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Theorem lneq2at 35852
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 1264 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ HL)
2 simp12 1265 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋𝐵)
31, 2jca 507 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ HL ∧ 𝑋𝐵))
4 simp13 1266 . . 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 35850 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠))))
1110biimpd 221 . . 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 1263 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑟 𝑠))
14 simp111 1405 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝐾 ∈ HL)
15 simp121 1408 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝐴)
16 simp122 1409 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑄𝐴)
1715, 16jca 507 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃𝐴𝑄𝐴))
18 simp2 1171 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑟𝐴𝑠𝐴))
1914, 17, 183jca 1162 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)))
20 simp123 1410 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝑄)
2119, 20jca 507 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄))
221hllatd 35438 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ Lat)
23 simp21 1267 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐴)
245, 7atbase 35363 . . . . . . . . . . . 12 (𝑃𝐴𝑃𝐵)
2523, 24syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐵)
26 simp22 1268 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐴)
275, 7atbase 35363 . . . . . . . . . . . 12 (𝑄𝐴𝑄𝐵)
2826, 27syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐵)
2925, 28, 23jca 1162 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃𝐵𝑄𝐵𝑋𝐵))
3022, 29jca 507 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)))
31 simp3 1172 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃 𝑋𝑄 𝑋))
32 lneq2at.l . . . . . . . . . . 11 = (le‘𝐾)
335, 32, 6latjle12 17422 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃 𝑄) 𝑋))
3433biimpd 221 . . . . . . . . 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 1167 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) 𝑋)
3736, 13breqtrd 4901 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) (𝑟 𝑠))
38 simpl1 1246 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
39 simpl2l 1301 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝐴)
40 simpl2r 1303 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑄𝐴)
41 simpr 479 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝑄)
42 simpl3 1250 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → (𝑟𝐴𝑠𝐴))
4332, 6, 7ps-1 35551 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑟𝐴𝑠𝐴)) → ((𝑃 𝑄) (𝑟 𝑠) ↔ (𝑃 𝑄) = (𝑟 𝑠)))
4438, 39, 40, 41, 42, 43syl131anc 1506 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → ((𝑃 𝑄) (𝑟 𝑠) ↔ (𝑃 𝑄) = (𝑟 𝑠)))
4544biimpd 221 . . . . . 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 2864 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑃 𝑄))
48473exp 1152 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → ((𝑟𝐴𝑠𝐴) → ((𝑟𝑠𝑋 = (𝑟 𝑠)) → 𝑋 = (𝑃 𝑄))))
4948rexlimdvv 3247 . 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 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wrex 3118   class class class wbr 4875  cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  Latclat 17405  Atomscatm 35337  HLchlt 35424  Linesclines 35568  pmapcpmap 35571
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 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
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 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-lat 17406  df-clat 17468  df-oposet 35250  df-ol 35252  df-oml 35253  df-covers 35340  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425  df-lines 35575  df-pmap 35578
This theorem is referenced by:  lnjatN  35854  lncmp  35857  cdlema1N  35865
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