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Theorem lncvrelatN 34582
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 34165 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1080 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → 𝐾 ∈ Lat)
3 eqid 2621 . . . . 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 34575 . . . 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 1098 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝐾 ∈ HL)
10 simpll2 1099 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑋𝐵)
119, 1syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝐾 ∈ Lat)
12 simplrl 799 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑞𝐴)
13 lncvrelat.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1413, 4atbase 34091 . . . . . . . . 9 (𝑞𝐴𝑞𝐵)
1512, 14syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑞𝐵)
16 simplrr 800 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐴)
1713, 4atbase 34091 . . . . . . . . 9 (𝑟𝐴𝑟𝐵)
1816, 17syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐵)
1913, 3latjcl 16983 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2011, 15, 18, 19syl3anc 1323 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2113, 6pmap11 34563 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑞(join‘𝐾)𝑟) ∈ 𝐵) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
229, 10, 20, 21syl3anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
23 breq2 4622 . . . . . . . 8 (𝑋 = (𝑞(join‘𝐾)𝑟) → (𝑃𝐶𝑋𝑃𝐶(𝑞(join‘𝐾)𝑟)))
2423biimpd 219 . . . . . . 7 (𝑋 = (𝑞(join‘𝐾)𝑟) → (𝑃𝐶𝑋𝑃𝐶(𝑞(join‘𝐾)𝑟)))
259adantr 481 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝐾 ∈ HL)
26 simpll3 1100 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑃𝐵)
2726, 12, 163jca 1240 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑃𝐵𝑞𝐴𝑟𝐴))
2827adantr 481 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → (𝑃𝐵𝑞𝐴𝑟𝐴))
29 simplr 791 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑞𝑟)
30 simpr 477 . . . . . . . . 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 34230 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐵𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑃𝐶(𝑞(join‘𝐾)𝑟))) → 𝑃𝐴)
3325, 28, 29, 30, 32syl112anc 1327 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑃𝐴)
3433ex 450 . . . . . . 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 230 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) → (𝑃𝐶𝑋𝑃𝐴)))
3736expimpd 628 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) → ((𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
3837rexlimdvva 3032 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
398, 38sylbid 230 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → ((𝑀𝑋) ∈ 𝑁 → (𝑃𝐶𝑋𝑃𝐴)))
4039imp32 449 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ ((𝑀𝑋) ∈ 𝑁𝑃𝐶𝑋)) → 𝑃𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  joincjn 16876  Latclat 16977  ccvr 34064  Atomscatm 34065  HLchlt 34152  Linesclines 34295  pmapcpmap 34298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-lat 16978  df-clat 17040  df-oposet 33978  df-ol 33980  df-oml 33981  df-covers 34068  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153  df-lines 34302  df-pmap 34305
This theorem is referenced by: (None)
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