncvrelat.c . . . . . . . . . 10 𝐶 = ( ⋖ ‘𝐾)
3213, 3, 31, 4cvrat2 36564 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐵𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑃𝐶(𝑞(join‘𝐾)𝑟))) → 𝑃𝐴)
3325, 28, 29, 30, 32syl112anc 1370 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑃𝐴)
3433ex 415 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑃𝐶(𝑞(join‘𝐾)𝑟) → 𝑃𝐴))
3524, 34syl9r 78 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑋 = (𝑞(join‘𝐾)𝑟) → (𝑃𝐶𝑋𝑃𝐴)))
3622, 35sylbid 242 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) → (𝑃𝐶𝑋𝑃𝐴)))
3736expimpd 456 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) → ((𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
3837rexlimdvva 3294 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
398, 38sylbid 242 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → ((𝑀𝑋) ∈ 𝑁 → (𝑃𝐶𝑋𝑃𝐴)))
4039imp32 421 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ ((𝑀𝑋) ∈ 𝑁𝑃𝐶𝑋)) → 𝑃𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016  wrex 3139   class class class wbr 5065  cfv 6354  (class class class)co 7155  Basecbs 16482  joincjn 17553  Latclat 17654  ccvr 36397  Atomscatm 36398  HLchlt 36485  Linesclines 36629  pmapcpmap 36632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-proset 17537  df-poset 17555  df-plt 17567  df-lub 17583  df-glb 17584  df-join 17585  df-meet 17586  df-p0 17648  df-lat 17655  df-clat 17717  df-oposet 36311  df-ol 36313  df-oml 36314  df-covers 36401  df-ats 36402  df-atl 36433  df-cvlat 36457  df-hlat 36486  df-lines 36636  df-pmap 36639
This theorem is referenced by: (None)
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