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Theorem lncvrelatN 38244
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 37825 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1133 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → 𝐾 ∈ Lat)
3 eqid 2736 . . . . 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 38237 . . . 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 1212 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝐾 ∈ HL)
10 simpll2 1213 . . . . . . 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 37751 . . . . . . . . 9 (𝑞𝐴𝑞𝐵)
1512, 14syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑞𝐵)
16 simplrr 776 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐴)
1713, 4atbase 37751 . . . . . . . . 9 (𝑟𝐴𝑟𝐵)
1816, 17syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑟𝐵)
1913, 3latjcl 18328 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2011, 15, 18, 19syl3anc 1371 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑞(join‘𝐾)𝑟) ∈ 𝐵)
2113, 6pmap11 38225 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑞(join‘𝐾)𝑟) ∈ 𝐵) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
229, 10, 20, 21syl3anc 1371 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → ((𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟)) ↔ 𝑋 = (𝑞(join‘𝐾)𝑟)))
23 breq2 5109 . . . . . . . 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 481 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝐾 ∈ HL)
26 simpll3 1214 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → 𝑃𝐵)
2726, 12, 163jca 1128 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) → (𝑃𝐵𝑞𝐴𝑟𝐴))
2827adantr 481 . . . . . . . . 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 485 . . . . . . . . 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 37892 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐵𝑞𝐴𝑟𝐴) ∧ (𝑞𝑟𝑃𝐶(𝑞(join‘𝐾)𝑟))) → 𝑃𝐴)
3325, 28, 29, 30, 32syl112anc 1374 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) ∧ 𝑞𝑟) ∧ 𝑃𝐶(𝑞(join‘𝐾)𝑟)) → 𝑃𝐴)
3433ex 413 . . . . . . 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 454 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ (𝑞𝐴𝑟𝐴)) → ((𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
3837rexlimdvva 3205 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟 ∧ (𝑀𝑋) = (𝑀‘(𝑞(join‘𝐾)𝑟))) → (𝑃𝐶𝑋𝑃𝐴)))
398, 38sylbid 239 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) → ((𝑀𝑋) ∈ 𝑁 → (𝑃𝐶𝑋𝑃𝐴)))
4039imp32 419 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐵) ∧ ((𝑀𝑋) ∈ 𝑁𝑃𝐶𝑋)) → 𝑃𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  joincjn 18200  Latclat 18320  ccvr 37724  Atomscatm 37725  HLchlt 37812  Linesclines 37957  pmapcpmap 37960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-lines 37964  df-pmap 37967
This theorem is referenced by: (None)
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