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Theorem lneq2at 37437
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 1204 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ HL)
2 simp12 1205 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋𝐵)
31, 2jca 515 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ HL ∧ 𝑋𝐵))
4 simp13 1206 . . 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 37435 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟 𝑠))))
1110biimpd 232 . . 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 1203 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑟 𝑠))
14 simp111 1303 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝐾 ∈ HL)
15 simp121 1306 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝐴)
16 simp122 1307 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑄𝐴)
1715, 16jca 515 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃𝐴𝑄𝐴))
18 simp2 1138 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑟𝐴𝑠𝐴))
1914, 17, 183jca 1129 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)))
20 simp123 1308 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑃𝑄)
2119, 20jca 515 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄))
221hllatd 37023 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝐾 ∈ Lat)
23 simp21 1207 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐴)
245, 7atbase 36948 . . . . . . . . . . . 12 (𝑃𝐴𝑃𝐵)
2523, 24syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑃𝐵)
26 simp22 1208 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐴)
275, 7atbase 36948 . . . . . . . . . . . 12 (𝑄𝐴𝑄𝐵)
2826, 27syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑄𝐵)
2925, 28, 23jca 1129 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃𝐵𝑄𝐵𝑋𝐵))
3022, 29jca 515 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)))
31 simp3 1139 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → (𝑃 𝑋𝑄 𝑋))
32 lneq2at.l . . . . . . . . . . 11 = (le‘𝐾)
335, 32, 6latjle12 17790 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐵𝑄𝐵𝑋𝐵)) → ((𝑃 𝑋𝑄 𝑋) ↔ (𝑃 𝑄) 𝑋))
3433biimpd 232 . . . . . . . . 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 1134 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) 𝑋)
3736, 13breqtrd 5056 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → (𝑃 𝑄) (𝑟 𝑠))
38 simpl1 1192 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
39 simpl2l 1227 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝐴)
40 simpl2r 1228 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑄𝐴)
41 simpr 488 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → 𝑃𝑄)
42 simpl3 1194 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → (𝑟𝐴𝑠𝐴))
4332, 6, 7ps-1 37136 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑟𝐴𝑠𝐴)) → ((𝑃 𝑄) (𝑟 𝑠) ↔ (𝑃 𝑄) = (𝑟 𝑠)))
4438, 39, 40, 41, 42, 43syl131anc 1384 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑠𝐴)) ∧ 𝑃𝑄) → ((𝑃 𝑄) (𝑟 𝑠) ↔ (𝑃 𝑄) = (𝑟 𝑠)))
4544biimpd 232 . . . . . 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 2776 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟 𝑠))) → 𝑋 = (𝑃 𝑄))
48473exp 1120 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → ((𝑟𝐴𝑠𝐴) → ((𝑟𝑠𝑋 = (𝑟 𝑠)) → 𝑋 = (𝑃 𝑄))))
4948rexlimdvv 3203 . 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2934  wrex 3054   class class class wbr 5030  cfv 6339  (class class class)co 7172  Basecbs 16588  lecple 16677  joincjn 17672  Latclat 17773  Atomscatm 36922  HLchlt 37009  Linesclines 37153  pmapcpmap 37156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-proset 17656  df-poset 17674  df-plt 17686  df-lub 17702  df-glb 17703  df-join 17704  df-meet 17705  df-p0 17767  df-lat 17774  df-clat 17836  df-oposet 36835  df-ol 36837  df-oml 36838  df-covers 36925  df-ats 36926  df-atl 36957  df-cvlat 36981  df-hlat 37010  df-lines 37160  df-pmap 37163
This theorem is referenced by:  lnjatN  37439  lncmp  37442  cdlema1N  37450
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