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Theorem lnatexN 35557
Description: There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnatex.b 𝐵 = (Base‘𝐾)
lnatex.l = (le‘𝐾)
lnatex.a 𝐴 = (Atoms‘𝐾)
lnatex.n 𝑁 = (Lines‘𝐾)
lnatex.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
lnatexN ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
Distinct variable groups:   𝐴,𝑞   ,𝑞   𝑃,𝑞   𝑋,𝑞
Allowed substitution hints:   𝐵(𝑞)   𝐾(𝑞)   𝑀(𝑞)   𝑁(𝑞)

Proof of Theorem lnatexN
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnatex.b . . . 4 𝐵 = (Base‘𝐾)
2 eqid 2806 . . . 4 (join‘𝐾) = (join‘𝐾)
3 lnatex.a . . . 4 𝐴 = (Atoms‘𝐾)
4 lnatex.n . . . 4 𝑁 = (Lines‘𝐾)
5 lnatex.m . . . 4 𝑀 = (pmap‘𝐾)
61, 2, 3, 4, 5isline3 35554 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))))
76biimp3a 1586 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠)))
8 simpl2r 1292 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠𝐴)
9 simpl3l 1294 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑟𝑠)
109necomd 3033 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠𝑟)
11 simpr 473 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑟 = 𝑃)
1210, 11neeqtrd 3047 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠𝑃)
13 simpl11 1322 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝐾 ∈ HL)
14 simpl2l 1290 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑟𝐴)
15 lnatex.l . . . . . . . . 9 = (le‘𝐾)
1615, 2, 3hlatlej2 35154 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑟𝐴𝑠𝐴) → 𝑠 (𝑟(join‘𝐾)𝑠))
1713, 14, 8, 16syl3anc 1483 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠 (𝑟(join‘𝐾)𝑠))
18 simpl3r 1296 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑋 = (𝑟(join‘𝐾)𝑠))
1917, 18breqtrrd 4872 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠 𝑋)
20 neeq1 3040 . . . . . . . 8 (𝑞 = 𝑠 → (𝑞𝑃𝑠𝑃))
21 breq1 4847 . . . . . . . 8 (𝑞 = 𝑠 → (𝑞 𝑋𝑠 𝑋))
2220, 21anbi12d 618 . . . . . . 7 (𝑞 = 𝑠 → ((𝑞𝑃𝑞 𝑋) ↔ (𝑠𝑃𝑠 𝑋)))
2322rspcev 3502 . . . . . 6 ((𝑠𝐴 ∧ (𝑠𝑃𝑠 𝑋)) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
248, 12, 19, 23syl12anc 856 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
25 simpl2l 1290 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑟𝐴)
26 simpr 473 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑟𝑃)
27 simpl11 1322 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝐾 ∈ HL)
28 simpl2r 1292 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑠𝐴)
2915, 2, 3hlatlej1 35153 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑟𝐴𝑠𝐴) → 𝑟 (𝑟(join‘𝐾)𝑠))
3027, 25, 28, 29syl3anc 1483 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑟 (𝑟(join‘𝐾)𝑠))
31 simpl3r 1296 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑋 = (𝑟(join‘𝐾)𝑠))
3230, 31breqtrrd 4872 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑟 𝑋)
33 neeq1 3040 . . . . . . . 8 (𝑞 = 𝑟 → (𝑞𝑃𝑟𝑃))
34 breq1 4847 . . . . . . . 8 (𝑞 = 𝑟 → (𝑞 𝑋𝑟 𝑋))
3533, 34anbi12d 618 . . . . . . 7 (𝑞 = 𝑟 → ((𝑞𝑃𝑞 𝑋) ↔ (𝑟𝑃𝑟 𝑋)))
3635rspcev 3502 . . . . . 6 ((𝑟𝐴 ∧ (𝑟𝑃𝑟 𝑋)) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
3725, 26, 32, 36syl12anc 856 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
3824, 37pm2.61dane 3065 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
39383exp 1141 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ((𝑟𝐴𝑠𝐴) → ((𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠)) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))))
4039rexlimdvv 3225 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → (∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠)) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋)))
417, 40mpd 15 1 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wrex 3097   class class class wbr 4844  cfv 6097  (class class class)co 6870  Basecbs 16064  lecple 16156  joincjn 17145  Atomscatm 35041  HLchlt 35128  Linesclines 35272  pmapcpmap 35275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-proset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-lat 17247  df-clat 17309  df-oposet 34954  df-ol 34956  df-oml 34957  df-covers 35044  df-ats 35045  df-atl 35076  df-cvlat 35100  df-hlat 35129  df-lines 35279  df-pmap 35282
This theorem is referenced by:  lnjatN  35558
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