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Theorem lncvrelatN 35849
 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 35431 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1167 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → 𝐾 ∈ Lat)
3 eqid 2825 . . . . 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 35842 . . . 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 1273 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝐾 ∈ HL)
10 simpll2 1275 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑋𝐵)
119, 1syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝐾 ∈ Lat)
12 simplrl 795 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑞𝐴)
13 lncvrelat.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1413, 4atbase 35357 . . . . . . . . 9 (𝑞𝐴𝑞𝐵)
1512, 14syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑞𝐵)
16 simplrr 796 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐴)
1713, 4atbase 35357 . . . . . . . . 9 (𝑟𝐴𝑟𝐵)
1816, 17syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐵)
1913, 3latjcl 17404 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2011, 15, 18, 19syl3anc 1494 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2113, 6pmap11 35830 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑞(join‘𝐾)𝑟) ∈ 𝐵) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
229, 10, 20, 21syl3anc 1494 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
23 breq2 4877 . . . . . . . 8 (𝑋 = (𝑞(join‘𝐾)𝑟) → (𝑃𝐶𝑋𝑃𝐶(𝑞(join‘𝐾)𝑟)))
2423biimpd 221 . . . . . . 7 (𝑋 = (𝑞(join‘𝐾)𝑟) → (𝑃𝐶𝑋𝑃𝐶(𝑞(join‘𝐾)𝑟)))
259adantr 474 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝐾 ∈ HL)
26 simpll3 1277 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑃𝐵)
2726, 12, 163jca 1162 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑃𝐵𝑞𝐴𝑟𝐴))
2827adantr 474 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → (𝑃𝐵𝑞𝐴𝑟𝐴))
29 simplr 785 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑞𝑟)
30 simpr 479 . . . . . . . . 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 35497 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐵𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑃𝐶(𝑞(join‘𝐾)𝑟))) → 𝑃𝐴)
3325, 28, 29, 30, 32syl112anc 1497 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑃𝐴)
3433ex 403 . . . . . . 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 232 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) → (𝑃𝐶𝑋𝑃𝐴)))
3736expimpd 447 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) → ((𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
3837rexlimdvva 3248 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
398, 38sylbid 232 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → ((𝑀𝑋) ∈ 𝑁 → (𝑃𝐶𝑋𝑃𝐴)))
4039imp32 411 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ ((𝑀𝑋) ∈ 𝑁𝑃𝐶𝑋)) → 𝑃𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∃wrex 3118   class class class wbr 4873  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  joincjn 17297  Latclat 17398   ⋖ ccvr 35330  Atomscatm 35331  HLchlt 35418  Linesclines 35562  pmapcpmap 35565 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-lat 17399  df-clat 17461  df-oposet 35244  df-ol 35246  df-oml 35247  df-covers 35334  df-ats 35335  df-atl 35366  df-cvlat 35390  df-hlat 35419  df-lines 35569  df-pmap 35572 This theorem is referenced by: (None)
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