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Theorem lncvrelatN 37722
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 37304 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1131 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → 𝐾 ∈ Lat)
3 eqid 2738 . . . . 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 37715 . . . 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 1210 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝐾 ∈ HL)
10 simpll2 1211 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑋𝐵)
119, 1syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝐾 ∈ Lat)
12 simplrl 773 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑞𝐴)
13 lncvrelat.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1413, 4atbase 37230 . . . . . . . . 9 (𝑞𝐴𝑞𝐵)
1512, 14syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑞𝐵)
16 simplrr 774 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐴)
1713, 4atbase 37230 . . . . . . . . 9 (𝑟𝐴𝑟𝐵)
1816, 17syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐵)
1913, 3latjcl 18072 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2011, 15, 18, 19syl3anc 1369 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2113, 6pmap11 37703 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑞(join‘𝐾)𝑟) ∈ 𝐵) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
229, 10, 20, 21syl3anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
23 breq2 5074 . . . . . . . 8 (𝑋 = (𝑞(join‘𝐾)𝑟) → (𝑃𝐶𝑋𝑃𝐶(𝑞(join‘𝐾)𝑟)))
2423biimpd 228 . . . . . . 7 (𝑋 = (𝑞(join‘𝐾)𝑟) → (𝑃𝐶𝑋𝑃𝐶(𝑞(join‘𝐾)𝑟)))
259adantr 480 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝐾 ∈ HL)
26 simpll3 1212 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑃𝐵)
2726, 12, 163jca 1126 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑃𝐵𝑞𝐴𝑟𝐴))
2827adantr 480 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → (𝑃𝐵𝑞𝐴𝑟𝐴))
29 simplr 765 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑞𝑟)
30 simpr 484 . . . . . . . . 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 37370 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐵𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑃𝐶(𝑞(join‘𝐾)𝑟))) → 𝑃𝐴)
3325, 28, 29, 30, 32syl112anc 1372 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑃𝐴)
3433ex 412 . . . . . . 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 239 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) → (𝑃𝐶𝑋𝑃𝐴)))
3736expimpd 453 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) → ((𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
3837rexlimdvva 3222 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
398, 38sylbid 239 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → ((𝑀𝑋) ∈ 𝑁 → (𝑃𝐶𝑋𝑃𝐴)))
4039imp32 418 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ ((𝑀𝑋) ∈ 𝑁𝑃𝐶𝑋)) → 𝑃𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1539  wcel 2108  wne 2942  wrex 3064   class class class wbr 5070  cfv 6418  (class class class)co 7255  Basecbs 16840  joincjn 17944  Latclat 18064  ccvr 37203  Atomscatm 37204  HLchlt 37291  Linesclines 37435  pmapcpmap 37438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-lines 37442  df-pmap 37445
This theorem is referenced by: (None)
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