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Theorem lneq2at 39381
Description: A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)
Hypotheses
Ref Expression
lneq2at.b 𝐵 = (Base‘𝐾)
lneq2at.l = (le‘𝐾)
lneq2at.j = (join‘𝐾)
lneq2at.a 𝐴 = (Atoms‘𝐾)
lneq2at.n 𝑁 = (Lines‘𝐾)
lneq2at.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
lneq2at (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋 = (𝑃 𝑄))

Proof of Theorem lneq2at
Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1200 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ HL)
2 simp12 1201 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋𝐵)
31, 2jca 510 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ HL ∧ 𝑋𝐵))
4 simp13 1202 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑀𝑋) ∈ 𝑁)
5 lneq2at.b . . . . 5 𝐵 = (Base‘𝐾)
6 lneq2at.j . . . . 5 = (join‘𝐾)
7 lneq2at.a . . . . 5 𝐴 = (Atoms‘𝐾)
8 lneq2at.n . . . . 5 𝑁 = (Lines‘𝐾)
9 lneq2at.m . . . . 5 𝑀 = (pmap‘𝐾)
105, 6, 7, 8, 9isline3 39379 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠))))
1110biimpd 228 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 → ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠))))
123, 4, 11sylc 65 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠)))
13 simp3r 1199 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑟 𝑠))
14 simp111 1299 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝐾 ∈ HL)
15 simp121 1302 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝐴)
16 simp122 1303 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑄𝐴)
1715, 16jca 510 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃𝐴𝑄𝐴))
18 simp2 1134 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑟𝐴𝑠𝐴))
1914, 17, 183jca 1125 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)))
20 simp123 1304 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝑄)
2119, 20jca 510 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄))
221hllatd 38966 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ Lat)
23 simp21 1203 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐴)
245, 7atbase 38891 . . . . . . . . . . . 12 (𝑃𝐴𝑃𝐵)
2523, 24syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐵)
26 simp22 1204 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐴)
275, 7atbase 38891 . . . . . . . . . . . 12 (𝑄𝐴𝑄𝐵)
2826, 27syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐵)
2925, 28, 23jca 1125 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃𝐵𝑄𝐵𝑋𝐵))
3022, 29jca 510 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)))
31 simp3 1135 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃 𝑋𝑄 𝑋))
32 lneq2at.l . . . . . . . . . . 11 = (le‘𝐾)
335, 32, 6latjle12 18445 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃 𝑄) 𝑋))
3433biimpd 228 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) → (𝑃 𝑄) 𝑋))
3530, 31, 34sylc 65 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃 𝑄) 𝑋)
36353ad2ant1 1130 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) 𝑋)
3736, 13breqtrd 5175 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) (𝑟 𝑠))
38 simpl1 1188 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
39 simpl2l 1223 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝐴)
40 simpl2r 1224 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑄𝐴)
41 simpr 483 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝑄)
42 simpl3 1190 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → (𝑟𝐴𝑠𝐴))
4332, 6, 7ps-1 39080 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑟𝐴𝑠𝐴)) → ((𝑃 𝑄) (𝑟 𝑠) ↔ (𝑃 𝑄) = (𝑟 𝑠)))
4438, 39, 40, 41, 42, 43syl131anc 1380 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → ((𝑃 𝑄) (𝑟 𝑠) ↔ (𝑃 𝑄) = (𝑟 𝑠)))
4544biimpd 228 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → ((𝑃 𝑄) (𝑟 𝑠) → (𝑃 𝑄) = (𝑟 𝑠)))
4621, 37, 45sylc 65 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) = (𝑟 𝑠))
4713, 46eqtr4d 2768 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑃 𝑄))
48473exp 1116 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → ((𝑟𝐴𝑠𝐴) → ((𝑟𝑠𝑋 = (𝑟 𝑠)) → 𝑋 = (𝑃 𝑄))))
4948rexlimdvv 3200 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠)) → 𝑋 = (𝑃 𝑄)))
5012, 49mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋 = (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wrex 3059   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  lecple 17243  joincjn 18306  Latclat 18426  Atomscatm 38865  HLchlt 38952  Linesclines 39097  pmapcpmap 39100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-proset 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-lat 18427  df-clat 18494  df-oposet 38778  df-ol 38780  df-oml 38781  df-covers 38868  df-ats 38869  df-atl 38900  df-cvlat 38924  df-hlat 38953  df-lines 39104  df-pmap 39107
This theorem is referenced by:  lnjatN  39383  lncmp  39386  cdlema1N  39394
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