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Theorem lneq2at 38241
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 1203 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ HL)
2 simp12 1204 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋𝐵)
31, 2jca 512 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ HL ∧ 𝑋𝐵))
4 simp13 1205 . . 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 38239 . . . 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 1202 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑟 𝑠))
14 simp111 1302 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝐾 ∈ HL)
15 simp121 1305 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝐴)
16 simp122 1306 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑄𝐴)
1715, 16jca 512 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃𝐴𝑄𝐴))
18 simp2 1137 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑟𝐴𝑠𝐴))
1914, 17, 183jca 1128 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)))
20 simp123 1307 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝑄)
2119, 20jca 512 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄))
221hllatd 37826 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ Lat)
23 simp21 1206 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐴)
245, 7atbase 37751 . . . . . . . . . . . 12 (𝑃𝐴𝑃𝐵)
2523, 24syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐵)
26 simp22 1207 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐴)
275, 7atbase 37751 . . . . . . . . . . . 12 (𝑄𝐴𝑄𝐵)
2826, 27syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐵)
2925, 28, 23jca 1128 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃𝐵𝑄𝐵𝑋𝐵))
3022, 29jca 512 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)))
31 simp3 1138 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃 𝑋𝑄 𝑋))
32 lneq2at.l . . . . . . . . . . 11 = (le‘𝐾)
335, 32, 6latjle12 18339 . . . . . . . . . 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 1133 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) 𝑋)
3736, 13breqtrd 5131 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) (𝑟 𝑠))
38 simpl1 1191 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
39 simpl2l 1226 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝐴)
40 simpl2r 1227 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑄𝐴)
41 simpr 485 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝑄)
42 simpl3 1193 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → (𝑟𝐴𝑠𝐴))
4332, 6, 7ps-1 37940 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑟𝐴𝑠𝐴)) → ((𝑃 𝑄) (𝑟 𝑠) ↔ (𝑃 𝑄) = (𝑟 𝑠)))
4438, 39, 40, 41, 42, 43syl131anc 1383 . . . . . . 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 2779 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑃 𝑄))
48473exp 1119 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → ((𝑟𝐴𝑠𝐴) → ((𝑟𝑠𝑋 = (𝑟 𝑠)) → 𝑋 = (𝑃 𝑄))))
4948rexlimdvv 3204 . 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 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  lecple 17140  joincjn 18200  Latclat 18320  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:  lnjatN  38243  lncmp  38246  cdlema1N  38254
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